The Multiplication Principle
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Permutations - Objects Ordered in a Row
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Permutations with Identical Objects
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Permutations with and without Returns
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Non-Sequential Sampling without Returns
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Non-Sequential Sampling with Returns
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Summary Questions
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- Tutorial 1
- Exercise 1 - Parts a-c
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6 - Parts a-e
- Exercise 6 - Parts f-g
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10 - Parts a-d
- Exercise 11 - Parts a-c
- Exercise 11 - Parts d-h
- Exercise 12 - Parts a-c
- Exercise 13 - Parts a-c
- Exercise 14
- Exercise 15 - Parts a-c
- Exercise 16
- Exercise 17 - Parts a-b
- Exercise 18 - Parts a-d
- Exercise 19 - Parts a-c
- Exercise 20
- Exercise 21 - Parts a-c
- Exercise 22 - Part a
- Exercise 22 - Part b
- Exercise 22 - Part c
- Exercise 23

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Now what is that?"},{"Start":"00:09.180 ","End":"00:13.140","Text":"The multiplication principle is a rule that can be used to"},{"Start":"00:13.140 ","End":"00:18.795","Text":"calculate either the size of an event or the size of a sample space."},{"Start":"00:18.795 ","End":"00:22.755","Text":"Assume that we have a trial here and the trial has"},{"Start":"00:22.755 ","End":"00:28.920","Text":"k stages and we have n_1 possible results in the first stage."},{"Start":"00:28.920 ","End":"00:31.530","Text":"We can get n_2 possible results in"},{"Start":"00:31.530 ","End":"00:36.490","Text":"the second stage up to n_k possible results in the k_th stage."},{"Start":"00:36.490 ","End":"00:38.870","Text":"Now, the number of possible results for"},{"Start":"00:38.870 ","End":"00:43.370","Text":"the entire trial will be n_1 times n_2 times n_3,"},{"Start":"00:43.370 ","End":"00:45.365","Text":"and so on and so forth times n_k."},{"Start":"00:45.365 ","End":"00:49.100","Text":"This multiplication, this product right here,"},{"Start":"00:49.100 ","End":"00:52.820","Text":"will be the size of the sample space."},{"Start":"00:52.820 ","End":"00:55.775","Text":"Now, this seems pretty simple,"},{"Start":"00:55.775 ","End":"01:00.720","Text":"but we need a lot of exercises to let this thing sink in."},{"Start":"01:00.720 ","End":"01:04.470","Text":"First of all, let\u0027s take a look at several examples."}],"ID":12777},{"Watched":false,"Name":"Example 1","Duration":"2m 9s","ChapterTopicVideoID":12299,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12299.jpeg","UploadDate":"2018-07-22T16:34:46.8430000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this example, we\u0027re asked how many possibilities are there in"},{"Start":"00:03.180 ","End":"00:06.660","Text":"a game in which both a die and a coin are thrown?"},{"Start":"00:06.660 ","End":"00:08.820","Text":"Well, let\u0027s see."},{"Start":"00:08.820 ","End":"00:11.820","Text":"This is a trial where we have 2 stages."},{"Start":"00:11.820 ","End":"00:18.420","Text":"The first stage is where we roll a die and the second stage is where we toss a coin."},{"Start":"00:18.420 ","End":"00:22.560","Text":"How many possible results can we get when we roll a die?"},{"Start":"00:22.560 ","End":"00:24.525","Text":"Well, we have 6 results."},{"Start":"00:24.525 ","End":"00:26.730","Text":"It\u0027ll be either 1, or 2,"},{"Start":"00:26.730 ","End":"00:28.590","Text":"or 3, or 4,"},{"Start":"00:28.590 ","End":"00:30.465","Text":"or 5, or 6."},{"Start":"00:30.465 ","End":"00:32.220","Text":"When we toss a coin, well,"},{"Start":"00:32.220 ","End":"00:34.320","Text":"how many possible results can we have?"},{"Start":"00:34.320 ","End":"00:35.965","Text":"Well, we have 2 results."},{"Start":"00:35.965 ","End":"00:39.250","Text":"It can be either a head or a tail."},{"Start":"00:39.250 ","End":"00:44.510","Text":"What would be the number of possible results"},{"Start":"00:44.510 ","End":"00:49.655","Text":"that I can get when I roll a die and throw a coin or toss a coin?"},{"Start":"00:49.655 ","End":"00:54.020","Text":"Well, that would be the size of my sample space."},{"Start":"00:54.020 ","End":"00:59.330","Text":"That would be the maximum number of combinations of"},{"Start":"00:59.330 ","End":"01:02.210","Text":"the results of getting a number when I roll"},{"Start":"01:02.210 ","End":"01:06.320","Text":"a die and getting either a heads or tails when I toss a coin."},{"Start":"01:06.320 ","End":"01:09.485","Text":"That\u0027ll be 6 times 2."},{"Start":"01:09.485 ","End":"01:11.720","Text":"That\u0027ll be n_1 times n_2;"},{"Start":"01:11.720 ","End":"01:13.190","Text":"that\u0027ll be 6 times 2,"},{"Start":"01:13.190 ","End":"01:14.960","Text":"and that equals to 12."},{"Start":"01:14.960 ","End":"01:17.995","Text":"Let\u0027s just take a look at this graphically."},{"Start":"01:17.995 ","End":"01:22.220","Text":"Here we have the whole possibilities here."},{"Start":"01:22.220 ","End":"01:26.930","Text":"The blue lines right here are the results of rolling a die."},{"Start":"01:26.930 ","End":"01:28.790","Text":"Here I\u0027ll get the result 1."},{"Start":"01:28.790 ","End":"01:30.710","Text":"This is 2, this is 3,"},{"Start":"01:30.710 ","End":"01:33.335","Text":"this is 4, 5, and 6."},{"Start":"01:33.335 ","End":"01:36.790","Text":"The orange lines would be tossing a coin,"},{"Start":"01:36.790 ","End":"01:41.360","Text":"so I\u0027d have here heads and tails when I get 1 with a dye,"},{"Start":"01:41.360 ","End":"01:43.595","Text":"heads and tails when I get 2,"},{"Start":"01:43.595 ","End":"01:46.205","Text":"heads and tails when I get 3,"},{"Start":"01:46.205 ","End":"01:48.275","Text":"heads and tails for 4,"},{"Start":"01:48.275 ","End":"01:49.808","Text":"heads and tails for 5,"},{"Start":"01:49.808 ","End":"01:52.160","Text":"and heads and tails for 6."},{"Start":"01:52.160 ","End":"01:56.509","Text":"If we count how many heads and tails we have,"},{"Start":"01:56.509 ","End":"01:58.610","Text":"well, we have 12."},{"Start":"01:58.610 ","End":"02:02.217","Text":"We have 6 times 2,"},{"Start":"02:02.217 ","End":"02:09.210","Text":"so this then would be the size of our sample space in this trial."}],"ID":12778},{"Watched":false,"Name":"Example 2","Duration":"2m 31s","ChapterTopicVideoID":12300,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12300.jpeg","UploadDate":"2018-07-22T16:34:53.8230000","DurationForVideoObject":"PT2M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this example, we\u0027re asked how many license plates with 5 characters"},{"Start":"00:03.750 ","End":"00:07.530","Text":"can be created when the first character is an English letter,"},{"Start":"00:07.530 ","End":"00:10.680","Text":"and the other characters are numerical digits?"},{"Start":"00:10.680 ","End":"00:14.415","Text":"First of all, let\u0027s take a look at our license plate."},{"Start":"00:14.415 ","End":"00:16.920","Text":"Here, this is our license plate,"},{"Start":"00:16.920 ","End":"00:19.035","Text":"and we have 5 characters here."},{"Start":"00:19.035 ","End":"00:21.255","Text":"The first character right here,"},{"Start":"00:21.255 ","End":"00:26.085","Text":"that\u0027s a letter, and all the other characters are numbers."},{"Start":"00:26.085 ","End":"00:28.080","Text":"This is a number, number,"},{"Start":"00:28.080 ","End":"00:32.995","Text":"number and number, and all the numbers here are from 0-9."},{"Start":"00:32.995 ","End":"00:38.545","Text":"Now, basically we have a trial here with 5 steps,"},{"Start":"00:38.545 ","End":"00:43.745","Text":"the first step is finding or calculating"},{"Start":"00:43.745 ","End":"00:49.505","Text":"the total number of possible values that I can get in this first character."},{"Start":"00:49.505 ","End":"00:53.370","Text":"The second that\u0027ll be n_1."},{"Start":"00:53.440 ","End":"01:00.964","Text":"N_2 would be the total number of possibilities that I can get for the second character,"},{"Start":"01:00.964 ","End":"01:02.360","Text":"that\u0027ll be n_2,"},{"Start":"01:02.360 ","End":"01:03.560","Text":"and so on,"},{"Start":"01:03.560 ","End":"01:05.510","Text":"and so forth, that\u0027ll be 3,"},{"Start":"01:05.510 ","End":"01:07.050","Text":"that will be n_4,"},{"Start":"01:07.050 ","End":"01:08.550","Text":"that\u0027ll be n_5,"},{"Start":"01:08.550 ","End":"01:10.275","Text":"so that\u0027ll be n_2,"},{"Start":"01:10.275 ","End":"01:14.200","Text":"n_3, n_4, and n_5."},{"Start":"01:14.200 ","End":"01:18.065","Text":"Now, since the first character is a letter,"},{"Start":"01:18.065 ","End":"01:20.585","Text":"how many letters do I have in the alphabet?"},{"Start":"01:20.585 ","End":"01:22.850","Text":"Well, I have 26 letters."},{"Start":"01:22.850 ","End":"01:26.330","Text":"What about the second character right here?"},{"Start":"01:26.330 ","End":"01:28.325","Text":"Well, that\u0027s a number,"},{"Start":"01:28.325 ","End":"01:30.380","Text":"and the number here is from 0-9,"},{"Start":"01:30.380 ","End":"01:33.285","Text":"so how many possibilities do I have here?"},{"Start":"01:33.285 ","End":"01:35.120","Text":"Well I have 10 possibilities,"},{"Start":"01:35.120 ","End":"01:39.200","Text":"and the same number of possibilities for n_3,"},{"Start":"01:39.200 ","End":"01:42.485","Text":"and n-4, and n_5."},{"Start":"01:42.485 ","End":"01:51.965","Text":"What\u0027s the total number of license plates that I can get when I have this rule here,"},{"Start":"01:51.965 ","End":"01:53.540","Text":"where I have 5 characters,"},{"Start":"01:53.540 ","End":"01:55.475","Text":"and the first character is a letter,"},{"Start":"01:55.475 ","End":"01:57.905","Text":"and all the other characters are numbers?"},{"Start":"01:57.905 ","End":"02:01.565","Text":"Well, that would be the size of my sample space,"},{"Start":"02:01.565 ","End":"02:03.200","Text":"and that would be equal to"},{"Start":"02:03.200 ","End":"02:12.195","Text":"26 n_1 times 10 times n_2 times n_3 times n_4 times n_5."},{"Start":"02:12.195 ","End":"02:19.485","Text":"Now, that comes out to 260,000 license plates."},{"Start":"02:19.485 ","End":"02:22.850","Text":"That\u0027ll be the size of my sample space,"},{"Start":"02:22.850 ","End":"02:31.590","Text":"and that\u0027s also the number of possible license plates that I can create using this rule."}],"ID":12779},{"Watched":false,"Name":"Exrcise 1 - Parts a-d","Duration":"3m 41s","ChapterTopicVideoID":12301,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12301.jpeg","UploadDate":"2018-07-22T16:35:01.0800000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this question, we\u0027re asked to calculate the number of"},{"Start":"00:02.040 ","End":"00:04.130","Text":"possibilities for the following trials."},{"Start":"00:04.130 ","End":"00:07.290","Text":"The 1st trial is throwing a die twice."},{"Start":"00:07.290 ","End":"00:09.780","Text":"Well, there are 2 stages in this trial."},{"Start":"00:09.780 ","End":"00:13.020","Text":"The 1st stage is throwing a die for the 1st time."},{"Start":"00:13.020 ","End":"00:17.160","Text":"Now how many possible results can I get when I throw the die?"},{"Start":"00:17.160 ","End":"00:20.985","Text":"6 possible results from 1-6."},{"Start":"00:20.985 ","End":"00:23.594","Text":"When I throw the die again,"},{"Start":"00:23.594 ","End":"00:25.425","Text":"well, that\u0027s the next stage."},{"Start":"00:25.425 ","End":"00:27.345","Text":"How many possible results here?"},{"Start":"00:27.345 ","End":"00:28.640","Text":"Well, that\u0027s also 6."},{"Start":"00:28.640 ","End":"00:32.750","Text":"How many possible results can I get when I throw the dice twice?"},{"Start":"00:32.750 ","End":"00:39.770","Text":"That\u0027ll be 6 times 6 or 36 possible possible results."},{"Start":"00:39.770 ","End":"00:46.340","Text":"In this trial, we\u0027re asked to calculate the number of 3-digit numbers that I can write."},{"Start":"00:46.340 ","End":"00:52.365","Text":"First of all, let\u0027s take a look at the format of a 3-digit number. Well, here it is."},{"Start":"00:52.365 ","End":"00:54.440","Text":"Here we have the 100s place,"},{"Start":"00:54.440 ","End":"00:55.850","Text":"here we have the 10s place,"},{"Start":"00:55.850 ","End":"00:57.605","Text":"and here we have the units place."},{"Start":"00:57.605 ","End":"01:02.795","Text":"Now, each place here can have a number from 0-9,"},{"Start":"01:02.795 ","End":"01:06.035","Text":"except for the 100s place. Now why is that?"},{"Start":"01:06.035 ","End":"01:13.465","Text":"Well, a 100s place can have the numbers from 1-9, not from 0-9."},{"Start":"01:13.465 ","End":"01:15.470","Text":"If the 100s place had a 0,"},{"Start":"01:15.470 ","End":"01:18.340","Text":"then this would be a 2-digit number."},{"Start":"01:18.340 ","End":"01:24.635","Text":"N_1 here, the number of possibilities for the 100s place, that\u0027s 9."},{"Start":"01:24.635 ","End":"01:28.535","Text":"Now, how many possibilities do we have for the 10s place?"},{"Start":"01:28.535 ","End":"01:31.265","Text":"Well, that\u0027s 10, that\u0027s from 0-9."},{"Start":"01:31.265 ","End":"01:32.780","Text":"For the units place,"},{"Start":"01:32.780 ","End":"01:35.645","Text":"they\u0027ll be n_3, that\u0027s also 10."},{"Start":"01:35.645 ","End":"01:42.315","Text":"How many possible numbers can I write when I have a 3-digit number?"},{"Start":"01:42.315 ","End":"01:45.635","Text":"Well, that would be 9 times 10 times 10,"},{"Start":"01:45.635 ","End":"01:49.550","Text":"and that equals to 900 numbers."},{"Start":"01:49.550 ","End":"01:54.065","Text":"In this section, we\u0027re asked to calculate the number of possibilities of selecting"},{"Start":"01:54.065 ","End":"01:58.730","Text":"1 boy and 1 girl from a class with 7 boys and 10 girls."},{"Start":"01:58.730 ","End":"02:02.090","Text":"Well, this is a trial with 2 steps."},{"Start":"02:02.090 ","End":"02:04.730","Text":"The 1st step is to select a boy."},{"Start":"02:04.730 ","End":"02:07.550","Text":"Now, how many possibilities do I have selecting"},{"Start":"02:07.550 ","End":"02:11.150","Text":"1 boy out of a class with 7 boys and 10 girls?"},{"Start":"02:11.150 ","End":"02:12.880","Text":"Well, 7."},{"Start":"02:12.880 ","End":"02:16.160","Text":"The 2nd step is to select and girl."},{"Start":"02:16.160 ","End":"02:19.250","Text":"Now, how many possibilities do I have selecting"},{"Start":"02:19.250 ","End":"02:22.865","Text":"1 girl from a class of 7 boys and 10 girls?"},{"Start":"02:22.865 ","End":"02:25.700","Text":"Well, 10. In total,"},{"Start":"02:25.700 ","End":"02:28.220","Text":"I have 7 times 10,"},{"Start":"02:28.220 ","End":"02:29.870","Text":"which is equal to 70,"},{"Start":"02:29.870 ","End":"02:35.045","Text":"and that\u0027s the number of possibilities of selecting 1 boy and 1 girl."},{"Start":"02:35.045 ","End":"02:38.300","Text":"In this section, we\u0027re asked to calculate the number of possibilities of"},{"Start":"02:38.300 ","End":"02:41.600","Text":"distributing 2 different prizes to 10 different people,"},{"Start":"02:41.600 ","End":"02:44.930","Text":"where each person can\u0027t receive more than 1 prize."},{"Start":"02:44.930 ","End":"02:47.555","Text":"Well, this has 2 steps in it,"},{"Start":"02:47.555 ","End":"02:51.575","Text":"distributing the 1st prize and distributing the 2nd price."},{"Start":"02:51.575 ","End":"02:59.390","Text":"Now, how many possibilities do I have of distributing the 1st-price? That\u0027ll be 10."},{"Start":"02:59.390 ","End":"03:01.190","Text":"I have 10 different people,"},{"Start":"03:01.190 ","End":"03:04.580","Text":"so I have to choose 1 person and give him the price."},{"Start":"03:04.580 ","End":"03:07.040","Text":"What about the 2nd prize? That\u0027ll be n_2."},{"Start":"03:07.040 ","End":"03:10.115","Text":"How many possibilities of distributing the 2nd prize?"},{"Start":"03:10.115 ","End":"03:16.155","Text":"Well, here I have only 9 people that I can distribute to. Why is that?"},{"Start":"03:16.155 ","End":"03:19.610","Text":"Well, the 10th person already received the prize."},{"Start":"03:19.610 ","End":"03:21.815","Text":"You can\u0027t receive more than 1 price."},{"Start":"03:21.815 ","End":"03:25.235","Text":"How many possibilities in total do I have?"},{"Start":"03:25.235 ","End":"03:27.935","Text":"Well, I have 10 times 9,"},{"Start":"03:27.935 ","End":"03:29.750","Text":"that equals to 90."},{"Start":"03:29.750 ","End":"03:36.395","Text":"So I have 90 possibilities of distributing 2 different prices to 10 different people,"},{"Start":"03:36.395 ","End":"03:41.789","Text":"where a person can\u0027t have more than 1 price."}],"ID":12780},{"Watched":false,"Name":"Exrcise 2 - Parts a-b","Duration":"2m 57s","ChapterTopicVideoID":12302,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12302.jpeg","UploadDate":"2018-07-22T16:35:07.9730000","DurationForVideoObject":"PT2M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"In this question, we\u0027re told that a restaurant offers"},{"Start":"00:02.130 ","End":"00:04.440","Text":"a business meal that consists of a first course,"},{"Start":"00:04.440 ","End":"00:06.150","Text":"a main course, and a beverage."},{"Start":"00:06.150 ","End":"00:09.345","Text":"The options for the first course are a vegetable salad,"},{"Start":"00:09.345 ","End":"00:10.485","Text":"a pasta salad,"},{"Start":"00:10.485 ","End":"00:12.000","Text":"and the soup of the day."},{"Start":"00:12.000 ","End":"00:14.550","Text":"The options for the main course are steak,"},{"Start":"00:14.550 ","End":"00:17.264","Text":"chicken, meat lasagna, and vegetarian lasagna."},{"Start":"00:17.264 ","End":"00:21.150","Text":"The beverage options are coffee, tea, and lemonade."},{"Start":"00:21.150 ","End":"00:25.500","Text":"Now we\u0027re asked, how many different meals can be assembled with this menu."},{"Start":"00:25.500 ","End":"00:31.470","Text":"Well, we\u0027re looking for the total number of combinations of choices for first course,"},{"Start":"00:31.470 ","End":"00:33.520","Text":"main course, and beverage."},{"Start":"00:33.520 ","End":"00:35.540","Text":"Well, for the first course,"},{"Start":"00:35.540 ","End":"00:37.205","Text":"well, how many options do I have?"},{"Start":"00:37.205 ","End":"00:39.770","Text":"Well, I have here a vegetable salad,"},{"Start":"00:39.770 ","End":"00:42.470","Text":"a pasta salad, and a soup of the day. That\u0027s 3."},{"Start":"00:42.470 ","End":"00:43.850","Text":"For the main course,"},{"Start":"00:43.850 ","End":"00:46.610","Text":"that\u0027s n_2, that\u0027ll be steak,"},{"Start":"00:46.610 ","End":"00:47.840","Text":"chicken, meat lasagna,"},{"Start":"00:47.840 ","End":"00:49.910","Text":"and vegetarian lasagna, that\u0027s 4,"},{"Start":"00:49.910 ","End":"00:52.400","Text":"and for the beverage that\u0027s n_3,"},{"Start":"00:52.400 ","End":"00:55.330","Text":"that\u0027s coffee, tea, and lemonade, that\u0027s 3."},{"Start":"00:55.330 ","End":"00:59.960","Text":"So the total number of combinations would be 3 times 4 times 3,"},{"Start":"00:59.960 ","End":"01:02.780","Text":"and that equals to 36."},{"Start":"01:02.780 ","End":"01:05.210","Text":"In this section, we\u0027re told that a person orders a meal"},{"Start":"01:05.210 ","End":"01:08.495","Text":"randomly and we\u0027re asked to calculate the following probabilities."},{"Start":"01:08.495 ","End":"01:11.885","Text":"The first probability is a meal with a vegetable salad,"},{"Start":"01:11.885 ","End":"01:13.610","Text":"a meat lasagna, and lemonade."},{"Start":"01:13.610 ","End":"01:17.310","Text":"So we\u0027re looking for the probability of a veg salad,"},{"Start":"01:17.310 ","End":"01:24.665","Text":"a meat lasagna, and lemonade."},{"Start":"01:24.665 ","End":"01:30.470","Text":"Now, we know that the total number of combinations to choose from is 36."},{"Start":"01:30.470 ","End":"01:38.435","Text":"How many combinations do I have of choosing a vegetable salad from the first course."},{"Start":"01:38.435 ","End":"01:40.790","Text":"Well, that\u0027s 1 times,"},{"Start":"01:40.790 ","End":"01:43.970","Text":"how many combinations or how many possibilities do I have to"},{"Start":"01:43.970 ","End":"01:47.330","Text":"choose a meat lasagna from the second course?"},{"Start":"01:47.330 ","End":"01:48.680","Text":"Well, that\u0027s 1 out of 4,"},{"Start":"01:48.680 ","End":"01:51.920","Text":"so that\u0027s 1, and a lemonade for beverage."},{"Start":"01:51.920 ","End":"01:53.300","Text":"Well again that\u0027s 1."},{"Start":"01:53.300 ","End":"01:56.975","Text":"That equals to 1 divided by 36."},{"Start":"01:56.975 ","End":"01:59.500","Text":"This is 1. Now 2,"},{"Start":"01:59.500 ","End":"02:05.374","Text":"we\u0027re asked about the probability of a meal with a salad,"},{"Start":"02:05.374 ","End":"02:09.200","Text":"lasagna, and tea."},{"Start":"02:09.200 ","End":"02:12.635","Text":"Well, here, if we notice,"},{"Start":"02:12.635 ","End":"02:14.450","Text":"I didn\u0027t say which salad."},{"Start":"02:14.450 ","End":"02:19.220","Text":"It could have been a vegetable salad or a pasta salad,"},{"Start":"02:19.220 ","End":"02:22.850","Text":"so how many possibilities do I have for the first course?"},{"Start":"02:22.850 ","End":"02:27.590","Text":"I have 2, and here we have a lasagna for the main course."},{"Start":"02:27.590 ","End":"02:29.390","Text":"Well, again I didn\u0027t say which one."},{"Start":"02:29.390 ","End":"02:32.660","Text":"It could have been a meat lasagna or a vegetable lasagna."},{"Start":"02:32.660 ","End":"02:34.475","Text":"So again, we have here 2,"},{"Start":"02:34.475 ","End":"02:37.695","Text":"and for the beverage I choose tea,"},{"Start":"02:37.695 ","End":"02:43.490","Text":"so that\u0027s 1 out of a total of 36 different possibilities."},{"Start":"02:43.490 ","End":"02:45.890","Text":"That\u0027s 4 divided by 36,"},{"Start":"02:45.890 ","End":"02:49.970","Text":"that\u0027s the probability of choosing a meal with salad, lasagna, and tea."},{"Start":"02:49.970 ","End":"02:52.100","Text":"If we want to reduce this,"},{"Start":"02:52.100 ","End":"02:53.375","Text":"you don\u0027t really have to,"},{"Start":"02:53.375 ","End":"02:57.150","Text":"but this can be reduced to 1/9."}],"ID":12781},{"Watched":false,"Name":"Exrcise 3 - Parts a-d","Duration":"7m 10s","ChapterTopicVideoID":12303,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12303.jpeg","UploadDate":"2018-07-22T16:35:25.7870000","DurationForVideoObject":"PT7M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this question, we\u0027re told that a 5 digit numbers selected"},{"Start":"00:03.030 ","End":"00:06.420","Text":"randomly and we\u0027re asked to calculate the following probabilities,"},{"Start":"00:06.420 ","End":"00:10.650","Text":"where the first probability is that the number is even."},{"Start":"00:10.650 ","End":"00:13.440","Text":"Well, before we calculate this probability,"},{"Start":"00:13.440 ","End":"00:17.625","Text":"let\u0027s look at the format of a 5 digit number."},{"Start":"00:17.625 ","End":"00:22.410","Text":"Here it is, and here we have 5 places here."},{"Start":"00:22.410 ","End":"00:24.900","Text":"We have the ones, the tens to hundreds,"},{"Start":"00:24.900 ","End":"00:27.450","Text":"the thousands and tens of thousands place."},{"Start":"00:27.450 ","End":"00:33.615","Text":"What we\u0027re looking for is to see how many digits from 0 to 9,"},{"Start":"00:33.615 ","End":"00:39.105","Text":"what is the possible number of digits that can be put in each 1 of the places here."},{"Start":"00:39.105 ","End":"00:42.195","Text":"Well, here we have n1, n2,"},{"Start":"00:42.195 ","End":"00:46.020","Text":"n3, n4 and n5."},{"Start":"00:46.020 ","End":"00:49.370","Text":"Now remember, each 1 of the n\u0027s here is"},{"Start":"00:49.370 ","End":"00:54.480","Text":"the number of possible digits that I can put in each 1 of the places."},{"Start":"00:54.480 ","End":"00:58.610","Text":"In n1, in the tens of thousands of place well,"},{"Start":"00:58.610 ","End":"01:00.350","Text":"how many digits can I put here?"},{"Start":"01:00.350 ","End":"01:02.165","Text":"How many numbers can I place here?"},{"Start":"01:02.165 ","End":"01:05.405","Text":"Well, 9 numbers. Why is that?"},{"Start":"01:05.405 ","End":"01:11.430","Text":"Well, usually in all the other places I have 10 digits that I can choose from,"},{"Start":"01:11.430 ","End":"01:15.920","Text":"from 0-9 but here I can\u0027t use the digit 0."},{"Start":"01:15.920 ","End":"01:18.350","Text":"I can\u0027t use the number 0 and why is that?"},{"Start":"01:18.350 ","End":"01:20.150","Text":"Well if I do have 0 here,"},{"Start":"01:20.150 ","End":"01:22.565","Text":"then this would be a 4 digit number."},{"Start":"01:22.565 ","End":"01:26.165","Text":"N1 has 9 different possibilities."},{"Start":"01:26.165 ","End":"01:27.485","Text":"What about n2?"},{"Start":"01:27.485 ","End":"01:32.555","Text":"Well, that has 10 different possibilities and n3 also 10,"},{"Start":"01:32.555 ","End":"01:37.825","Text":"n4 also 10 and n5 also 10."},{"Start":"01:37.825 ","End":"01:41.540","Text":"Now, that means that when we\u0027re looking for"},{"Start":"01:41.540 ","End":"01:47.105","Text":"the total number of 5 digits numbers that I can have,"},{"Start":"01:47.105 ","End":"01:53.525","Text":"that\u0027ll be equal to the total sample size here and they\u0027ll be 9 times 10 times 10,"},{"Start":"01:53.525 ","End":"01:58.915","Text":"times 10, times 10 and that equals to 90,000."},{"Start":"01:58.915 ","End":"02:02.120","Text":"Now that I know what my sample sizes,"},{"Start":"02:02.120 ","End":"02:07.130","Text":"let\u0027s calculate the probability that this number is even."},{"Start":"02:07.130 ","End":"02:10.265","Text":"Now what does this mean that a number is even?"},{"Start":"02:10.265 ","End":"02:15.170","Text":"That means that in the ones place we have an even number because we"},{"Start":"02:15.170 ","End":"02:20.956","Text":"want all this number to be divisible by 2 without any remainders."},{"Start":"02:20.956 ","End":"02:24.755","Text":"What would be the probability here?"},{"Start":"02:24.755 ","End":"02:30.670","Text":"In the denominator we have the size of my sample space, that\u0027ll be 90,000."},{"Start":"02:30.670 ","End":"02:35.465","Text":"Now what about the size of this event that the number is even?"},{"Start":"02:35.465 ","End":"02:39.680","Text":"Well, that\u0027ll be 9 times 10 times 10 times"},{"Start":"02:39.680 ","End":"02:45.320","Text":"10 because in each 1 of the numbers I can have either even or odd numbers,"},{"Start":"02:45.320 ","End":"02:49.724","Text":"but only the ones place I need for the number to be even."},{"Start":"02:49.724 ","End":"02:55.350","Text":"How many even numbers do I have from the list of numbers from 0-9?"},{"Start":"02:55.350 ","End":"02:57.470","Text":"Well, I have 5 numbers."},{"Start":"02:57.470 ","End":"03:07.465","Text":"That means that this probability is 45,000 divided by 90,000 and that equals to a 1/2."},{"Start":"03:07.465 ","End":"03:10.510","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"03:10.510 ","End":"03:13.390","Text":"that all the digits in the number are different?"},{"Start":"03:13.390 ","End":"03:18.361","Text":"We\u0027re looking for the probability of digits in the number being different."},{"Start":"03:18.361 ","End":"03:22.180","Text":"Well, we know that"},{"Start":"03:22.180 ","End":"03:26.905","Text":"the sample space or the size of the sample space that equals to 90,000,"},{"Start":"03:26.905 ","End":"03:29.565","Text":"so we have 90,000 numbers to choose from."},{"Start":"03:29.565 ","End":"03:33.965","Text":"Now, what\u0027s the size of this event were all the numbers here are different?"},{"Start":"03:33.965 ","End":"03:36.650","Text":"Well, let\u0027s take a look at the number of"},{"Start":"03:36.650 ","End":"03:39.725","Text":"digits that I can have in the tens of thousands place."},{"Start":"03:39.725 ","End":"03:44.420","Text":"Well, I have 9 digits that I can choose from, from 1-9."},{"Start":"03:44.420 ","End":"03:48.035","Text":"Now what about the number of digits in this place?"},{"Start":"03:48.035 ","End":"03:52.430","Text":"Well, I can now choose a 0 but I can\u0027t choose"},{"Start":"03:52.430 ","End":"03:56.996","Text":"the number that\u0027s here in the tens of thousands place."},{"Start":"03:56.996 ","End":"04:00.800","Text":"Here I also have 9 numbers to choose from."},{"Start":"04:00.800 ","End":"04:05.180","Text":"In here, I had 10 digits to choose from,"},{"Start":"04:05.180 ","End":"04:10.343","Text":"from 0-9, but I can\u0027t use the number here and the number here."},{"Start":"04:10.343 ","End":"04:13.854","Text":"That means I have 8 digits to choose from here."},{"Start":"04:13.854 ","End":"04:20.075","Text":"Here I have 7 digits to choose from and here I have 6 digits to choose from."},{"Start":"04:20.075 ","End":"04:24.025","Text":"All in all, this ratio is equal to"},{"Start":"04:24.025 ","End":"04:32.980","Text":"0.3024 so that\u0027s the probability that all the digits in this number are different."},{"Start":"04:32.980 ","End":"04:34.760","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"04:34.760 ","End":"04:37.760","Text":"the probability that all the digits in the number are the same?"},{"Start":"04:37.760 ","End":"04:42.620","Text":"We\u0027re looking for the probability that all the digits are the same."},{"Start":"04:42.620 ","End":"04:45.020","Text":"Now, in the denominator,"},{"Start":"04:45.020 ","End":"04:49.220","Text":"we have the size of our sample space that will be 90,000."},{"Start":"04:49.220 ","End":"04:55.060","Text":"Now, what\u0027s the size of this event right here where all the digits are the same?"},{"Start":"04:55.060 ","End":"04:58.250","Text":"Well, let\u0027s take a look at the tens of thousands place."},{"Start":"04:58.250 ","End":"05:01.580","Text":"Now, how many numbers can I choose from here?"},{"Start":"05:01.580 ","End":"05:04.185","Text":"Well, I have 9 numbers to choose from here,"},{"Start":"05:04.185 ","End":"05:08.460","Text":"but if all the numbers are the same then in the thousands place,"},{"Start":"05:08.460 ","End":"05:10.710","Text":"I have 1 number to choose from,"},{"Start":"05:10.710 ","End":"05:13.150","Text":"that\u0027s the number that\u0027s in here and,"},{"Start":"05:13.150 ","End":"05:15.080","Text":"in the hundreds place, well,"},{"Start":"05:15.080 ","End":"05:18.950","Text":"I have that 1 number as well and in the tens place I"},{"Start":"05:18.950 ","End":"05:23.015","Text":"have that 1 number as well and in the ones place I have that number as well."},{"Start":"05:23.015 ","End":"05:30.650","Text":"In essence, this probability is equal to 9 divided by 90,000"},{"Start":"05:30.650 ","End":"05:39.250","Text":"and that equals to 0.0001 a very small probability."},{"Start":"05:39.250 ","End":"05:42.530","Text":"In this section, we\u0027re asked for the probability that"},{"Start":"05:42.530 ","End":"05:45.935","Text":"this number has at least 2 different digits?"},{"Start":"05:45.935 ","End":"05:51.769","Text":"Let\u0027s just define the event at least 2 different digits."},{"Start":"05:51.769 ","End":"05:57.500","Text":"We\u0027ll define that as A and we\u0027re looking for the probability of A."},{"Start":"05:57.500 ","End":"06:03.425","Text":"Now, this is a very difficult probability to calculate,"},{"Start":"06:03.425 ","End":"06:06.380","Text":"like this when we define A like this."},{"Start":"06:06.380 ","End":"06:11.000","Text":"But what happens if we take the complimentary set of this?"},{"Start":"06:11.000 ","End":"06:12.920","Text":"What\u0027s the complimentary set of this?"},{"Start":"06:12.920 ","End":"06:17.450","Text":"At least 2 different digits with a complimentary set of that,"},{"Start":"06:17.450 ","End":"06:22.000","Text":"well that would be that all digits are the same."},{"Start":"06:22.000 ","End":"06:26.780","Text":"We know from De Morgan\u0027s rule that the probability of"},{"Start":"06:26.780 ","End":"06:32.665","Text":"A that equals to 1 minus the probability of the complimentary set of A,"},{"Start":"06:32.665 ","End":"06:36.650","Text":"but let\u0027s take a look at the complimentary set of A."},{"Start":"06:36.650 ","End":"06:39.110","Text":"We\u0027ve calculated this probability,"},{"Start":"06:39.110 ","End":"06:41.240","Text":"the probability of A bar,"},{"Start":"06:41.240 ","End":"06:47.245","Text":"that all the digits are the same but that equals to 0.0001."},{"Start":"06:47.245 ","End":"06:50.010","Text":"We\u0027ve calculated that in the last section,"},{"Start":"06:50.010 ","End":"06:53.040","Text":"so the probability of A that we have"},{"Start":"06:53.040 ","End":"06:57.120","Text":"at least 2 different digits in this number that equals to"},{"Start":"06:57.120 ","End":"07:04.920","Text":"1 minus the probability of A bar that equals to 1 minus 0.0001,"},{"Start":"07:04.920 ","End":"07:10.360","Text":"and that equals to 0.9999."}],"ID":12782},{"Watched":false,"Name":"Exrcise 3 - Parts e-f","Duration":"3m 2s","ChapterTopicVideoID":12304,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12304.jpeg","UploadDate":"2018-07-22T16:35:32.5770000","DurationForVideoObject":"PT3M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this section, we\u0027re asked to calculate the probability that"},{"Start":"00:02.670 ","End":"00:05.775","Text":"this number has at least 2 identical digits."},{"Start":"00:05.775 ","End":"00:14.160","Text":"So let\u0027s define A as the event that this number has at least 2 identical digits."},{"Start":"00:14.160 ","End":"00:18.825","Text":"Okay. We\u0027re looking for the probability of A."},{"Start":"00:18.825 ","End":"00:22.290","Text":"When A is defined like this,"},{"Start":"00:22.290 ","End":"00:24.510","Text":"then the probability of A, well,"},{"Start":"00:24.510 ","End":"00:26.535","Text":"that\u0027s very difficult to calculate,"},{"Start":"00:26.535 ","End":"00:31.455","Text":"but let\u0027s see if we can calculate it using the complementary set."},{"Start":"00:31.455 ","End":"00:34.500","Text":"What would be the complimentary set of A"},{"Start":"00:34.500 ","End":"00:39.340","Text":"that this number has at least 2 identical digits."},{"Start":"00:39.340 ","End":"00:44.425","Text":"Well, the complementary set of that would be that all digits are different."},{"Start":"00:44.425 ","End":"00:47.780","Text":"This would be the complimentary set."},{"Start":"00:47.780 ","End":"00:52.070","Text":"We know that the probability of A that equals"},{"Start":"00:52.070 ","End":"00:56.630","Text":"to 1 minus the probability of the complimentary set of A."},{"Start":"00:56.630 ","End":"01:00.265","Text":"So what is the probability of the complimentary set available?"},{"Start":"01:00.265 ","End":"01:01.940","Text":"We\u0027ve calculated that."},{"Start":"01:01.940 ","End":"01:06.160","Text":"That equals to 0.3024."},{"Start":"01:07.150 ","End":"01:13.120","Text":"Now, that means that the probability of A that equals to 1"},{"Start":"01:13.120 ","End":"01:17.470","Text":"minus 0.3024 and that"},{"Start":"01:17.470 ","End":"01:23.840","Text":"equals to 0.6976."},{"Start":"01:23.840 ","End":"01:25.220","Text":"This section we\u0027re asked,"},{"Start":"01:25.220 ","End":"01:28.754","Text":"what\u0027s the probability that this number is a palindrome?"},{"Start":"01:28.754 ","End":"01:32.210","Text":"Palindrome is a number that can be read from"},{"Start":"01:32.210 ","End":"01:35.690","Text":"left to right or right to left and it\u0027ll be the same number,"},{"Start":"01:35.690 ","End":"01:40.050","Text":"for example, the number 12,3211."},{"Start":"01:40.190 ","End":"01:46.760","Text":"We can read it from left to right or from right to left and it\u0027ll be the same number."},{"Start":"01:46.760 ","End":"01:56.070","Text":"Now, what\u0027s the probability that the number selected is a palindrome?"},{"Start":"01:56.070 ","End":"01:58.880","Text":"Well, in the denominator,"},{"Start":"01:58.880 ","End":"02:02.030","Text":"we have the size of our sample space."},{"Start":"02:02.030 ","End":"02:08.690","Text":"How do we calculate the size of this event right here with the numbers are palindromes?"},{"Start":"02:08.690 ","End":"02:12.800","Text":"How many digits do I to choose from in"},{"Start":"02:12.800 ","End":"02:18.085","Text":"this place right here in the tens of thousands of place where I can have 9 numbers here?"},{"Start":"02:18.085 ","End":"02:20.960","Text":"How many digits can I have here?"},{"Start":"02:20.960 ","End":"02:23.105","Text":"Well, I can have 10 to choose from,"},{"Start":"02:23.105 ","End":"02:24.635","Text":"and here, again,"},{"Start":"02:24.635 ","End":"02:26.135","Text":"10 to choose from."},{"Start":"02:26.135 ","End":"02:33.845","Text":"But here, this number right here has to be the same as this number right here, so,"},{"Start":"02:33.845 ","End":"02:38.675","Text":"I have only 1 number to choose from, and here,"},{"Start":"02:38.675 ","End":"02:43.160","Text":"I have only 1 number to choose from because this number has to be the same as"},{"Start":"02:43.160 ","End":"02:49.735","Text":"this number in order for this whole number to be a palindrome."},{"Start":"02:49.735 ","End":"02:57.470","Text":"The probability here would be 900 divided by 90,000."},{"Start":"02:57.470 ","End":"02:58.910","Text":"Now, let\u0027s reduce that."},{"Start":"02:58.910 ","End":"03:02.310","Text":"That equals to 0.01."}],"ID":12783},{"Watched":false,"Name":"Exrcise 4 - Parts a-d","Duration":"4m 44s","ChapterTopicVideoID":12305,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12305.jpeg","UploadDate":"2018-07-22T16:35:44.0570000","DurationForVideoObject":"PT4M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this question, we\u0027re told that 5 random people enter an elevator in"},{"Start":"00:03.510 ","End":"00:07.665","Text":"a building with 8 floors and we\u0027re asked to calculate the following probabilities."},{"Start":"00:07.665 ","End":"00:09.195","Text":"Now, before we do that,"},{"Start":"00:09.195 ","End":"00:12.240","Text":"let\u0027s just calculate the size of a sample space."},{"Start":"00:12.240 ","End":"00:18.210","Text":"That means how many combinations do we have of people getting off on what floor?"},{"Start":"00:18.210 ","End":"00:20.865","Text":"What person gets off on what floor?"},{"Start":"00:20.865 ","End":"00:24.870","Text":"The size of our sample space is this."},{"Start":"00:24.870 ","End":"00:26.160","Text":"We have to ask ourselves,"},{"Start":"00:26.160 ","End":"00:30.045","Text":"on how many floors a person can get off?"},{"Start":"00:30.045 ","End":"00:32.850","Text":"Well, a person can get off on 1 of 8 floors,"},{"Start":"00:32.850 ","End":"00:34.650","Text":"so it has 8 choices."},{"Start":"00:34.650 ","End":"00:38.480","Text":"Now, the second person can also have 8 choices,"},{"Start":"00:38.480 ","End":"00:40.725","Text":"and the third, and the fourth,"},{"Start":"00:40.725 ","End":"00:42.465","Text":"and the fifth person."},{"Start":"00:42.465 ","End":"00:45.900","Text":"In all, we have 8 to the power of"},{"Start":"00:45.900 ","End":"00:52.915","Text":"5 possibilities of 5 people getting off on different floors."},{"Start":"00:52.915 ","End":"00:59.420","Text":"Now we want to calculate the probability that all the people got off on the 5th floor."},{"Start":"00:59.420 ","End":"01:01.310","Text":"Well, if that\u0027s the case,"},{"Start":"01:01.310 ","End":"01:03.455","Text":"well then in the denominator,"},{"Start":"01:03.455 ","End":"01:06.770","Text":"what we have are the size of our sample space,"},{"Start":"01:06.770 ","End":"01:08.765","Text":"which is 8 to the power of 5."},{"Start":"01:08.765 ","End":"01:10.580","Text":"Now, in the numerator,"},{"Start":"01:10.580 ","End":"01:13.385","Text":"we have to calculate the size of this event that"},{"Start":"01:13.385 ","End":"01:16.745","Text":"all the people got off at the fifth floor."},{"Start":"01:16.745 ","End":"01:20.390","Text":"Well, if that\u0027s the case then the first person well,"},{"Start":"01:20.390 ","End":"01:22.240","Text":"he didn\u0027t have that much choice."},{"Start":"01:22.240 ","End":"01:25.910","Text":"He got off on 1 of the 8 floors,"},{"Start":"01:25.910 ","End":"01:30.560","Text":"that\u0027s the fifth floor but the second person got off on the same floor,"},{"Start":"01:30.560 ","End":"01:35.855","Text":"the fifth floor, and the same for the third and the fourth and the fifth person."},{"Start":"01:35.855 ","End":"01:40.415","Text":"We\u0027re looking at 1 to the power of 5 divided by 8 to the power of 5,"},{"Start":"01:40.415 ","End":"01:42.950","Text":"which is equal to 1 divided"},{"Start":"01:42.950 ","End":"01:45.560","Text":"by 8 to the power of 5 that\u0027s"},{"Start":"01:45.560 ","End":"01:49.910","Text":"the probability that all the people got off on the fifth floor."},{"Start":"01:49.910 ","End":"01:52.595","Text":"Here we\u0027re asked to calculate the probability that"},{"Start":"01:52.595 ","End":"01:55.400","Text":"all the people got off at the same floor."},{"Start":"01:55.400 ","End":"02:00.965","Text":"Well, again, we have in the denominator the size of our sample space."},{"Start":"02:00.965 ","End":"02:04.355","Text":"Now, what\u0027s the size of this event that the people,"},{"Start":"02:04.355 ","End":"02:06.860","Text":"all the people got off at the same floor?"},{"Start":"02:06.860 ","End":"02:11.380","Text":"Well, the first person has 1 of 8 choices."},{"Start":"02:11.380 ","End":"02:14.300","Text":"He can choose 1 of 8 floors to get up,"},{"Start":"02:14.300 ","End":"02:17.420","Text":"but the others don\u0027t have a choice."},{"Start":"02:17.420 ","End":"02:21.680","Text":"They have to get off at that same floor so the next person"},{"Start":"02:21.680 ","End":"02:27.040","Text":"gets off at that floor and the next and the next and the next."},{"Start":"02:27.040 ","End":"02:30.360","Text":"That means that we have 8 divided"},{"Start":"02:30.360 ","End":"02:36.480","Text":"by 8 to the power of 5 that means that we have 1 divided by 8 to the power of 4."},{"Start":"02:36.760 ","End":"02:40.100","Text":"In this section, we\u0027re asked for the probability that"},{"Start":"02:40.100 ","End":"02:43.280","Text":"all the people got off at different floors."},{"Start":"02:43.280 ","End":"02:45.830","Text":"Again, in the denominator,"},{"Start":"02:45.830 ","End":"02:48.275","Text":"we have the size of our sample space."},{"Start":"02:48.275 ","End":"02:51.170","Text":"Now, if all the people got off from the different floors, well,"},{"Start":"02:51.170 ","End":"02:55.805","Text":"the first person had 8 floors to choose from,"},{"Start":"02:55.805 ","End":"02:59.600","Text":"but the next person had 7 floors to choose from"},{"Start":"02:59.600 ","End":"03:04.040","Text":"because he couldn\u0027t get off on the floor that the first person got off."},{"Start":"03:04.040 ","End":"03:09.300","Text":"The next person had 6 floors to choose from, the next 1,"},{"Start":"03:09.300 ","End":"03:11.960","Text":"had 5 floors to choose from,"},{"Start":"03:11.960 ","End":"03:15.980","Text":"and the last person had 4 floors to choose from."},{"Start":"03:15.980 ","End":"03:22.620","Text":"Now, that comes out to approximately 0.205."},{"Start":"03:22.900 ","End":"03:28.220","Text":"In this section, we\u0027re asked for the probability that Mike and Danny got off at"},{"Start":"03:28.220 ","End":"03:32.975","Text":"the sixth floor and that the other people got off at the other floors,"},{"Start":"03:32.975 ","End":"03:34.945","Text":"not on the 6th floor."},{"Start":"03:34.945 ","End":"03:38.615","Text":"Again, let\u0027s calculate the severity."},{"Start":"03:38.615 ","End":"03:40.229","Text":"On the denominator,"},{"Start":"03:40.229 ","End":"03:42.500","Text":"we have the size of a sample space,"},{"Start":"03:42.500 ","End":"03:44.495","Text":"that\u0027s 8 to the power of 5."},{"Start":"03:44.495 ","End":"03:47.630","Text":"Now, what about this event right here?"},{"Start":"03:47.630 ","End":"03:51.230","Text":"Well, if 2 of the 5 people got off at the sixth floor,"},{"Start":"03:51.230 ","End":"03:57.585","Text":"then they had a choice of 1 out of 8 floors to get off."},{"Start":"03:57.585 ","End":"04:03.277","Text":"That Mike got off on the sixth floor so he had only 1 choice set of 8"},{"Start":"04:03.277 ","End":"04:09.095","Text":"and Danny got off at the same floor so he also had 1 choice."},{"Start":"04:09.095 ","End":"04:12.860","Text":"Now, all the other people got off on the other floors,"},{"Start":"04:12.860 ","End":"04:17.690","Text":"not at the 6th floor so they had 7 out of 8 floors to choose from."},{"Start":"04:17.690 ","End":"04:21.730","Text":"That would be 7 times 7 times 7."},{"Start":"04:21.730 ","End":"04:25.320","Text":"Each person had 7 floors to choose from."},{"Start":"04:25.320 ","End":"04:32.880","Text":"That comes out to 0.0105 approximately."},{"Start":"04:32.880 ","End":"04:37.460","Text":"This then is the probability that Mike and Danny got off at"},{"Start":"04:37.460 ","End":"04:44.140","Text":"the sixth floor and the 3 other people got off at different floors, other floors."}],"ID":12784},{"Watched":false,"Name":"Exrcise 5","Duration":"2m 57s","ChapterTopicVideoID":12306,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12306.jpeg","UploadDate":"2018-07-22T16:35:51.4930000","DurationForVideoObject":"PT2M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"In this question, we\u0027re given a political party that has 15 members."},{"Start":"00:03.675 ","End":"00:07.230","Text":"Now, 3 of the members are selected for 3 different positions,"},{"Start":"00:07.230 ","End":"00:12.135","Text":"and we\u0027re asked in how many ways can the positions be filled if"},{"Start":"00:12.135 ","End":"00:13.410","Text":"a member can serve in"},{"Start":"00:13.410 ","End":"00:17.580","Text":"more than 1 position or a member cannot serve in more than 1 position?"},{"Start":"00:17.580 ","End":"00:21.930","Text":"Well, here, we have to note that we\u0027re dealing with the problem"},{"Start":"00:21.930 ","End":"00:26.250","Text":"in combinatorics and not in probability. Why is that?"},{"Start":"00:26.250 ","End":"00:31.430","Text":"Well, a question of probability asks us to calculate the size"},{"Start":"00:31.430 ","End":"00:37.620","Text":"of a specific event and then divide that by the size of the sample space,"},{"Start":"00:37.620 ","End":"00:39.260","Text":"and that will give us the probability."},{"Start":"00:39.260 ","End":"00:41.990","Text":"Here we\u0027re talking about combinatorics."},{"Start":"00:41.990 ","End":"00:45.995","Text":"We\u0027re asked in how many ways can the positions be filled?"},{"Start":"00:45.995 ","End":"00:52.200","Text":"That means that we\u0027re looking for the count of the possible combinations."},{"Start":"00:53.150 ","End":"00:56.884","Text":"We have here a trial in 3 stages."},{"Start":"00:56.884 ","End":"01:00.845","Text":"The first stage is choosing a person for the first position,"},{"Start":"01:00.845 ","End":"01:04.190","Text":"the second stage is choosing a person for the second position,"},{"Start":"01:04.190 ","End":"01:07.985","Text":"and the third stage is choosing a person for the third position."},{"Start":"01:07.985 ","End":"01:11.325","Text":"Now, if we\u0027re looking at n_1,"},{"Start":"01:11.325 ","End":"01:16.190","Text":"that\u0027s the number of people that I have to choose from for the first position."},{"Start":"01:16.190 ","End":"01:19.655","Text":"Well, I have 15 people to choose from for the first position."},{"Start":"01:19.655 ","End":"01:22.610","Text":"Now, what about the second position?"},{"Start":"01:22.610 ","End":"01:24.485","Text":"That\u0027s the second stage here."},{"Start":"01:24.485 ","End":"01:27.200","Text":"Well, how many people do I have to choose from"},{"Start":"01:27.200 ","End":"01:29.840","Text":"if I choose a person for the second position?"},{"Start":"01:29.840 ","End":"01:34.125","Text":"Now because a member can serve in more than 1 position,"},{"Start":"01:34.125 ","End":"01:40.640","Text":"then I can choose the same person that I chose in position number 1."},{"Start":"01:40.640 ","End":"01:44.120","Text":"I still have 15 people to choose from,"},{"Start":"01:44.120 ","End":"01:49.285","Text":"and the same argument for the third position."},{"Start":"01:49.285 ","End":"01:58.550","Text":"All in all, I have 15 times 15 times 15 ways that I can choose 3 people for 3 positions."},{"Start":"01:58.550 ","End":"02:04.705","Text":"Now that equals to 15 cubed and that equals to 3,375."},{"Start":"02:04.705 ","End":"02:06.880","Text":"Now, what about b?"},{"Start":"02:06.880 ","End":"02:10.310","Text":"B asks us in how many ways can the position to be"},{"Start":"02:10.310 ","End":"02:14.180","Text":"filled if a member cannot serve in more than 1 position?"},{"Start":"02:14.180 ","End":"02:16.555","Text":"Well, again, we have 3 stages."},{"Start":"02:16.555 ","End":"02:20.360","Text":"The first stage, how many people do I have to choose from?"},{"Start":"02:20.360 ","End":"02:24.575","Text":"Well, I can choose from 15 people."},{"Start":"02:24.575 ","End":"02:26.735","Text":"Now in the second stage,"},{"Start":"02:26.735 ","End":"02:28.820","Text":"how many people do I have to choose from?"},{"Start":"02:28.820 ","End":"02:33.620","Text":"Now I have only 14 people to choose from. Why is that?"},{"Start":"02:33.620 ","End":"02:38.700","Text":"Because the 15th person was put in position number 1."},{"Start":"02:38.700 ","End":"02:40.905","Text":"For position number 3,"},{"Start":"02:40.905 ","End":"02:43.815","Text":"I have now 13 people to choose from."},{"Start":"02:43.815 ","End":"02:47.225","Text":"Again, according to the multiplication principle,"},{"Start":"02:47.225 ","End":"02:51.770","Text":"that equals to 15 times 14 times 13,"},{"Start":"02:51.770 ","End":"02:58.140","Text":"and that equals to 2,730."}],"ID":12785},{"Watched":false,"Name":"Exrcise 6 - Parts a-d","Duration":"5m 54s","ChapterTopicVideoID":12307,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12307.jpeg","UploadDate":"2018-07-22T16:36:05.8900000","DurationForVideoObject":"PT5M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this question, we\u0027re given that a die is thrown 4 times and"},{"Start":"00:03.480 ","End":"00:07.845","Text":"we\u0027re asked what\u0027s the probability that all 4 results are the same?"},{"Start":"00:07.845 ","End":"00:12.180","Text":"First of all, I want to calculate the size of my sample space."},{"Start":"00:12.180 ","End":"00:15.915","Text":"I\u0027m dealing with calculating probabilities."},{"Start":"00:15.915 ","End":"00:19.320","Text":"What is the size of my sample space?"},{"Start":"00:19.320 ","End":"00:20.605","Text":"Well, in order to do that,"},{"Start":"00:20.605 ","End":"00:23.075","Text":"I have to know what type of trial do I have?"},{"Start":"00:23.075 ","End":"00:26.060","Text":"Well, I have a trial with 4 stages."},{"Start":"00:26.060 ","End":"00:28.835","Text":"Each stage is a throw of a die."},{"Start":"00:28.835 ","End":"00:32.570","Text":"Now, how many possible results do I have in each throw?"},{"Start":"00:32.570 ","End":"00:38.240","Text":"Well, I have 6 possible results in each throw and I\u0027m throwing the die 4 times,"},{"Start":"00:38.240 ","End":"00:46.850","Text":"so I have 6 times 6 times 6 times 6 possible combinations of results."},{"Start":"00:46.850 ","End":"00:53.000","Text":"That means that I have 6^4 possible combinations."},{"Start":"00:53.000 ","End":"00:55.355","Text":"Now, A asks,"},{"Start":"00:55.355 ","End":"00:58.745","Text":"what\u0027s the probability that all 4 results are the same?"},{"Start":"00:58.745 ","End":"01:05.820","Text":"We\u0027re looking for the probability that all the results are the same."},{"Start":"01:06.740 ","End":"01:11.400","Text":"We know that when we\u0027re dealing with probabilities,"},{"Start":"01:11.400 ","End":"01:12.630","Text":"in the denominator,"},{"Start":"01:12.630 ","End":"01:15.570","Text":"I have the size of my sample space,"},{"Start":"01:15.570 ","End":"01:17.780","Text":"so that\u0027ll be 6^4."},{"Start":"01:17.780 ","End":"01:21.085","Text":"Now, let\u0027s take a look at the numerator."},{"Start":"01:21.085 ","End":"01:27.199","Text":"The numerator gives me all the possible combinations when all the results are the same."},{"Start":"01:27.199 ","End":"01:28.925","Text":"In the first throw,"},{"Start":"01:28.925 ","End":"01:31.890","Text":"I can have 6 possible results,"},{"Start":"01:31.890 ","End":"01:34.635","Text":"I\u0027m not bound by any constraints."},{"Start":"01:34.635 ","End":"01:36.450","Text":"What about the second throw?"},{"Start":"01:36.450 ","End":"01:42.565","Text":"The result of the second throw has to be the same as the result in the first throw."},{"Start":"01:42.565 ","End":"01:45.730","Text":"I have here only 1 possible result to choose from."},{"Start":"01:45.730 ","End":"01:49.770","Text":"The same goes for the third and the fourth throws."},{"Start":"01:49.770 ","End":"02:00.135","Text":"That means that the probability that all results are the same would be 6 divided by 6^4,"},{"Start":"02:00.135 ","End":"02:03.720","Text":"and that is equal to"},{"Start":"02:03.720 ","End":"02:10.560","Text":"1/6^3 or we can write this out like this this,"},{"Start":"02:10.560 ","End":"02:14.310","Text":"1 divided by 216."},{"Start":"02:14.310 ","End":"02:16.100","Text":"In section B, we\u0027re asked,"},{"Start":"02:16.100 ","End":"02:19.685","Text":"what\u0027s the probability that all the results are different?"},{"Start":"02:19.685 ","End":"02:22.580","Text":"What\u0027s an example of this event?"},{"Start":"02:22.580 ","End":"02:24.440","Text":"Well, I could get, let\u0027s say,"},{"Start":"02:24.440 ","End":"02:25.775","Text":"a 1, 2,"},{"Start":"02:25.775 ","End":"02:27.710","Text":"4, and 5."},{"Start":"02:27.710 ","End":"02:34.480","Text":"That\u0027s 4 throws where each result is different from the next,"},{"Start":"02:34.480 ","End":"02:36.330","Text":"or I can get a 5,"},{"Start":"02:36.330 ","End":"02:38.385","Text":"3, 1, and 2."},{"Start":"02:38.385 ","End":"02:42.235","Text":"Again, another example where all the results are different."},{"Start":"02:42.235 ","End":"02:44.060","Text":"When we\u0027re looking for"},{"Start":"02:44.060 ","End":"02:47.870","Text":"the probability of this event where all the results are different, well,"},{"Start":"02:47.870 ","End":"02:54.335","Text":"in the denominator, we have the size of the sample space which is 6^4."},{"Start":"02:54.335 ","End":"02:57.200","Text":"Now, how many combinations of"},{"Start":"02:57.200 ","End":"03:01.235","Text":"possible results can I have where all the results here are different for each throw?"},{"Start":"03:01.235 ","End":"03:03.025","Text":"In the first throw,"},{"Start":"03:03.025 ","End":"03:05.880","Text":"I can have 6 possible results,"},{"Start":"03:05.880 ","End":"03:07.905","Text":"but in the second throw,"},{"Start":"03:07.905 ","End":"03:10.904","Text":"I have 5 possible results."},{"Start":"03:10.904 ","End":"03:12.570","Text":"In the third throw,"},{"Start":"03:12.570 ","End":"03:14.640","Text":"I have 4 possible results,"},{"Start":"03:14.640 ","End":"03:17.945","Text":"and in the fourth throw, I have 3 possible results."},{"Start":"03:17.945 ","End":"03:24.599","Text":"Now, that comes out to 5 divided by 18."},{"Start":"03:24.599 ","End":"03:26.480","Text":"In this section, we\u0027re asked,"},{"Start":"03:26.480 ","End":"03:30.365","Text":"what\u0027s the probability that at least 2 results are the same?"},{"Start":"03:30.365 ","End":"03:33.500","Text":"An example of that would be, let\u0027s say,"},{"Start":"03:33.500 ","End":"03:38.330","Text":"getting a 2 and a 3 and then another 2 and a 4."},{"Start":"03:38.330 ","End":"03:41.165","Text":"We have at least 2 results that are the same,"},{"Start":"03:41.165 ","End":"03:42.560","Text":"or I can get a 1,"},{"Start":"03:42.560 ","End":"03:44.750","Text":"5, 5, and 5."},{"Start":"03:44.750 ","End":"03:50.345","Text":"Again here, this is the same event that at least 2 results are the same."},{"Start":"03:50.345 ","End":"03:55.775","Text":"There are a lot of combinations here and it\u0027s very difficult to count them all,"},{"Start":"03:55.775 ","End":"03:59.690","Text":"but let\u0027s now take a look at this event right here."},{"Start":"03:59.690 ","End":"04:01.730","Text":"At least 2 results are the same."},{"Start":"04:01.730 ","End":"04:04.400","Text":"Now what\u0027s the complimentary event here?"},{"Start":"04:04.400 ","End":"04:08.515","Text":"The complimentary event of at least 2 results are"},{"Start":"04:08.515 ","End":"04:12.845","Text":"the same would be that all the results would be different."},{"Start":"04:12.845 ","End":"04:17.059","Text":"This probability, that\u0027s equal to 1 minus"},{"Start":"04:17.059 ","End":"04:23.925","Text":"the probability that all the results are different."},{"Start":"04:23.925 ","End":"04:28.275","Text":"Now, we\u0027ve calculated that. That\u0027s right here."},{"Start":"04:28.275 ","End":"04:32.644","Text":"That\u0027ll be equal to 1 minus 5 divided by 18,"},{"Start":"04:32.644 ","End":"04:36.250","Text":"and that equals to 13 divided by 18."},{"Start":"04:36.250 ","End":"04:41.105","Text":"That\u0027s the probability that at least 2 results are the same."},{"Start":"04:41.105 ","End":"04:42.650","Text":"In this section, we\u0027re asked,"},{"Start":"04:42.650 ","End":"04:46.730","Text":"what\u0027s the probability that at least 2 results are different?"},{"Start":"04:47.070 ","End":"04:51.500","Text":"First of all, let\u0027s take a look at some examples of this event right here."},{"Start":"04:51.500 ","End":"04:54.050","Text":"I could get a 1, a 2, a 3,"},{"Start":"04:54.050 ","End":"04:57.195","Text":"and a 3, or a 4, and a 1,"},{"Start":"04:57.195 ","End":"04:58.935","Text":"and a 5, and a 6,"},{"Start":"04:58.935 ","End":"05:00.900","Text":"or a 1, and a 2,"},{"Start":"05:00.900 ","End":"05:03.035","Text":"and a 1, and a 1, and so on and so forth."},{"Start":"05:03.035 ","End":"05:09.545","Text":"All these results here belong to this set that at least 2 results are different."},{"Start":"05:09.545 ","End":"05:12.680","Text":"Now, I can have many combinations right here."},{"Start":"05:12.680 ","End":"05:15.065","Text":"Let\u0027s take a look at the complimentary set."},{"Start":"05:15.065 ","End":"05:17.300","Text":"The complementary set of"},{"Start":"05:17.300 ","End":"05:23.280","Text":"at least 2 results are different would be that all the results are the same."},{"Start":"05:23.280 ","End":"05:27.800","Text":"This probability right here would be equal to 1 minus"},{"Start":"05:27.800 ","End":"05:34.430","Text":"the probability that all the results are the same."},{"Start":"05:34.430 ","End":"05:38.315","Text":"Well, we\u0027ve calculated that. It\u0027s right here."},{"Start":"05:38.315 ","End":"05:42.680","Text":"That equals to 1 minus 1 divided by 216,"},{"Start":"05:42.680 ","End":"05:48.125","Text":"and that equals to 215 divided by 216."},{"Start":"05:48.125 ","End":"05:55.530","Text":"This then would be the probability of the event that at least 2 results are different."}],"ID":12786},{"Watched":false,"Name":"Exrcise 7 - Parts a-d","Duration":"7m 6s","ChapterTopicVideoID":12308,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12308.jpeg","UploadDate":"2018-07-22T16:36:22.0200000","DurationForVideoObject":"PT7M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"In this question, we\u0027re asked to spell a 5 letter word from the alphabet,"},{"Start":"00:03.585 ","End":"00:05.130","Text":"having 26 letters,"},{"Start":"00:05.130 ","End":"00:08.760","Text":"and the word doesn\u0027t have to have any meaning."},{"Start":"00:08.760 ","End":"00:10.020","Text":"Now, we\u0027re asked,"},{"Start":"00:10.020 ","End":"00:13.080","Text":"what\u0027s the probability that there\u0027s no D,"},{"Start":"00:13.080 ","End":"00:14.625","Text":"A, or L in the word?"},{"Start":"00:14.625 ","End":"00:18.645","Text":"Well, the first thing that I want to do when we\u0027re asked to calculate probabilities,"},{"Start":"00:18.645 ","End":"00:21.270","Text":"is to calculate the size of my sample space."},{"Start":"00:21.270 ","End":"00:25.845","Text":"We\u0027re looking for this value right here."},{"Start":"00:25.845 ","End":"00:29.520","Text":"Now, how many letters do I have to choose from?"},{"Start":"00:29.520 ","End":"00:31.560","Text":"In the first letter of the word, well,"},{"Start":"00:31.560 ","End":"00:33.750","Text":"I have 26 letters to choose from."},{"Start":"00:33.750 ","End":"00:37.755","Text":"In the second letter of the word, again, 26 letters."},{"Start":"00:37.755 ","End":"00:41.955","Text":"26 times 26 times 26."},{"Start":"00:41.955 ","End":"00:44.989","Text":"Do I have a trial with 5 stages?"},{"Start":"00:44.989 ","End":"00:49.250","Text":"Each stage is choosing a letter in the word."},{"Start":"00:49.250 ","End":"00:51.410","Text":"The first letter, the second letter in the word,"},{"Start":"00:51.410 ","End":"00:52.490","Text":"the third letter in the word,"},{"Start":"00:52.490 ","End":"00:53.870","Text":"and so on and so forth."},{"Start":"00:53.870 ","End":"00:55.700","Text":"I have 5 letters,"},{"Start":"00:55.700 ","End":"00:57.080","Text":"and in each of the letters,"},{"Start":"00:57.080 ","End":"01:00.365","Text":"I have 26 possible letters to choose from."},{"Start":"01:00.365 ","End":"01:05.930","Text":"All in all, I have 26^5 possible combinations,"},{"Start":"01:05.930 ","End":"01:09.960","Text":"or 26^5 possible words."},{"Start":"01:09.960 ","End":"01:12.260","Text":"That\u0027s the size of my sample space."},{"Start":"01:12.260 ","End":"01:15.470","Text":"Now, in A, we\u0027re asked what\u0027s the probability that"},{"Start":"01:15.470 ","End":"01:19.535","Text":"in the 5 letter word there\u0027s no D, A, or L?"},{"Start":"01:19.535 ","End":"01:24.335","Text":"Well, when we\u0027re asked to calculate the probability in the denominator,"},{"Start":"01:24.335 ","End":"01:29.750","Text":"I\u0027ll have the size of the sample space, which is 26^5."},{"Start":"01:29.750 ","End":"01:32.640","Text":"Now, if I have no D,"},{"Start":"01:32.640 ","End":"01:33.980","Text":"A, or L on the word,"},{"Start":"01:33.980 ","End":"01:36.530","Text":"then that means that in the first letter of the word,"},{"Start":"01:36.530 ","End":"01:39.005","Text":"well, how many letters do I have to choose from?"},{"Start":"01:39.005 ","End":"01:41.390","Text":"I have 23 letters to choose from."},{"Start":"01:41.390 ","End":"01:44.669","Text":"26 minus 3 letters."},{"Start":"01:44.669 ","End":"01:50.825","Text":"I have 23 letters to choose from in the first letter of the word."},{"Start":"01:50.825 ","End":"01:52.745","Text":"In the second letter of the word,"},{"Start":"01:52.745 ","End":"01:55.160","Text":"I also have 23 letters to choose from,"},{"Start":"01:55.160 ","End":"01:59.510","Text":"and so on and so forth for all 5 letters in the word. I have 23^5."},{"Start":"01:59.510 ","End":"02:06.105","Text":"That\u0027s the number of combinations."},{"Start":"02:06.105 ","End":"02:10.520","Text":"The number of words that they can have without the letters D, A,"},{"Start":"02:10.520 ","End":"02:17.060","Text":"or L. The probability would be 23^5 divided by 26^5."},{"Start":"02:17.060 ","End":"02:20.360","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"02:20.360 ","End":"02:24.810","Text":"the probability that all the letters in the word are identical?"},{"Start":"02:26.030 ","End":"02:30.320","Text":"Again, when we\u0027re dealing with a probability in the denominator,"},{"Start":"02:30.320 ","End":"02:35.360","Text":"we have the size of the sample space which is 26^5."},{"Start":"02:35.360 ","End":"02:41.090","Text":"Now, how many combinations of this event do I have?"},{"Start":"02:41.090 ","End":"02:45.140","Text":"Well, in the first letter of the word,"},{"Start":"02:45.140 ","End":"02:47.270","Text":"how many letters do I have to choose from?"},{"Start":"02:47.270 ","End":"02:50.090","Text":"Well, I have 26 letters to choose from."},{"Start":"02:50.090 ","End":"02:53.185","Text":"Now, what about the second letter of the word?"},{"Start":"02:53.185 ","End":"02:55.760","Text":"Well, I only have 1 letter to choose from."},{"Start":"02:55.760 ","End":"02:58.925","Text":"That\u0027s the letter that\u0027s in the first position,"},{"Start":"02:58.925 ","End":"03:03.600","Text":"and the same for all other letters in the word."},{"Start":"03:03.600 ","End":"03:09.965","Text":"All in all, I have 26 divided by 26 to the power of 5,"},{"Start":"03:09.965 ","End":"03:15.955","Text":"and that equals to 1 divided by 26 to the power of 4."},{"Start":"03:15.955 ","End":"03:19.575","Text":"Now, an example of this event would be A,"},{"Start":"03:19.575 ","End":"03:21.735","Text":"A, A, A,"},{"Start":"03:21.735 ","End":"03:24.420","Text":"A, or B, B,"},{"Start":"03:24.420 ","End":"03:27.330","Text":"B, B, B, and so on and so forth."},{"Start":"03:27.330 ","End":"03:33.690","Text":"How many possible words do I have of this form?"},{"Start":"03:33.690 ","End":"03:37.530","Text":"Well, I have only 26 combinations."},{"Start":"03:37.530 ","End":"03:40.475","Text":"That\u0027s what\u0027s in the numerator."},{"Start":"03:40.475 ","End":"03:42.020","Text":"In this section we\u0027re asked,"},{"Start":"03:42.020 ","End":"03:46.970","Text":"what\u0027s the probability that at least 2 letters in the word are different from each other?"},{"Start":"03:46.970 ","End":"03:49.010","Text":"An example of this would be,"},{"Start":"03:49.010 ","End":"03:51.845","Text":"let\u0027s say A,"},{"Start":"03:51.845 ","End":"03:54.620","Text":"B, C, D, E."},{"Start":"03:54.620 ","End":"04:00.545","Text":"We have here a word that at least 2 letters are different from each other, or C,"},{"Start":"04:00.545 ","End":"04:02.280","Text":"E, C, C,"},{"Start":"04:02.280 ","End":"04:07.190","Text":"C. That\u0027s also a word where at least 2 letters are different from each other."},{"Start":"04:07.190 ","End":"04:10.940","Text":"Now, I can have many different combinations here,"},{"Start":"04:10.940 ","End":"04:12.860","Text":"and it\u0027s very difficult to count them up,"},{"Start":"04:12.860 ","End":"04:16.550","Text":"but let\u0027s take a look at the complementary set right here."},{"Start":"04:16.550 ","End":"04:21.140","Text":"What\u0027s the complementary set of at least 2 letters are different?"},{"Start":"04:21.140 ","End":"04:24.920","Text":"Well, that would be that all the letters are the same."},{"Start":"04:24.920 ","End":"04:28.340","Text":"The probability here would be equal to 1"},{"Start":"04:28.340 ","End":"04:33.500","Text":"minus the probability that all the letters are the same."},{"Start":"04:33.500 ","End":"04:38.780","Text":"Now, we\u0027ve calculated this probability in the last section that equals to"},{"Start":"04:38.780 ","End":"04:46.230","Text":"1 minus 1 divided by 26^4."},{"Start":"04:46.310 ","End":"04:52.090","Text":"That\u0027s the probability that at least 2 letters are different."},{"Start":"04:52.090 ","End":"04:54.790","Text":"In this section we\u0027re asked,"},{"Start":"04:54.790 ","End":"04:57.280","Text":"what\u0027s the probability that the word is a palindrome?"},{"Start":"04:57.280 ","End":"04:59.410","Text":"That means that the word is the same,"},{"Start":"04:59.410 ","End":"05:02.470","Text":"whether read from right to left or left to right."},{"Start":"05:02.470 ","End":"05:05.990","Text":"An example of that would be, let\u0027s say,"},{"Start":"05:05.990 ","End":"05:10.500","Text":"A, B, B, B, A."},{"Start":"05:10.500 ","End":"05:13.939","Text":"I can read that from right to left,"},{"Start":"05:13.939 ","End":"05:16.315","Text":"or from left to right and would be the same."},{"Start":"05:16.315 ","End":"05:18.600","Text":"Or let\u0027s say C,"},{"Start":"05:18.600 ","End":"05:21.330","Text":"E, L, E,"},{"Start":"05:21.330 ","End":"05:23.220","Text":"C. C, E,"},{"Start":"05:23.220 ","End":"05:24.285","Text":"L, E, C,"},{"Start":"05:24.285 ","End":"05:26.100","Text":"that\u0027s from left to right, and C,"},{"Start":"05:26.100 ","End":"05:28.380","Text":"E, L, E, C from right to left."},{"Start":"05:28.380 ","End":"05:30.140","Text":"These are the examples here."},{"Start":"05:30.140 ","End":"05:32.750","Text":"Now, what\u0027s the probability?"},{"Start":"05:32.750 ","End":"05:34.415","Text":"Well, in the denominator,"},{"Start":"05:34.415 ","End":"05:40.010","Text":"I have the size of my sample space which is 26^5."},{"Start":"05:40.010 ","End":"05:48.590","Text":"Now, let\u0027s take a look at the number of combinations for a word that\u0027s a palindrome."},{"Start":"05:48.590 ","End":"05:51.060","Text":"On the first letter of the word,"},{"Start":"05:51.060 ","End":"05:53.970","Text":"well, I have 26 letters to choose from."},{"Start":"05:53.970 ","End":"05:59.085","Text":"In the second letter of the word,"},{"Start":"05:59.085 ","End":"06:02.384","Text":"I also have 26 letters to choose from,"},{"Start":"06:02.384 ","End":"06:06.290","Text":"and the same for the third letter in the word."},{"Start":"06:06.290 ","End":"06:13.040","Text":"Now, the fourth letter in the word has to be the same as the second letter in the word."},{"Start":"06:13.040 ","End":"06:15.350","Text":"I don\u0027t have that much to choose from."},{"Start":"06:15.350 ","End":"06:18.815","Text":"I only have 1 letter to choose from out of the whole alphabet,"},{"Start":"06:18.815 ","End":"06:24.620","Text":"and the same for the fifth letter that has to be equal to the first letter."},{"Start":"06:24.620 ","End":"06:29.660","Text":"Again, here, I have 26 letters to choose from."},{"Start":"06:29.660 ","End":"06:32.300","Text":"Here, I have 26 letters to choose from,"},{"Start":"06:32.300 ","End":"06:34.550","Text":"and here I have 26 letters to choose from."},{"Start":"06:34.550 ","End":"06:36.920","Text":"Here, I don\u0027t have any choice."},{"Start":"06:36.920 ","End":"06:39.580","Text":"This has to be equal to this letter,"},{"Start":"06:39.580 ","End":"06:42.740","Text":"and here this has to be equal to this letter."},{"Start":"06:42.740 ","End":"06:49.940","Text":"This would be the number of possible combinations for a palindrome."},{"Start":"06:49.940 ","End":"06:57.040","Text":"That means I have 26^3 divided by 26^5."},{"Start":"06:57.040 ","End":"07:00.020","Text":"That\u0027s the probability of getting a palindrome."},{"Start":"07:00.020 ","End":"07:05.910","Text":"Now, that equals to 1 divided by 26 squared."}],"ID":12787},{"Watched":false,"Name":"Exrcise 8 - Parts a-c","Duration":"4m 39s","ChapterTopicVideoID":12309,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12309.jpeg","UploadDate":"2018-07-22T16:36:32.5900000","DurationForVideoObject":"PT4M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this question, we\u0027re told that a code with a digits is created."},{"Start":"00:03.570 ","End":"00:06.840","Text":"Now, each digit can be used more than once in the code."},{"Start":"00:06.840 ","End":"00:11.640","Text":"We\u0027re asked to calculate probabilities about the code here."},{"Start":"00:11.640 ","End":"00:14.790","Text":"Now, when we\u0027re asked to calculate probabilities,"},{"Start":"00:14.790 ","End":"00:19.005","Text":"the first thing that I want to do is to calculate the size of my sample space."},{"Start":"00:19.005 ","End":"00:23.985","Text":"Now, we have here a trial with a steps. Why is that?"},{"Start":"00:23.985 ","End":"00:27.660","Text":"Well, we have a code with a numbers, this is the first number,"},{"Start":"00:27.660 ","End":"00:29.070","Text":"this is the second number,"},{"Start":"00:29.070 ","End":"00:31.410","Text":"and this is the last number."},{"Start":"00:31.410 ","End":"00:34.305","Text":"We have a numbers here."},{"Start":"00:34.305 ","End":"00:40.445","Text":"The first step is choosing a number from 0-9 in the first place in the code."},{"Start":"00:40.445 ","End":"00:44.870","Text":"The second step is choosing a number from 0-9 in the second place,"},{"Start":"00:44.870 ","End":"00:46.630","Text":"and so on and so forth."},{"Start":"00:46.630 ","End":"00:51.590","Text":"All in all, the size of my sample space would be"},{"Start":"00:51.590 ","End":"00:54.050","Text":"10 numbers to choose from in"},{"Start":"00:54.050 ","End":"00:58.115","Text":"the first place times 10 numbers to choose in the second place,"},{"Start":"00:58.115 ","End":"01:00.290","Text":"and so on and so forth a times."},{"Start":"01:00.290 ","End":"01:04.000","Text":"Now, that equals to 10 to the power of a."},{"Start":"01:04.000 ","End":"01:08.150","Text":"Now, the first probability that we\u0027re asked to calculate is that"},{"Start":"01:08.150 ","End":"01:13.360","Text":"the code that does not contain the digit 5."},{"Start":"01:13.360 ","End":"01:20.270","Text":"Well, so we know that in the denominator we have the size of my sample space,"},{"Start":"01:20.270 ","End":"01:22.475","Text":"that\u0027ll be 10 to the power of a."},{"Start":"01:22.475 ","End":"01:27.170","Text":"Now, if I don\u0027t have the number 5 in all of the code,"},{"Start":"01:27.170 ","End":"01:31.010","Text":"then basically, I have only 9 numbers to"},{"Start":"01:31.010 ","End":"01:35.620","Text":"choose from and not 10 numbers to choose from in the first place."},{"Start":"01:35.620 ","End":"01:38.900","Text":"How many numbers do I have to choose from in the second place?"},{"Start":"01:38.900 ","End":"01:40.865","Text":"Well, again, they\u0027ll be 9 numbers."},{"Start":"01:40.865 ","End":"01:43.655","Text":"The numbers from 0-9,"},{"Start":"01:43.655 ","End":"01:46.375","Text":"not including the number 5,"},{"Start":"01:46.375 ","End":"01:48.335","Text":"and so on and so forth,"},{"Start":"01:48.335 ","End":"01:50.870","Text":"until the last place in the code."},{"Start":"01:50.870 ","End":"01:57.320","Text":"Now that equals to 9 to the power of a divided by 10 to the power of a."},{"Start":"01:57.320 ","End":"02:00.830","Text":"Now that equals to 0.9 to the power of a."},{"Start":"02:00.830 ","End":"02:06.710","Text":"Now, that\u0027s the probability that the code does not contain the digit 5."},{"Start":"02:06.710 ","End":"02:12.260","Text":"In this section, we\u0027re asked for the probability that the code contains the digit 3."},{"Start":"02:12.260 ","End":"02:14.345","Text":"Well, when we say that,"},{"Start":"02:14.345 ","End":"02:21.400","Text":"we mean that the number 3 can appear once or twice or more in the code."},{"Start":"02:21.400 ","End":"02:25.940","Text":"In essence, what we\u0027re saying is that we\u0027re looking for"},{"Start":"02:25.940 ","End":"02:31.805","Text":"the probability that the number 3 appears at least once in the code."},{"Start":"02:31.805 ","End":"02:35.600","Text":"Well, there are a lot of combinations and a lot of"},{"Start":"02:35.600 ","End":"02:40.035","Text":"codes where the number 3 appears at least once."},{"Start":"02:40.035 ","End":"02:45.650","Text":"What we need to do is we need to look at the complimentary set to make things easier."},{"Start":"02:45.650 ","End":"02:50.930","Text":"Now, what\u0027s the complimentary set of there\u0027s a number 3 in the code."},{"Start":"02:50.930 ","End":"02:56.180","Text":"Well, that means the complimentary set of that is that there\u0027s no number 3 in the code."},{"Start":"02:56.180 ","End":"02:58.385","Text":"When we\u0027re dealing with probabilities,"},{"Start":"02:58.385 ","End":"03:01.250","Text":"that means that we\u0027re looking at 1 minus"},{"Start":"03:01.250 ","End":"03:07.630","Text":"the probability that there\u0027s no number 3 in code."},{"Start":"03:07.630 ","End":"03:11.090","Text":"Now, what\u0027s this probability right here?"},{"Start":"03:11.090 ","End":"03:13.700","Text":"Well, this probability equals to"},{"Start":"03:13.700 ","End":"03:18.545","Text":"the probability that there\u0027s no number 5 in the code or 6 in the code."},{"Start":"03:18.545 ","End":"03:20.560","Text":"We\u0027ve calculated that."},{"Start":"03:20.560 ","End":"03:23.145","Text":"That equals to 1 minus,"},{"Start":"03:23.145 ","End":"03:24.810","Text":"now in the denominator,"},{"Start":"03:24.810 ","End":"03:26.865","Text":"that\u0027ll be 10 to the power of a."},{"Start":"03:26.865 ","End":"03:30.350","Text":"If there\u0027s no number 3 in the code, well, again,"},{"Start":"03:30.350 ","End":"03:33.140","Text":"we have 9 digits to choose from in"},{"Start":"03:33.140 ","End":"03:36.920","Text":"the first place and 9 digits to choose from in the second place."},{"Start":"03:36.920 ","End":"03:41.490","Text":"That\u0027s digits from 0 to 9 without the number 3."},{"Start":"03:41.490 ","End":"03:46.170","Text":"That would be 9 times 9 and so on and so forth times 9, 8 times."},{"Start":"03:46.170 ","End":"03:52.375","Text":"That means that we\u0027re looking at 1 minus 0.9 to the power of a."},{"Start":"03:52.375 ","End":"03:59.400","Text":"In this section, we\u0027re looking for the probability that the code contains no odd digits."},{"Start":"03:59.560 ","End":"04:03.560","Text":"Again, in the denominator we have the size of"},{"Start":"04:03.560 ","End":"04:07.005","Text":"the sample space that\u0027ll be 10 to the power of a."},{"Start":"04:07.005 ","End":"04:10.190","Text":"Now, when we say that there\u0027s no odd numbers, well,"},{"Start":"04:10.190 ","End":"04:13.970","Text":"that means that we have 5 digits to choose from: 0,"},{"Start":"04:13.970 ","End":"04:17.175","Text":"2, 4, 6, and 8."},{"Start":"04:17.175 ","End":"04:18.930","Text":"In the first place,"},{"Start":"04:18.930 ","End":"04:21.800","Text":"we have 5 digits to choose from,"},{"Start":"04:21.800 ","End":"04:24.830","Text":"and in the second we have 5 digits to choose from,"},{"Start":"04:24.830 ","End":"04:26.625","Text":"and so on and so forth,"},{"Start":"04:26.625 ","End":"04:28.910","Text":"until the last place in the code."},{"Start":"04:28.910 ","End":"04:34.210","Text":"That means that we\u0027re looking at 5 to the power of a divided by 10 to the power of a,"},{"Start":"04:34.210 ","End":"04:39.150","Text":"and that equals to 0.5 to the power of a."}],"ID":12788},{"Watched":false,"Name":"Exrcise 9","Duration":"1m 9s","ChapterTopicVideoID":12310,"CourseChapterTopicPlaylistID":264986,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12310.jpeg","UploadDate":"2018-07-22T16:36:35.1630000","DurationForVideoObject":"PT1M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In a game of chance, a form with n cells is filled out."},{"Start":"00:02.940 ","End":"00:06.460","Text":"Each cell is marked with either a check mark or an x."},{"Start":"00:06.460 ","End":"00:10.920","Text":"We\u0027re asked how many different ways can the form for the game be filled out?"},{"Start":"00:10.920 ","End":"00:18.015","Text":"Here, we have a trial with n stages where each stage is filling out 1 cell in the form."},{"Start":"00:18.015 ","End":"00:25.080","Text":"We have here a form with the various cells and we have n cells here."},{"Start":"00:25.080 ","End":"00:28.910","Text":"We can either fill out the cells with a check mark or an x,"},{"Start":"00:28.910 ","End":"00:31.250","Text":"a check mark or an x, and so on and so forth."},{"Start":"00:31.250 ","End":"00:37.610","Text":"How many possible values or how many possible symbols can we fill out for each cell?"},{"Start":"00:37.610 ","End":"00:41.180","Text":"We have 2 possible values or symbols."},{"Start":"00:41.180 ","End":"00:43.025","Text":"It can either be a check mark or an x."},{"Start":"00:43.025 ","End":"00:45.445","Text":"We have 2 possibilities here,"},{"Start":"00:45.445 ","End":"00:48.755","Text":"2 possibilities here, and so on and so forth."},{"Start":"00:48.755 ","End":"00:55.930","Text":"We have 2 times 2 times 2 times 2, n times."},{"Start":"00:55.930 ","End":"01:00.645","Text":"That means that we\u0027re looking at 2^n."},{"Start":"01:00.645 ","End":"01:04.550","Text":"This would be the number of different ways that we"},{"Start":"01:04.550 ","End":"01:09.600","Text":"can fill out the form with either a check mark or an x."}],"ID":12789}],"Thumbnail":null,"ID":264986},{"Name":"Permutations - Objects Ordered in a Row","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"50s","ChapterTopicVideoID":12311,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12311.jpeg","UploadDate":"2018-07-22T16:42:53.9370000","DurationForVideoObject":"PT50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this chapter, I\u0027ll be talking about combinatorics, specifically permutations."},{"Start":"00:04.410 ","End":"00:08.880","Text":"Permutations is just ordering objects in a row."},{"Start":"00:08.880 ","End":"00:15.985","Text":"Let\u0027s take a look at the mathematical definition of a permutation."},{"Start":"00:15.985 ","End":"00:21.826","Text":"Well, that\u0027s the number of possibilities for ordering n different objects in a row."},{"Start":"00:21.826 ","End":"00:25.340","Text":"Now that\u0027s written as n with an exclamation mark,"},{"Start":"00:25.340 ","End":"00:27.185","Text":"and that\u0027s called factorials."},{"Start":"00:27.185 ","End":"00:29.420","Text":"We call this n factorial."},{"Start":"00:29.420 ","End":"00:35.000","Text":"Now that\u0027s defined as being equal to 1 times 2 times 3 and so on and"},{"Start":"00:35.000 ","End":"00:41.000","Text":"so forth times n. We also have to note that 0 factorial,"},{"Start":"00:41.000 ","End":"00:43.825","Text":"that\u0027s defined as 1."},{"Start":"00:43.825 ","End":"00:45.405","Text":"Enough theory."},{"Start":"00:45.405 ","End":"00:47.370","Text":"This is the theory of permutations,"},{"Start":"00:47.370 ","End":"00:50.460","Text":"now let\u0027s just take a look at an example."}],"ID":12790},{"Watched":false,"Name":"Tutorial 2","Duration":"1m 29s","ChapterTopicVideoID":12312,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12312.jpeg","UploadDate":"2018-07-22T16:42:57.9270000","DurationForVideoObject":"PT1M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this example, we\u0027re asked in how many different ways can the letters a,"},{"Start":"00:04.560 ","End":"00:06.780","Text":"b, c, and d be ordered?"},{"Start":"00:06.780 ","End":"00:09.870","Text":"Let\u0027s take a look what we have here."},{"Start":"00:09.870 ","End":"00:11.705","Text":"We have 4 letters."},{"Start":"00:11.705 ","End":"00:15.150","Text":"We have 4 positions."},{"Start":"00:15.150 ","End":"00:18.635","Text":"Because we want to order the 4 letters."},{"Start":"00:18.635 ","End":"00:22.985","Text":"If we would have used the multiplication principle,"},{"Start":"00:22.985 ","End":"00:25.895","Text":"then we would go like this,"},{"Start":"00:25.895 ","End":"00:27.590","Text":"in the first position,"},{"Start":"00:27.590 ","End":"00:29.660","Text":"how many letters can I have here?"},{"Start":"00:29.660 ","End":"00:32.072","Text":"I can have 4 letters."},{"Start":"00:32.072 ","End":"00:33.620","Text":"In the second position,"},{"Start":"00:33.620 ","End":"00:35.990","Text":"I\u0027d have 3 letters to choose from."},{"Start":"00:35.990 ","End":"00:37.220","Text":"In the third place,"},{"Start":"00:37.220 ","End":"00:39.480","Text":"I\u0027d have 2 letters to choose from,"},{"Start":"00:39.480 ","End":"00:41.795","Text":"and in the last place,"},{"Start":"00:41.795 ","End":"00:44.450","Text":"I\u0027d have only 1 letter to choose from."},{"Start":"00:44.450 ","End":"00:50.375","Text":"How many possible words can I make out of 4 letters by ordering them?"},{"Start":"00:50.375 ","End":"00:55.970","Text":"Then I\u0027d have 4 times 3 times, 2 times 1."},{"Start":"00:55.970 ","End":"01:01.400","Text":"That equals to 12, 24 letters."},{"Start":"01:01.400 ","End":"01:06.750","Text":"If I use the permutation equation,"},{"Start":"01:06.750 ","End":"01:09.986","Text":"or how do I calculate permutations,"},{"Start":"01:09.986 ","End":"01:13.340","Text":"all I have to say is that I have 4 letters,"},{"Start":"01:13.340 ","End":"01:16.940","Text":"how many ways can I order them in a row?"},{"Start":"01:16.940 ","End":"01:19.280","Text":"That\u0027s 4 factorial."},{"Start":"01:19.280 ","End":"01:24.431","Text":"We know that that\u0027s defined as 1 times 2 times 3 times 4,"},{"Start":"01:24.431 ","End":"01:29.620","Text":"and again, that equals to 24."}],"ID":12791},{"Watched":false,"Name":"Tutorial 3","Duration":"2m 17s","ChapterTopicVideoID":12313,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12313.jpeg","UploadDate":"2018-07-22T16:43:02.8170000","DurationForVideoObject":"PT2M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In this question we\u0027re asked,"},{"Start":"00:01.260 ","End":"00:03.600","Text":"in how many different ways can the letters a, b,"},{"Start":"00:03.600 ","End":"00:05.115","Text":"c, and d be ordered,"},{"Start":"00:05.115 ","End":"00:07.875","Text":"so that a and b are adjacent to each other."},{"Start":"00:07.875 ","End":"00:12.285","Text":"Well, let\u0027s first of all take a look at an example of what we mean."},{"Start":"00:12.285 ","End":"00:15.480","Text":"Let\u0027s say we have an example of c,"},{"Start":"00:15.480 ","End":"00:18.060","Text":"a, b, and d. Well,"},{"Start":"00:18.060 ","End":"00:23.370","Text":"we see here that a and b are adjacent to each other or we can have the example of d,"},{"Start":"00:23.370 ","End":"00:25.965","Text":"c, b, and a and again,"},{"Start":"00:25.965 ","End":"00:28.395","Text":"a and b are adjacent to each other."},{"Start":"00:28.395 ","End":"00:30.435","Text":"I didn\u0027t specify the order."},{"Start":"00:30.435 ","End":"00:32.945","Text":"I didn\u0027t say that a has to be before b."},{"Start":"00:32.945 ","End":"00:36.545","Text":"All I said was that a and b are adjacent to each other."},{"Start":"00:36.545 ","End":"00:37.910","Text":"Now, if that\u0027s the case,"},{"Start":"00:37.910 ","End":"00:40.190","Text":"how do we approach this type of question?"},{"Start":"00:40.190 ","End":"00:44.014","Text":"Well, we do so by thinking of groups."},{"Start":"00:44.014 ","End":"00:49.523","Text":"We have here c as a group with 1 letter,"},{"Start":"00:49.523 ","End":"00:52.230","Text":"we have d as a group with 1 letter,"},{"Start":"00:52.230 ","End":"00:56.025","Text":"and we have ab as a group with 2 letters."},{"Start":"00:56.025 ","End":"01:04.275","Text":"Now, how many ways can we order these 3 groups?"},{"Start":"01:04.275 ","End":"01:06.975","Text":"Well, 3 factorial."},{"Start":"01:06.975 ","End":"01:10.710","Text":"I can first have c and then d and ab,"},{"Start":"01:10.710 ","End":"01:13.350","Text":"or c, ab and d,"},{"Start":"01:13.350 ","End":"01:16.620","Text":"or d, c and ab, d,"},{"Start":"01:16.620 ","End":"01:18.300","Text":"a b and c and ab,"},{"Start":"01:18.300 ","End":"01:21.310","Text":"c and ab and d. Now,"},{"Start":"01:21.310 ","End":"01:28.320","Text":"this isn\u0027t all because it didn\u0027t specify what the order is of a and"},{"Start":"01:28.320 ","End":"01:31.550","Text":"b in this group then I have to calculate"},{"Start":"01:31.550 ","End":"01:35.570","Text":"the permutations of ordering the letters within each group."},{"Start":"01:35.570 ","End":"01:40.715","Text":"Within this group where the permutations is 1. There\u0027s only 1 letter."},{"Start":"01:40.715 ","End":"01:44.820","Text":"Here the permutation is 1 because again there\u0027s only 1 letter,"},{"Start":"01:44.820 ","End":"01:47.010","Text":"but here there are 2 letters."},{"Start":"01:47.010 ","End":"01:49.550","Text":"How many permutations can I have here?"},{"Start":"01:49.550 ","End":"01:52.945","Text":"That\u0027ll be 2 factorial."},{"Start":"01:52.945 ","End":"01:56.060","Text":"In essence, what do I have here?"},{"Start":"01:56.060 ","End":"01:58.685","Text":"I have 1 times 2 times 3."},{"Start":"01:58.685 ","End":"02:03.455","Text":"That\u0027s 3 factorial times 1 times 2, that\u0027s 2 factorial."},{"Start":"02:03.455 ","End":"02:07.700","Text":"This would be the number of permutations of ordering"},{"Start":"02:07.700 ","End":"02:13.635","Text":"the groups and this would be the number of permutations within each group."},{"Start":"02:13.635 ","End":"02:17.830","Text":"That equals to 12."}],"ID":12792},{"Watched":false,"Name":"Tutorial 4","Duration":"1m 58s","ChapterTopicVideoID":12314,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12314.jpeg","UploadDate":"2018-07-22T16:43:10.5000000","DurationForVideoObject":"PT1M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"In this example, we\u0027re asked in how many ways can we order the letters a, b, c,"},{"Start":"00:04.920 ","End":"00:10.035","Text":"and d so that the letters a and b appear in the order ba?"},{"Start":"00:10.035 ","End":"00:12.720","Text":"That means that b has to come before a."},{"Start":"00:12.720 ","End":"00:19.500","Text":"For example, cbad or dcba."},{"Start":"00:19.500 ","End":"00:22.920","Text":"See here that we have an order."},{"Start":"00:22.920 ","End":"00:24.810","Text":"Again, let\u0027s think about groups."},{"Start":"00:24.810 ","End":"00:27.390","Text":"We have c as a group,"},{"Start":"00:27.390 ","End":"00:29.490","Text":"we have d as a group,"},{"Start":"00:29.490 ","End":"00:31.980","Text":"and we have ba as a group."},{"Start":"00:31.980 ","End":"00:34.935","Text":"Now here, the order does matter."},{"Start":"00:34.935 ","End":"00:38.120","Text":"Again, we have 3 factorial,"},{"Start":"00:38.120 ","End":"00:41.720","Text":"that\u0027s the number of permutations of ordering the groups."},{"Start":"00:41.720 ","End":"00:49.190","Text":"But now, we don\u0027t have to calculate the internal permutations of the groups here."},{"Start":"00:49.190 ","End":"00:50.960","Text":"Why? Because this is fixed."},{"Start":"00:50.960 ","End":"00:53.525","Text":"This is like saying ba."},{"Start":"00:53.525 ","End":"00:57.605","Text":"Thinking of ba as the letter t for example."},{"Start":"00:57.605 ","End":"01:04.080","Text":"We only have 3 letters instead of 4."},{"Start":"01:04.080 ","End":"01:08.270","Text":"The number of permutations here would be 6."},{"Start":"01:08.270 ","End":"01:11.120","Text":"That equals to 1 times 2 times 3."},{"Start":"01:11.120 ","End":"01:14.990","Text":"Again, let\u0027s just make sure that this is correct."},{"Start":"01:14.990 ","End":"01:23.430","Text":"For example, we can have ba c d, ba d c,"},{"Start":"01:23.950 ","End":"01:30.690","Text":"we can have c ba d,"},{"Start":"01:30.690 ","End":"01:33.350","Text":"c d ba,"},{"Start":"01:33.350 ","End":"01:37.030","Text":"and we can have d c ba,"},{"Start":"01:37.030 ","End":"01:41.010","Text":"and we can have d ba,"},{"Start":"01:41.010 ","End":"01:42.720","Text":"and c. We see here 1,"},{"Start":"01:42.720 ","End":"01:44.955","Text":"2, 3, 4, 5, and 6."},{"Start":"01:44.955 ","End":"01:51.765","Text":"These are the only permutations that I can make where I have 4 letters,"},{"Start":"01:51.765 ","End":"01:58.290","Text":"where 2 of the letters have to be grouped in this specific order."}],"ID":12793},{"Watched":false,"Name":"Exrcise 1 - Parts a-b","Duration":"1m 28s","ChapterTopicVideoID":12315,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12315.jpeg","UploadDate":"2018-07-22T16:43:13.4370000","DurationForVideoObject":"PT1M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"In this question we\u0027re asked in how many ways can 4 books be ordered on a shelf?"},{"Start":"00:04.995 ","End":"00:10.560","Text":"Well, whenever we think or whenever we\u0027re asked about ordering things,"},{"Start":"00:10.560 ","End":"00:13.485","Text":"then we\u0027re talking about factorials."},{"Start":"00:13.485 ","End":"00:17.775","Text":"When we have n objects that we need to order in a line,"},{"Start":"00:17.775 ","End":"00:22.845","Text":"then we calculate the number of permutations by n factorial."},{"Start":"00:22.845 ","End":"00:28.200","Text":"Now here we\u0027re asked about 4 books that should be ordered on a shelf,"},{"Start":"00:28.200 ","End":"00:30.420","Text":"so we have 4 factorial."},{"Start":"00:30.420 ","End":"00:32.535","Text":"That\u0027s the number of permutations."},{"Start":"00:32.535 ","End":"00:38.225","Text":"That\u0027s the number of ways that we can order 4 books on a shelf."},{"Start":"00:38.225 ","End":"00:42.740","Text":"Now, that equals to 1 times 2 times 3 times 4."},{"Start":"00:42.740 ","End":"00:45.785","Text":"That\u0027s the definition of a factorial,"},{"Start":"00:45.785 ","End":"00:48.620","Text":"and that comes out to 24,"},{"Start":"00:48.620 ","End":"00:52.925","Text":"so there are 24 ways of ordering 4 books on a shelf."},{"Start":"00:52.925 ","End":"00:57.980","Text":"In this section we\u0027re asked in how many ways can 5 soldiers stand in a line?"},{"Start":"00:57.980 ","End":"01:01.100","Text":"Again, we have 5 objects, in our case,"},{"Start":"01:01.100 ","End":"01:04.600","Text":"soldiers that we have to order in a line."},{"Start":"01:04.600 ","End":"01:07.080","Text":"We have 5 factorial,"},{"Start":"01:07.080 ","End":"01:08.730","Text":"that\u0027s the number of permutations,"},{"Start":"01:08.730 ","End":"01:12.770","Text":"that\u0027s the number of ways that we can order 5 soldiers in"},{"Start":"01:12.770 ","End":"01:18.095","Text":"a line and that equals to 1 times 2 times 3 times 4 times 5,"},{"Start":"01:18.095 ","End":"01:22.370","Text":"and that comes out to 120."},{"Start":"01:22.370 ","End":"01:28.740","Text":"This is the number of ways that we can order 5 soldiers in a line."}],"ID":12794},{"Watched":false,"Name":"Exrcise 2 - Parts a-c","Duration":"4m 43s","ChapterTopicVideoID":12316,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12316.jpeg","UploadDate":"2018-07-22T16:43:24.0830000","DurationForVideoObject":"PT4M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.195","Text":"In this question, we have 10 disks that had been ordered randomly on a shelf,"},{"Start":"00:04.195 ","End":"00:05.855","Text":"2 of which are in French."},{"Start":"00:05.855 ","End":"00:10.710","Text":"We\u0027re asked, what\u0027s the probability that the French disks are next to each other?"},{"Start":"00:10.710 ","End":"00:14.550","Text":"Well, the first thing that I want to do when I\u0027m asked about"},{"Start":"00:14.550 ","End":"00:18.420","Text":"probability is to calculate the size of my sample space."},{"Start":"00:18.420 ","End":"00:21.310","Text":"That\u0027ll be the size of Omega right here."},{"Start":"00:21.310 ","End":"00:24.150","Text":"Now what\u0027s my sample space?"},{"Start":"00:24.150 ","End":"00:29.340","Text":"That\u0027s the total number of permutations or combinations"},{"Start":"00:29.340 ","End":"00:34.980","Text":"or ways that I can order 10 disks in a row and that\u0027s 10 factorial."},{"Start":"00:34.980 ","End":"00:38.960","Text":"We learned that. Now, let\u0027s take a look at this."},{"Start":"00:38.960 ","End":"00:40.699","Text":"We have 10 disks."},{"Start":"00:40.699 ","End":"00:44.195","Text":"This number 1, 2, 3, 4,"},{"Start":"00:44.195 ","End":"00:48.905","Text":"5, 6, 7, 8, 9, and 10."},{"Start":"00:48.905 ","End":"00:50.735","Text":"Now, 21 of them are French."},{"Start":"00:50.735 ","End":"00:54.245","Text":"Let\u0027s say this is French and this is French."},{"Start":"00:54.245 ","End":"01:01.230","Text":"We\u0027re interested in the probability that the French disks are next to each other."},{"Start":"01:01.460 ","End":"01:05.105","Text":"This is the probability that we\u0027re looking for."},{"Start":"01:05.105 ","End":"01:09.080","Text":"Now when we\u0027re looking for the probability of something,"},{"Start":"01:09.080 ","End":"01:11.345","Text":"well in the denominator,"},{"Start":"01:11.345 ","End":"01:13.940","Text":"I have the size of my sample space,"},{"Start":"01:13.940 ","End":"01:15.380","Text":"that\u0027ll be 10 factorial."},{"Start":"01:15.380 ","End":"01:22.820","Text":"Now what\u0027s the permutations of the 2 French disk being together?"},{"Start":"01:22.820 ","End":"01:25.550","Text":"Well, let\u0027s take a look at the 10 disks."},{"Start":"01:25.550 ","End":"01:28.490","Text":"Now if we wanted 2 French disks to be together,"},{"Start":"01:28.490 ","End":"01:31.825","Text":"then let\u0027s think of them as 1 group."},{"Start":"01:31.825 ","End":"01:33.890","Text":"How many groups do I have here?"},{"Start":"01:33.890 ","End":"01:35.300","Text":"Well, I have 1,"},{"Start":"01:35.300 ","End":"01:36.770","Text":"2, 3, 4, 5,"},{"Start":"01:36.770 ","End":"01:37.880","Text":"6, 7, 8, 9,"},{"Start":"01:37.880 ","End":"01:42.720","Text":"so I have 9 groups that I have to order in a line."},{"Start":"01:42.720 ","End":"01:44.930","Text":"That\u0027ll be 9 factorial."},{"Start":"01:44.930 ","End":"01:49.910","Text":"But I have to take into account the internal permutation of this group."},{"Start":"01:49.910 ","End":"01:53.150","Text":"Now how many permutations do I have here?"},{"Start":"01:53.150 ","End":"01:56.090","Text":"Well, I can order these disks as disk number 1,"},{"Start":"01:56.090 ","End":"02:01.080","Text":"disk number 2, or disk number 2 and disk number 1."},{"Start":"02:01.080 ","End":"02:04.155","Text":"I have 2 factorial."},{"Start":"02:04.155 ","End":"02:09.410","Text":"This then would be the number of permutations of"},{"Start":"02:09.410 ","End":"02:14.915","Text":"having 10 disks ordered in a row where the 2 French disks are together."},{"Start":"02:14.915 ","End":"02:18.305","Text":"Now, that comes out to 0.2."},{"Start":"02:18.305 ","End":"02:20.900","Text":"Now, how did I calculate this quickly?"},{"Start":"02:20.900 ","End":"02:22.670","Text":"Well, that\u0027s very easy."},{"Start":"02:22.670 ","End":"02:24.290","Text":"What\u0027s 9 factorial?"},{"Start":"02:24.290 ","End":"02:26.840","Text":"That\u0027s 1 times 2 times 3,"},{"Start":"02:26.840 ","End":"02:31.200","Text":"and so on and so forth until 9, times 2 factorial,"},{"Start":"02:31.200 ","End":"02:35.240","Text":"1 times 2 divided by 10 factorial, well,"},{"Start":"02:35.240 ","End":"02:38.300","Text":"10 factorial is 1 times 2 times 3,"},{"Start":"02:38.300 ","End":"02:39.575","Text":"and so on and so forth,"},{"Start":"02:39.575 ","End":"02:41.735","Text":"times 9 times 10."},{"Start":"02:41.735 ","End":"02:43.790","Text":"Now, let\u0027s take a look at this."},{"Start":"02:43.790 ","End":"02:46.460","Text":"This and this is the same,"},{"Start":"02:46.460 ","End":"02:48.260","Text":"so they cancel each other out."},{"Start":"02:48.260 ","End":"02:54.740","Text":"I\u0027m left with 2 divided by 10 and that equals to 0.2."},{"Start":"02:54.740 ","End":"02:56.765","Text":"In this section, we\u0027re asked,"},{"Start":"02:56.765 ","End":"03:01.730","Text":"what\u0027s the probability that the French disks are not next to each other?"},{"Start":"03:01.730 ","End":"03:06.050","Text":"Well, if this is the probability that we\u0027re interested in."},{"Start":"03:06.050 ","End":"03:11.345","Text":"Well, what\u0027s the complimentary set of the 2 French discs not together?"},{"Start":"03:11.345 ","End":"03:15.480","Text":"Well, last thing 2 French discs being together."},{"Start":"03:15.480 ","End":"03:23.315","Text":"The probability of this event is 1 minus the probability of the 2 disks,"},{"Start":"03:23.315 ","End":"03:31.169","Text":"2 French disks together."},{"Start":"03:31.840 ","End":"03:36.230","Text":"Now we\u0027ve calculated that in the last section,"},{"Start":"03:36.230 ","End":"03:38.330","Text":"that equals to 0.2."},{"Start":"03:38.330 ","End":"03:45.830","Text":"This whole probability right here is 1 minus 0.2 and that equals to 0.8."},{"Start":"03:45.830 ","End":"03:48.800","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"03:48.800 ","End":"03:52.600","Text":"the 2 French disks are an opposite ends of the shelf?"},{"Start":"03:52.600 ","End":"03:55.459","Text":"Here are 10 disks,"},{"Start":"03:55.459 ","End":"03:58.025","Text":"and we want this to be a French disk,"},{"Start":"03:58.025 ","End":"03:59.900","Text":"and this to be a French disk."},{"Start":"03:59.900 ","End":"04:03.320","Text":"Now we\u0027re interested in this probability right here."},{"Start":"04:03.320 ","End":"04:07.610","Text":"Now, whenever we have a probability in the denominator,"},{"Start":"04:07.610 ","End":"04:12.620","Text":"we have the size of our sample space and that equals to 10 factorial."},{"Start":"04:12.620 ","End":"04:20.810","Text":"Now, what about the number of permutations of having 2 French disks at both ends."},{"Start":"04:20.810 ","End":"04:24.575","Text":"Well, let\u0027s take a look at these 8 disks in the middle."},{"Start":"04:24.575 ","End":"04:28.500","Text":"Well, how many ways can we order these 8 disks in a row?"},{"Start":"04:28.500 ","End":"04:30.390","Text":"That\u0027s 8 factorial?"},{"Start":"04:30.390 ","End":"04:35.460","Text":"How many ways can we have the 2 French disks on both sides?"},{"Start":"04:35.460 ","End":"04:37.295","Text":"Well, that\u0027s 2 factorial."},{"Start":"04:37.295 ","End":"04:42.420","Text":"Now, this comes out to 1 divided by 45."}],"ID":12795},{"Watched":false,"Name":"Exrcise 3 - Parts a-b","Duration":"1m 52s","ChapterTopicVideoID":12317,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12317.jpeg","UploadDate":"2018-07-22T16:43:27.9930000","DurationForVideoObject":"PT1M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.125","Text":"In this question, we have 5 boys and 4 girls who wrote an exam,"},{"Start":"00:04.125 ","End":"00:07.350","Text":"and then they were ranked according to their marks."},{"Start":"00:07.350 ","End":"00:10.770","Text":"Now, assume that no students have the same mark."},{"Start":"00:10.770 ","End":"00:12.900","Text":"Now, why do we have the statement?"},{"Start":"00:12.900 ","End":"00:17.265","Text":"Just to emphasize the fact that we have 9 different objects,"},{"Start":"00:17.265 ","End":"00:20.665","Text":"boys and girls and we have to order them in a row."},{"Start":"00:20.665 ","End":"00:23.035","Text":"We\u0027re asked, how many rankings are possible?"},{"Start":"00:23.035 ","End":"00:25.320","Text":"That means how many ways can we order"},{"Start":"00:25.320 ","End":"00:29.340","Text":"the 5 boys and 4 girls in a row according to their marks?"},{"Start":"00:29.340 ","End":"00:32.280","Text":"Well, we have 9 people here."},{"Start":"00:32.280 ","End":"00:35.135","Text":"That\u0027ll be 9 factorial."},{"Start":"00:35.135 ","End":"00:38.900","Text":"Now, for those who are really interested in"},{"Start":"00:38.900 ","End":"00:46.720","Text":"the actual number 9 factorial equals to 362,880."},{"Start":"00:46.720 ","End":"00:56.285","Text":"There are 362,880 ways of ranking the 5 boys and 4 girls."},{"Start":"00:56.285 ","End":"00:59.480","Text":"In this section, we\u0027re asked how many rankings are possible"},{"Start":"00:59.480 ","End":"01:02.840","Text":"if the boys and girls are marked separately."},{"Start":"01:02.840 ","End":"01:09.090","Text":"We have 5 boys and we have 4 girls."},{"Start":"01:09.090 ","End":"01:11.165","Text":"Now, if we rank the boys separately,"},{"Start":"01:11.165 ","End":"01:14.270","Text":"then we have 5 factorial permutations."},{"Start":"01:14.270 ","End":"01:16.940","Text":"If we rank the girls separately,"},{"Start":"01:16.940 ","End":"01:19.955","Text":"then we have 4 factorial permutations."},{"Start":"01:19.955 ","End":"01:25.670","Text":"Now the combination of the boys rankings and"},{"Start":"01:25.670 ","End":"01:31.775","Text":"the girls rankings would that be 5 factorial times 4 factorial."},{"Start":"01:31.775 ","End":"01:37.100","Text":"Now, this is obviously not equal to 9 factorial."},{"Start":"01:37.100 ","End":"01:39.215","Text":"This thing right here,"},{"Start":"01:39.215 ","End":"01:43.740","Text":"that equals to 2,880."},{"Start":"01:43.740 ","End":"01:52.650","Text":"So there are 2,880 ways of ranking the boys separately with the girls separately."}],"ID":12796},{"Watched":false,"Name":"Exrcise 4 - Parts a-d","Duration":"4m 49s","ChapterTopicVideoID":12318,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12318.jpeg","UploadDate":"2018-07-22T16:43:39.5100000","DurationForVideoObject":"PT4M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.170","Text":"In this question, we have"},{"Start":"00:01.170 ","End":"00:04.095","Text":"10 different books that are ordered on the shelf and we\u0027re asked,"},{"Start":"00:04.095 ","End":"00:07.740","Text":"in how many different ways can the books be ordered on the shelf?"},{"Start":"00:07.740 ","End":"00:12.480","Text":"Well, we have 10 objects that are asked to be ordered in a line."},{"Start":"00:12.480 ","End":"00:15.000","Text":"Now, we\u0027re asked for the number of permutations."},{"Start":"00:15.000 ","End":"00:20.415","Text":"How many ways can we order the 10 books in a line or on the shelf?"},{"Start":"00:20.415 ","End":"00:25.230","Text":"Well, we have 10 factorial ways of ordering the books,"},{"Start":"00:25.230 ","End":"00:27.570","Text":"10 different books on a shelf."},{"Start":"00:27.570 ","End":"00:30.810","Text":"Now for those who really want to calculate this,"},{"Start":"00:30.810 ","End":"00:32.790","Text":"it\u0027s fine to leave the number like this,"},{"Start":"00:32.790 ","End":"00:39.090","Text":"but if you really want to know the number, that\u0027s 3,628,880."},{"Start":"00:39.410 ","End":"00:45.520","Text":"That\u0027s the number of ways that we can order 10 different books on the shelf."},{"Start":"00:45.520 ","End":"00:50.040","Text":"Now, we\u0027re told that 2 out of the 10 books are books about statistics."},{"Start":"00:50.040 ","End":"00:54.500","Text":"We\u0027re asked, what\u0027s the probability that if the books are put in random order,"},{"Start":"00:54.500 ","End":"00:58.285","Text":"the statistics books will be next to each other?"},{"Start":"00:58.285 ","End":"01:07.355","Text":"Here, we have the 10 books and we\u0027re looking for the statistics books to be together."},{"Start":"01:07.355 ","End":"01:10.355","Text":"Let\u0027s say we have 2 statistics books."},{"Start":"01:10.355 ","End":"01:12.095","Text":"This is a statistics book,"},{"Start":"01:12.095 ","End":"01:17.490","Text":"and this is a statistics book and we\u0027re looking for this probability."},{"Start":"01:19.910 ","End":"01:23.460","Text":"Let\u0027s take a look at these 2 books right here,"},{"Start":"01:23.460 ","End":"01:27.280","Text":"and let\u0027s look at them as a group."},{"Start":"01:27.280 ","End":"01:31.955","Text":"Let\u0027s consider them as 1 book instead of 2 books."},{"Start":"01:31.955 ","End":"01:36.205","Text":"When we\u0027re dealing with probabilities in the denominator,"},{"Start":"01:36.205 ","End":"01:39.550","Text":"where we have the size of our sample space,"},{"Start":"01:39.550 ","End":"01:42.145","Text":"which is 10 factorial."},{"Start":"01:42.145 ","End":"01:45.340","Text":"Now, how many groups do we have here?"},{"Start":"01:45.340 ","End":"01:47.095","Text":"Well, we have 9 groups,"},{"Start":"01:47.095 ","End":"01:49.045","Text":"1, 2, 3,"},{"Start":"01:49.045 ","End":"01:50.800","Text":"4, 5, 6, 7, 8, 9,"},{"Start":"01:50.800 ","End":"01:52.900","Text":"so that would be 9 factorial."},{"Start":"01:52.900 ","End":"01:59.815","Text":"I can arrange these 9 groups in 9 factorial ways."},{"Start":"01:59.815 ","End":"02:05.830","Text":"But I have to take into account the internal number of permutations."},{"Start":"02:05.830 ","End":"02:10.470","Text":"I have to multiply 9 factorial by 2 factorial."},{"Start":"02:10.470 ","End":"02:15.540","Text":"I have 2 books that I can order them in 2 different ways,"},{"Start":"02:15.540 ","End":"02:19.320","Text":"number 3 and number 4 or number 4 and number 3."},{"Start":"02:19.320 ","End":"02:23.500","Text":"This then would be the probability of"},{"Start":"02:23.500 ","End":"02:27.895","Text":"having the stat books together when I order 10 books."},{"Start":"02:27.895 ","End":"02:32.215","Text":"Now, that comes out to 0.2."},{"Start":"02:32.215 ","End":"02:33.850","Text":"In this section we\u0027re asked,"},{"Start":"02:33.850 ","End":"02:37.270","Text":"what\u0027s the probability that if the books are put on the shelf in"},{"Start":"02:37.270 ","End":"02:42.545","Text":"random order that the statistics books will not be next to each other?"},{"Start":"02:42.545 ","End":"02:49.670","Text":"We\u0027re looking for the probability of the statistics books not being together."},{"Start":"02:49.670 ","End":"02:52.055","Text":"Well, what does that equal to?"},{"Start":"02:52.055 ","End":"02:54.740","Text":"Well, in the previous section,"},{"Start":"02:54.740 ","End":"03:00.920","Text":"we\u0027ve calculated the probability of the 2 stat books being together."},{"Start":"03:00.920 ","End":"03:07.010","Text":"Now, this is the complimentary event of the 2 stat books not being together."},{"Start":"03:07.010 ","End":"03:14.365","Text":"This probability here would be 1 minus the probability of the stat books being together."},{"Start":"03:14.365 ","End":"03:17.535","Text":"As I said, we\u0027ve calculated that."},{"Start":"03:17.535 ","End":"03:21.680","Text":"This probability is 1 minus 0.2,"},{"Start":"03:21.680 ","End":"03:24.980","Text":"and that comes out to 0.8."},{"Start":"03:24.980 ","End":"03:27.860","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"03:27.860 ","End":"03:31.175","Text":"that the statistics books are on the ends of the shell,"},{"Start":"03:31.175 ","End":"03:33.595","Text":"each book at a different end?"},{"Start":"03:33.595 ","End":"03:39.255","Text":"Here are the 10 books on the shelf and what we want is that this book,"},{"Start":"03:39.255 ","End":"03:41.535","Text":"Book number 1 to be a statistics book,"},{"Start":"03:41.535 ","End":"03:44.190","Text":"and Book number 10 to be a statistics book,"},{"Start":"03:44.190 ","End":"03:48.660","Text":"and we\u0027re looking for the probability of this event occurring."},{"Start":"03:49.310 ","End":"03:53.250","Text":"Again, we have in the denominator,"},{"Start":"03:53.250 ","End":"03:57.890","Text":"we have the size of our sample space, that\u0027s 10 factorial."},{"Start":"03:57.890 ","End":"04:02.030","Text":"Now, let\u0027s take a look at this configuration right here,"},{"Start":"04:02.030 ","End":"04:04.760","Text":"and we\u0027ll look at the middle 8 books."},{"Start":"04:04.760 ","End":"04:08.600","Text":"Well, how many permutations do I have,"},{"Start":"04:08.600 ","End":"04:12.395","Text":"or how many ways can I order these 8 books together?"},{"Start":"04:12.395 ","End":"04:14.955","Text":"Well, 8 factorial."},{"Start":"04:14.955 ","End":"04:17.835","Text":"What about these 2 books, the stat books?"},{"Start":"04:17.835 ","End":"04:20.065","Text":"Well, that\u0027s 2 factorial."},{"Start":"04:20.065 ","End":"04:23.120","Text":"I can order the stat books in 2 different ways."},{"Start":"04:23.120 ","End":"04:27.965","Text":"I can have Book number 1 here and Book number 10 here,"},{"Start":"04:27.965 ","End":"04:29.330","Text":"or the other way around,"},{"Start":"04:29.330 ","End":"04:32.600","Text":"Book number 10 here and Book number 1 here."},{"Start":"04:32.600 ","End":"04:34.700","Text":"That\u0027s the 2 factorial right here."},{"Start":"04:34.700 ","End":"04:43.609","Text":"All this comes out to 1 divided by 45 and if we\u0027re looking to express this in decimal,"},{"Start":"04:43.609 ","End":"04:49.170","Text":"well, this is equal to 0.022."}],"ID":12797},{"Watched":false,"Name":"Exrcise 5 - Parts a-c","Duration":"5m 30s","ChapterTopicVideoID":12319,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12319.jpeg","UploadDate":"2018-07-22T16:43:56.1170000","DurationForVideoObject":"PT5M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this question, we have a person that creates a playlist with 12 different songs."},{"Start":"00:04.335 ","End":"00:06.360","Text":"Now, 4 of the songs are in Spanish,"},{"Start":"00:06.360 ","End":"00:08.565","Text":"5 are in English, and 3 are in French."},{"Start":"00:08.565 ","End":"00:13.020","Text":"The person plays the songs randomly from the playlist and we\u0027re asked,"},{"Start":"00:13.020 ","End":"00:16.350","Text":"what\u0027s the probability that all the songs in English"},{"Start":"00:16.350 ","End":"00:20.890","Text":"appear as a single group at the beginning of the list?"},{"Start":"00:20.890 ","End":"00:25.320","Text":"Again, whenever we have to calculate probabilities"},{"Start":"00:25.320 ","End":"00:30.420","Text":"then the first thing that we need to do is to calculate the size of our sample space."},{"Start":"00:30.420 ","End":"00:34.155","Text":"That\u0027ll be the size of Omega."},{"Start":"00:34.155 ","End":"00:36.150","Text":"Now, how many songs do we have?"},{"Start":"00:36.150 ","End":"00:38.610","Text":"Well, we have 12 different songs."},{"Start":"00:38.610 ","End":"00:42.385","Text":"How many permutations or how many combinations or how"},{"Start":"00:42.385 ","End":"00:46.910","Text":"ways can we order these 12 songs in a row?"},{"Start":"00:46.910 ","End":"00:50.415","Text":"Well, that\u0027s 12 factorial."},{"Start":"00:50.415 ","End":"00:52.745","Text":"Now that we have that,"},{"Start":"00:52.745 ","End":"00:54.770","Text":"then, let\u0027s answer the question."},{"Start":"00:54.770 ","End":"00:59.915","Text":"What\u0027s the probabilities that all the songs in English appear as a single group?"},{"Start":"00:59.915 ","End":"01:02.285","Text":"Well, if they appear in a single group,"},{"Start":"01:02.285 ","End":"01:06.630","Text":"then here we have a group with 5 songs,"},{"Start":"01:06.910 ","End":"01:14.785","Text":"because we have 5 songs in English and then we have 7 songs."},{"Start":"01:14.785 ","End":"01:19.875","Text":"These are the 4 songs in Spanish and 3 songs in French."},{"Start":"01:19.875 ","End":"01:23.180","Text":"We want this probability right here."},{"Start":"01:23.180 ","End":"01:30.435","Text":"Well, the probability of the 5 English songs first,"},{"Start":"01:30.435 ","End":"01:35.165","Text":"well, that equals to 5 factorial."},{"Start":"01:35.165 ","End":"01:42.670","Text":"This is the number of permutations of ordering the 5 English songs,"},{"Start":"01:42.670 ","End":"01:45.380","Text":"times 7 factorial,"},{"Start":"01:45.380 ","End":"01:52.595","Text":"that\u0027s the number of permutations of ordering the other 7 songs divided by 12 factorial,"},{"Start":"01:52.595 ","End":"01:55.565","Text":"which is the size of our sample space."},{"Start":"01:55.565 ","End":"02:04.130","Text":"Now, this comes out to 1 divided by 792."},{"Start":"02:04.130 ","End":"02:06.710","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"02:06.710 ","End":"02:09.635","Text":"all the songs in English appear consecutively?"},{"Start":"02:09.635 ","End":"02:11.345","Text":"That means that they\u0027re together."},{"Start":"02:11.345 ","End":"02:13.460","Text":"We have this type of configuration."},{"Start":"02:13.460 ","End":"02:18.960","Text":"We have the 5 songs here and they\u0027re in English."},{"Start":"02:18.960 ","End":"02:21.390","Text":"Now, really doesn\u0027t matter how many songs are before"},{"Start":"02:21.390 ","End":"02:24.253","Text":"this group and how many songs are after this group,"},{"Start":"02:24.253 ","End":"02:26.360","Text":"so out of the 12 songs that we have,"},{"Start":"02:26.360 ","End":"02:28.850","Text":"we have a group of 5 songs."},{"Start":"02:28.850 ","End":"02:32.960","Text":"We\u0027re looking at this probability right here."},{"Start":"02:32.960 ","End":"02:35.855","Text":"Now, how do we solve this question?"},{"Start":"02:35.855 ","End":"02:40.685","Text":"Well, let\u0027s take a look at the number of groups that I have."},{"Start":"02:40.685 ","End":"02:44.800","Text":"Let\u0027s take the 5 songs in English and we\u0027ll create"},{"Start":"02:44.800 ","End":"02:50.655","Text":"1 group and each other song is a group by itself."},{"Start":"02:50.655 ","End":"02:53.180","Text":"How many groups in total do I have?"},{"Start":"02:53.180 ","End":"02:58.220","Text":"Well, I have 1 group which is the 5 songs and the other 7 groups are the 7 songs,"},{"Start":"02:58.220 ","End":"03:01.050","Text":"so I have 8 groups in all."},{"Start":"03:02.390 ","End":"03:08.735","Text":"How many permutations do I have in ordering the 8 groups?"},{"Start":"03:08.735 ","End":"03:11.660","Text":"Well, 8 factorial."},{"Start":"03:11.660 ","End":"03:15.440","Text":"Now I have to consider the number of"},{"Start":"03:15.440 ","End":"03:20.560","Text":"permutations of ordering the 5 songs within this group."},{"Start":"03:20.560 ","End":"03:26.060","Text":"That\u0027s 5 factorial divided by the size of the sample space,"},{"Start":"03:26.060 ","End":"03:28.385","Text":"which is 12 factorial."},{"Start":"03:28.385 ","End":"03:34.040","Text":"This then would be the probability of having the 5 English songs together."},{"Start":"03:34.040 ","End":"03:38.915","Text":"By the way, this comes out to 1 divided by 99."},{"Start":"03:38.915 ","End":"03:40.580","Text":"In this section we\u0027re asked,"},{"Start":"03:40.580 ","End":"03:45.410","Text":"what\u0027s the probability that the songs in the same language appear consecutively?"},{"Start":"03:45.410 ","End":"03:48.215","Text":"That means that all the songs in English are consecutive,"},{"Start":"03:48.215 ","End":"03:50.540","Text":"all the songs in Spanish are consecutive and"},{"Start":"03:50.540 ","End":"03:53.570","Text":"other signs in French are played consecutively."},{"Start":"03:53.570 ","End":"03:55.640","Text":"Well, let\u0027s just recall."},{"Start":"03:55.640 ","End":"04:00.140","Text":"We have 4 Spanish songs,"},{"Start":"04:00.140 ","End":"04:05.855","Text":"5 English songs, and 3 French songs."},{"Start":"04:05.855 ","End":"04:09.810","Text":"Now we\u0027re looking for this probability."},{"Start":"04:10.030 ","End":"04:14.315","Text":"The first thing that I want to know is how many groups do I have?"},{"Start":"04:14.315 ","End":"04:15.995","Text":"Well, I have 3 groups."},{"Start":"04:15.995 ","End":"04:18.080","Text":"I have the Spanish group,"},{"Start":"04:18.080 ","End":"04:19.310","Text":"I have the English group,"},{"Start":"04:19.310 ","End":"04:20.855","Text":"and I have the French group."},{"Start":"04:20.855 ","End":"04:24.620","Text":"How many ways can I order these groups?"},{"Start":"04:24.620 ","End":"04:28.520","Text":"What\u0027s the number of permutations of ordering 3 groups?"},{"Start":"04:28.520 ","End":"04:30.860","Text":"Well, that\u0027s 3 factorial."},{"Start":"04:30.860 ","End":"04:37.085","Text":"Now I have to look at the number of permutations within each group."},{"Start":"04:37.085 ","End":"04:39.110","Text":"For the Spanish group,"},{"Start":"04:39.110 ","End":"04:41.540","Text":"I have 4 factorial,"},{"Start":"04:41.540 ","End":"04:46.160","Text":"I can order these 4 songs in 4 factorial ways."},{"Start":"04:46.160 ","End":"04:47.570","Text":"In the English group,"},{"Start":"04:47.570 ","End":"04:49.430","Text":"I have 5 factorial."},{"Start":"04:49.430 ","End":"04:52.610","Text":"In the French group I have 3 factorial,"},{"Start":"04:52.610 ","End":"04:56.750","Text":"and all this has to be divided by 12 factorial,"},{"Start":"04:56.750 ","End":"04:59.360","Text":"the size of my sample space."},{"Start":"04:59.360 ","End":"05:07.685","Text":"Now, this comes out to 1 divided by 4,620."},{"Start":"05:07.685 ","End":"05:12.965","Text":"There are 4,620 ways"},{"Start":"05:12.965 ","End":"05:18.960","Text":"of playing the songs in the same language together."},{"Start":"05:18.960 ","End":"05:22.190","Text":"We have the groupings of the Spanish, English,"},{"Start":"05:22.190 ","End":"05:25.100","Text":"and French and the Spanish songs are played together,"},{"Start":"05:25.100 ","End":"05:27.290","Text":"all the English sounds are prayed together,"},{"Start":"05:27.290 ","End":"05:30.540","Text":"and all the French songs are played together."}],"ID":12798},{"Watched":false,"Name":"Exrcise 6 - Parts a-d","Duration":"6m 27s","ChapterTopicVideoID":12320,"CourseChapterTopicPlaylistID":264987,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12320.jpeg","UploadDate":"2018-07-22T16:44:15.5170000","DurationForVideoObject":"PT6M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this question, we have 4 boys and 4 girls that sit randomly"},{"Start":"00:03.570 ","End":"00:07.185","Text":"in a row in a movie theater in seats 1-8."},{"Start":"00:07.185 ","End":"00:13.440","Text":"We\u0027re asked, what\u0027s the probability that John and Monica did not sit next to each other?"},{"Start":"00:13.440 ","End":"00:16.950","Text":"The first thing that we need to do is to define"},{"Start":"00:16.950 ","End":"00:20.550","Text":"or calculate the size of our sample space."},{"Start":"00:20.550 ","End":"00:30.135","Text":"We have here 8 people and they have to be put in order."},{"Start":"00:30.135 ","End":"00:33.090","Text":"How many possible ways can we order"},{"Start":"00:33.090 ","End":"00:37.230","Text":"the 4 boys and 4 girls when they\u0027re together as 1 group?"},{"Start":"00:37.230 ","End":"00:40.215","Text":"That\u0027s 8 factorial."},{"Start":"00:40.215 ","End":"00:45.590","Text":"Now, we\u0027re asked here, what\u0027s the probability that 1 specific boy,"},{"Start":"00:45.590 ","End":"00:47.930","Text":"1 specific girl, John and Monica,"},{"Start":"00:47.930 ","End":"00:51.160","Text":"did not sit next to each other?"},{"Start":"00:51.160 ","End":"00:54.285","Text":"This probability right here."},{"Start":"00:54.285 ","End":"01:01.220","Text":"It\u0027ll be much easier for us to calculate the probability of the complimentary set."},{"Start":"01:01.220 ","End":"01:04.205","Text":"That means that we\u0027re looking at 1 minus"},{"Start":"01:04.205 ","End":"01:09.990","Text":"the probability of John and Monica sitting together."},{"Start":"01:09.990 ","End":"01:12.225","Text":"What does that equal to?"},{"Start":"01:12.225 ","End":"01:15.250","Text":"It equals to 1 minus this probability."},{"Start":"01:15.250 ","End":"01:19.550","Text":"What\u0027s the probability of John and Monica sitting together?"},{"Start":"01:19.550 ","End":"01:24.785","Text":"In the denominator, we have the sample size,"},{"Start":"01:24.785 ","End":"01:26.360","Text":"the size of our sample space,"},{"Start":"01:26.360 ","End":"01:28.855","Text":"which is 8 factorial."},{"Start":"01:28.855 ","End":"01:32.179","Text":"If John and Monica are sitting together,"},{"Start":"01:32.179 ","End":"01:34.370","Text":"then let\u0027s group them together."},{"Start":"01:34.370 ","End":"01:39.815","Text":"Instead of 8 people or 8 objects,"},{"Start":"01:39.815 ","End":"01:42.260","Text":"then we have 7 objects."},{"Start":"01:42.260 ","End":"01:45.335","Text":"We have John and Monica together,"},{"Start":"01:45.335 ","End":"01:47.630","Text":"and then the rest of the guys."},{"Start":"01:47.630 ","End":"01:52.160","Text":"How many possible combinations or"},{"Start":"01:52.160 ","End":"01:57.995","Text":"how many permutations do we have of ordering 7 objects in a row?"},{"Start":"01:57.995 ","End":"02:00.760","Text":"That\u0027s 7 factorial."},{"Start":"02:00.760 ","End":"02:03.680","Text":"We have to multiply that by"},{"Start":"02:03.680 ","End":"02:09.565","Text":"the internal number of permutations of the group, John and Monica."},{"Start":"02:09.565 ","End":"02:14.180","Text":"What\u0027s the number of permutations of ordering John and Monica?"},{"Start":"02:14.180 ","End":"02:16.625","Text":"That\u0027s 2 factorial."},{"Start":"02:16.625 ","End":"02:18.800","Text":"We can have John and Monica,"},{"Start":"02:18.800 ","End":"02:20.800","Text":"or Monica and John."},{"Start":"02:20.800 ","End":"02:25.910","Text":"That equals to 1 minus 0.25."},{"Start":"02:25.910 ","End":"02:31.690","Text":"The whole probability here equals to 0.75."},{"Start":"02:31.690 ","End":"02:33.350","Text":"In this section, we\u0027re asked,"},{"Start":"02:33.350 ","End":"02:38.390","Text":"what\u0027s the probability that the boys sat in the odd-numbered seats?"},{"Start":"02:38.390 ","End":"02:40.780","Text":"What do we mean by that?"},{"Start":"02:40.780 ","End":"02:45.045","Text":"Here are the seats numbered 1-8 and"},{"Start":"02:45.045 ","End":"02:48.795","Text":"all the boys sat in the odd-numbered seat: seat number 1,"},{"Start":"02:48.795 ","End":"02:52.590","Text":"seat number 3, 5, and 7."},{"Start":"02:52.590 ","End":"03:01.310","Text":"The seats in the circle are boys and the seats that are not in the circle,"},{"Start":"03:01.310 ","End":"03:04.410","Text":"that\u0027ll be girls, right here."},{"Start":"03:05.420 ","End":"03:10.609","Text":"We\u0027re asked, what\u0027s the probability that the boys"},{"Start":"03:10.609 ","End":"03:15.260","Text":"sat in odd-numbered seats? What are we doing?"},{"Start":"03:15.260 ","End":"03:20.810","Text":"We\u0027re taking all the 4 boys and we\u0027re ordering them into the seats."},{"Start":"03:20.810 ","End":"03:26.010","Text":"We have 4 factorial ways of ordering the boys"},{"Start":"03:26.010 ","End":"03:31.620","Text":"into the odd number seats times 4 factorial,"},{"Start":"03:31.620 ","End":"03:38.885","Text":"that\u0027s the number of permutations of the girls being in the even number seats,"},{"Start":"03:38.885 ","End":"03:46.655","Text":"divided by the total number of people that we have or the size of our sample space."},{"Start":"03:46.655 ","End":"03:54.235","Text":"That comes out to 0.0143 approximately."},{"Start":"03:54.235 ","End":"03:55.895","Text":"In this section, we\u0027re asked,"},{"Start":"03:55.895 ","End":"04:01.360","Text":"what\u0027s the probability that all the boys sat next to each other? How would that look?"},{"Start":"04:01.360 ","End":"04:07.005","Text":"We want all the boys sitting together."},{"Start":"04:07.005 ","End":"04:10.390","Text":"It really doesn\u0027t matter how many girls we have"},{"Start":"04:10.390 ","End":"04:15.160","Text":"before this group and how many girls we have after this group."},{"Start":"04:15.160 ","End":"04:21.960","Text":"All that\u0027s interesting is for the boys to be sitting next to each other."},{"Start":"04:21.960 ","End":"04:25.360","Text":"What\u0027s this probability right here?"},{"Start":"04:25.460 ","End":"04:28.045","Text":"Let\u0027s take a look at this."},{"Start":"04:28.045 ","End":"04:32.205","Text":"Let\u0027s consider all the boys as 1 group."},{"Start":"04:32.205 ","End":"04:34.900","Text":"How many groups in total do I have to order?"},{"Start":"04:34.900 ","End":"04:38.715","Text":"I have the 4 girls plus the group of boys."},{"Start":"04:38.715 ","End":"04:41.280","Text":"I have 5 groups."},{"Start":"04:41.280 ","End":"04:43.800","Text":"I need to order 5 groups."},{"Start":"04:43.800 ","End":"04:46.170","Text":"That\u0027s 5 factorial."},{"Start":"04:46.170 ","End":"04:53.210","Text":"I also have to multiply the 5 factorial by the internal ordering of the boys."},{"Start":"04:53.210 ","End":"05:00.620","Text":"That\u0027ll be 4 factorial divided by the size of the sample space, that\u0027s 8 factorial."},{"Start":"05:00.620 ","End":"05:08.315","Text":"That comes out to 0.0714 in approximation."},{"Start":"05:08.315 ","End":"05:09.920","Text":"In this section, we\u0027re asked,"},{"Start":"05:09.920 ","End":"05:12.770","Text":"what\u0027s the probability that the boys sat next to each"},{"Start":"05:12.770 ","End":"05:16.355","Text":"other and the girl sat next to each other?"},{"Start":"05:16.355 ","End":"05:18.530","Text":"In the previous section,"},{"Start":"05:18.530 ","End":"05:21.110","Text":"we were interested in just the boys sitting"},{"Start":"05:21.110 ","End":"05:25.995","Text":"together and we didn\u0027t care if the girls sat together or not."},{"Start":"05:25.995 ","End":"05:30.410","Text":"We could have had girls on this side of the boys and on this side of the boys."},{"Start":"05:30.410 ","End":"05:32.770","Text":"But now, in this section,"},{"Start":"05:32.770 ","End":"05:35.900","Text":"we\u0027re asking for the boys to be sitting"},{"Start":"05:35.900 ","End":"05:40.550","Text":"together and the girls to be sitting together as well."},{"Start":"05:40.550 ","End":"05:46.560","Text":"This then would be the probability that we\u0027re looking for."},{"Start":"05:47.510 ","End":"05:52.415","Text":"Here, when we\u0027re looking at this configuration,"},{"Start":"05:52.415 ","End":"05:54.530","Text":"then we have 2 groups,"},{"Start":"05:54.530 ","End":"05:57.300","Text":"the boys group and the girls group."},{"Start":"05:57.300 ","End":"06:00.450","Text":"How many ways can we order these 2 groups?"},{"Start":"06:00.450 ","End":"06:02.610","Text":"Well, 2 factorial."},{"Start":"06:02.610 ","End":"06:06.740","Text":"We have to multiply this by the internal ordering of the boys,"},{"Start":"06:06.740 ","End":"06:09.050","Text":"that\u0027s 4 factorial,"},{"Start":"06:09.050 ","End":"06:11.510","Text":"times the internal ordering of the girls,"},{"Start":"06:11.510 ","End":"06:13.490","Text":"that\u0027s another 4 factorial,"},{"Start":"06:13.490 ","End":"06:17.210","Text":"divided by the size of the sample space,"},{"Start":"06:17.210 ","End":"06:19.115","Text":"which is 8 factorial."},{"Start":"06:19.115 ","End":"06:27.150","Text":"This comes out to 0.0286 approximately."}],"ID":12799}],"Thumbnail":null,"ID":264987},{"Name":"Permutations with Identical Objects","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"58s","ChapterTopicVideoID":12321,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12321.jpeg","UploadDate":"2018-07-22T16:45:55.2570000","DurationForVideoObject":"PT58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this chapter, we\u0027ll be talking about combinatorics,"},{"Start":"00:02.520 ","End":"00:05.880","Text":"specifically permutations with identical objects."},{"Start":"00:05.880 ","End":"00:08.265","Text":"When arranging items in a row,"},{"Start":"00:08.265 ","End":"00:11.430","Text":"if some of the objects to be ordered are identical,"},{"Start":"00:11.430 ","End":"00:16.140","Text":"the internal order of those objects is irrelevant and should be eliminated by"},{"Start":"00:16.140 ","End":"00:21.765","Text":"dividing the total number of arrangements by the internal order of the identical items."},{"Start":"00:21.765 ","End":"00:25.530","Text":"The number of ways to put n objects in a row of"},{"Start":"00:25.530 ","End":"00:28.905","Text":"which n_1 identical objects are of Type 1,"},{"Start":"00:28.905 ","End":"00:31.880","Text":"and 2 identical objects are of Type 2,"},{"Start":"00:31.880 ","End":"00:38.465","Text":"and so on and so forth until we have n_r identical objects that are of Type R. Well,"},{"Start":"00:38.465 ","End":"00:44.000","Text":"this then is the number of ways to put these type of objects in a row."},{"Start":"00:44.000 ","End":"00:51.380","Text":"We have n factorial divided by n_1 factorial times n_2 factorial,"},{"Start":"00:51.380 ","End":"00:52.730","Text":"and so on and so forth,"},{"Start":"00:52.730 ","End":"00:54.965","Text":"times n_ r factorial."},{"Start":"00:54.965 ","End":"00:56.900","Text":"This is the theory,"},{"Start":"00:56.900 ","End":"00:59.190","Text":"let\u0027s do an example."}],"ID":12800},{"Watched":false,"Name":"Tutorial 2","Duration":"2m 16s","ChapterTopicVideoID":12322,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12322.jpeg","UploadDate":"2018-07-22T16:46:00.1370000","DurationForVideoObject":"PT2M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this example, we\u0027re asked how many words can be formed from"},{"Start":"00:03.150 ","End":"00:07.095","Text":"the following letters, WWTT and KK."},{"Start":"00:07.095 ","End":"00:10.890","Text":"Well, let\u0027s first take a look at some examples of words."},{"Start":"00:10.890 ","End":"00:14.530","Text":"I can have WKWTTK,"},{"Start":"00:15.560 ","End":"00:18.990","Text":"or I can just switch these 2 letters around."},{"Start":"00:18.990 ","End":"00:22.090","Text":"That\u0027ll be KWWTTK."},{"Start":"00:22.430 ","End":"00:24.990","Text":"These are 2 different words."},{"Start":"00:24.990 ","End":"00:28.060","Text":"Now, if all the letters here would have been different,"},{"Start":"00:28.060 ","End":"00:32.360","Text":"then how many words can I form,"},{"Start":"00:32.360 ","End":"00:34.520","Text":"with the 6 letters?"},{"Start":"00:34.520 ","End":"00:37.820","Text":"Well, that would have been the size of my sample space."},{"Start":"00:37.820 ","End":"00:41.510","Text":"That would have been equal to 6 factorial."},{"Start":"00:41.510 ","End":"00:43.745","Text":"But that\u0027s not the case here."},{"Start":"00:43.745 ","End":"00:48.605","Text":"I have here 3 groups all here that equals to 3."},{"Start":"00:48.605 ","End":"00:53.090","Text":"In each group, I have 2 identical letters,"},{"Start":"00:53.090 ","End":"00:57.635","Text":"so what I have to do is I have to divide"},{"Start":"00:57.635 ","End":"01:03.200","Text":"the total size of my sample space by the following."},{"Start":"01:03.200 ","End":"01:05.045","Text":"Well, according to the formula,"},{"Start":"01:05.045 ","End":"01:08.925","Text":"that will be n factorial divided by"},{"Start":"01:08.925 ","End":"01:15.645","Text":"n_1 factorial times n_2 factorial times n_3 factorial."},{"Start":"01:15.645 ","End":"01:17.070","Text":"I have 3 groups."},{"Start":"01:17.070 ","End":"01:23.315","Text":"Now, n_1, how many identical objects do I have in this group right here?"},{"Start":"01:23.315 ","End":"01:26.225","Text":"Well, I have 2, and in n_2,"},{"Start":"01:26.225 ","End":"01:29.280","Text":"I have 2 and in n_3,"},{"Start":"01:29.280 ","End":"01:30.990","Text":"I also have 2."},{"Start":"01:30.990 ","End":"01:37.760","Text":"In our case, how many words can I form using these letters right here?"},{"Start":"01:37.760 ","End":"01:45.520","Text":"Well, that\u0027ll be 6 factorial divided by 2 factorial times 2 factorial."},{"Start":"01:45.520 ","End":"01:51.275","Text":"Now that equals to 90 as opposed to 6 factorial,"},{"Start":"01:51.275 ","End":"01:54.170","Text":"which equals to 720."},{"Start":"01:54.170 ","End":"01:58.370","Text":"If all of the letters would have been different then I would have had"},{"Start":"01:58.370 ","End":"02:03.830","Text":"720 different permutations or 720 different words."},{"Start":"02:03.830 ","End":"02:06.890","Text":"But because I have 3 groups,"},{"Start":"02:06.890 ","End":"02:10.610","Text":"where in each group I have 2 identical letters,"},{"Start":"02:10.610 ","End":"02:15.840","Text":"then the number of permutations drops to 90."}],"ID":12801},{"Watched":false,"Name":"Exrcise 1","Duration":"2m 35s","ChapterTopicVideoID":12323,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12323.jpeg","UploadDate":"2018-07-22T16:46:06.2300000","DurationForVideoObject":"PT2M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.275","Text":"In this question, we\u0027re asked,"},{"Start":"00:01.275 ","End":"00:05.490","Text":"what\u0027s the number of permutations in coloring 2 out of 5 cells."},{"Start":"00:05.490 ","End":"00:07.320","Text":"For example, here I have 5 cells,"},{"Start":"00:07.320 ","End":"00:08.880","Text":"and I want to color in,"},{"Start":"00:08.880 ","End":"00:11.400","Text":"for example, this cell right here,"},{"Start":"00:11.400 ","End":"00:13.710","Text":"and this cell right here."},{"Start":"00:13.710 ","End":"00:15.300","Text":"Well, first of all,"},{"Start":"00:15.300 ","End":"00:16.485","Text":"I have 5 cells,"},{"Start":"00:16.485 ","End":"00:20.070","Text":"n here, that equals to 5."},{"Start":"00:20.070 ","End":"00:22.935","Text":"Now if we take a look at this, well,"},{"Start":"00:22.935 ","End":"00:25.740","Text":"basically I have 2 groups."},{"Start":"00:25.740 ","End":"00:27.390","Text":"R here that equals to 2."},{"Start":"00:27.390 ","End":"00:30.930","Text":"The first group would be the cells that are not colored,"},{"Start":"00:30.930 ","End":"00:32.330","Text":"so n_1, well,"},{"Start":"00:32.330 ","End":"00:34.800","Text":"how many cells do I have that are not colored?"},{"Start":"00:34.800 ","End":"00:40.200","Text":"I have 3, and n_2 would be the number of cells that are colored with that\u0027s 2."},{"Start":"00:40.200 ","End":"00:42.815","Text":"That\u0027s what the question asks."},{"Start":"00:42.815 ","End":"00:46.370","Text":"How many combinations or how many permutations"},{"Start":"00:46.370 ","End":"00:50.315","Text":"can I have of coloring in 2 out of 5 cells?"},{"Start":"00:50.315 ","End":"00:52.010","Text":"Well, if we think about it,"},{"Start":"00:52.010 ","End":"00:55.505","Text":"this is just like having 5 letters."},{"Start":"00:55.505 ","End":"00:57.843","Text":"Let\u0027s split into 2 groups,"},{"Start":"00:57.843 ","End":"00:59.210","Text":"where the first group has,"},{"Start":"00:59.210 ","End":"01:03.410","Text":"let\u0027s say AAA and the second group has B and B."},{"Start":"01:03.410 ","End":"01:06.980","Text":"Well, how many ways can I order these letters?"},{"Start":"01:06.980 ","End":"01:09.800","Text":"Well, according to our formula,"},{"Start":"01:09.800 ","End":"01:17.805","Text":"that will be n-factorial divided by n_1-factorial times n_2-factorial."},{"Start":"01:17.805 ","End":"01:22.865","Text":"In our case, that\u0027ll be 5 factorial divided by"},{"Start":"01:22.865 ","End":"01:28.850","Text":"3 factorial times 2 factorial and that equals to 10."},{"Start":"01:28.850 ","End":"01:30.200","Text":"Again, what did they do?"},{"Start":"01:30.200 ","End":"01:35.825","Text":"I took the total number of permutations that I can get out of,"},{"Start":"01:35.825 ","End":"01:38.700","Text":"let\u0027s say, ordering 5 objects."},{"Start":"01:38.700 ","End":"01:42.665","Text":"If all these letters would have been different then"},{"Start":"01:42.665 ","End":"01:46.415","Text":"I would have 5 factorial ways of ordering them."},{"Start":"01:46.415 ","End":"01:48.290","Text":"But because I have 2 groups,"},{"Start":"01:48.290 ","End":"01:52.910","Text":"where in the first group I have 3 identical objects."},{"Start":"01:52.910 ","End":"01:55.535","Text":"In the second group I have 2 identical objects."},{"Start":"01:55.535 ","End":"02:02.070","Text":"Then I have to divide by the number of internal orderings."},{"Start":"02:02.070 ","End":"02:04.775","Text":"It really doesn\u0027t matter if this comes first,"},{"Start":"02:04.775 ","End":"02:07.385","Text":"this comes second, and this comes third."},{"Start":"02:07.385 ","End":"02:09.740","Text":"Or any other combinations,"},{"Start":"02:09.740 ","End":"02:11.880","Text":"irrelevant, it doesn\u0027t matter."},{"Start":"02:11.880 ","End":"02:20.690","Text":"I have to divide the 5 factorial by the number of internal orderings in each group."},{"Start":"02:20.690 ","End":"02:24.860","Text":"This is the 3 factorial and the 2 factorial here."},{"Start":"02:24.860 ","End":"02:35.550","Text":"All in all, I can have 10 different permutations of coloring in 2 cells out of 5 cells."}],"ID":12802},{"Watched":false,"Name":"Exrcise 2","Duration":"1m 45s","ChapterTopicVideoID":12324,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12324.jpeg","UploadDate":"2018-07-22T16:46:09.8670000","DurationForVideoObject":"PT1M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.240","Text":"In this question we\u0027re asked,"},{"Start":"00:01.240 ","End":"00:03.780","Text":"in how many different ways can the letters b,"},{"Start":"00:03.780 ","End":"00:04.830","Text":"m, m, b,"},{"Start":"00:04.830 ","End":"00:06.885","Text":"m, c be ordered?"},{"Start":"00:06.885 ","End":"00:10.700","Text":"First of all, I see that I have 6 letters,"},{"Start":"00:10.700 ","End":"00:13.460","Text":"so n that equals to 6."},{"Start":"00:13.460 ","End":"00:16.560","Text":"I can also identify 3 groups,"},{"Start":"00:16.560 ","End":"00:20.130","Text":"or r here that equals to 3."},{"Start":"00:20.130 ","End":"00:24.480","Text":"The first group, let\u0027s say that\u0027s the group that has the letters b."},{"Start":"00:24.480 ","End":"00:27.330","Text":"Well, n_1,"},{"Start":"00:27.330 ","End":"00:29.625","Text":"that\u0027s how many Bs I have."},{"Start":"00:29.625 ","End":"00:31.530","Text":"Well, I have 2 Bs."},{"Start":"00:31.530 ","End":"00:36.225","Text":"The second group has the letter n. That\u0027s n_2."},{"Start":"00:36.225 ","End":"00:39.690","Text":"How many Ms do I have? Well, I have 3."},{"Start":"00:39.690 ","End":"00:43.890","Text":"The 3rd group is the letter c,"},{"Start":"00:43.890 ","End":"00:45.660","Text":"so n_3, well,"},{"Start":"00:45.660 ","End":"00:46.920","Text":"how many Cs do I have?"},{"Start":"00:46.920 ","End":"00:49.795","Text":"I have 1 c. Now,"},{"Start":"00:49.795 ","End":"00:53.465","Text":"how many different ways can I order these letters?"},{"Start":"00:53.465 ","End":"00:56.780","Text":"Well, according to our formula,"},{"Start":"00:56.780 ","End":"01:04.560","Text":"that\u0027ll be 6 factorial divided by the number of internal orderings."},{"Start":"01:04.560 ","End":"01:10.245","Text":"That\u0027ll be 2 factorial times 3 factorial times 1 factorial."},{"Start":"01:10.245 ","End":"01:13.260","Text":"Then we can get rid of 1 factorial that equals to 1."},{"Start":"01:13.260 ","End":"01:16.085","Text":"Whenever we have a group with, let\u0027s say,"},{"Start":"01:16.085 ","End":"01:17.870","Text":"only 1 object in it,"},{"Start":"01:17.870 ","End":"01:19.715","Text":"we can disregard it."},{"Start":"01:19.715 ","End":"01:23.585","Text":"Now that equals to 60."},{"Start":"01:23.585 ","End":"01:28.550","Text":"I can have 60 different ways of ordering these letters."},{"Start":"01:28.550 ","End":"01:31.515","Text":"For example, I can have m,"},{"Start":"01:31.515 ","End":"01:33.045","Text":"b, m,"},{"Start":"01:33.045 ","End":"01:36.400","Text":"c, m, b."},{"Start":"01:37.070 ","End":"01:40.918","Text":"That\u0027s 1 way of ordering these 6 letters,"},{"Start":"01:40.918 ","End":"01:41.960","Text":"and so on and so forth."},{"Start":"01:41.960 ","End":"01:45.420","Text":"I have 60 ways of doing that."}],"ID":12803},{"Watched":false,"Name":"Exrcise 3","Duration":"1m 18s","ChapterTopicVideoID":12325,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12325.jpeg","UploadDate":"2018-07-22T16:46:12.9430000","DurationForVideoObject":"PT1M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this question, we\u0027re given that 6 lights are mounted on a ceiling rail: 2 blue,"},{"Start":"00:04.320 ","End":"00:05.760","Text":"2 yellow, and 2 red."},{"Start":"00:05.760 ","End":"00:09.180","Text":"We\u0027re asked, in how many ways can the lights be ordered."},{"Start":"00:09.180 ","End":"00:11.373","Text":"Here, we have our ceiling rail."},{"Start":"00:11.373 ","End":"00:12.975","Text":"Here, we have our lights."},{"Start":"00:12.975 ","End":"00:15.540","Text":"We see that we have the 2 blue lights,"},{"Start":"00:15.540 ","End":"00:17.910","Text":"2 yellow lights and 2 red lights."},{"Start":"00:17.910 ","End":"00:21.300","Text":"All in all, we have 6 lights."},{"Start":"00:21.300 ","End":"00:24.300","Text":"Now if all the lights would have been a different color,"},{"Start":"00:24.300 ","End":"00:31.725","Text":"then we would have 6 factorial ways of ordering them on the ceiling rail, but we don\u0027t."},{"Start":"00:31.725 ","End":"00:33.390","Text":"We have 3 groups,"},{"Start":"00:33.390 ","End":"00:36.930","Text":"so r here, that equals to 3."},{"Start":"00:36.930 ","End":"00:41.750","Text":"Let\u0027s say the first group would be the blue lights."},{"Start":"00:41.750 ","End":"00:43.505","Text":"How many blue lights do I have?"},{"Start":"00:43.505 ","End":"00:45.610","Text":"Well, that equals to 2."},{"Start":"00:45.610 ","End":"00:48.245","Text":"How many yellow lights do I have?"},{"Start":"00:48.245 ","End":"00:50.765","Text":"That\u0027s the second group, that\u0027s 2."},{"Start":"00:50.765 ","End":"00:53.180","Text":"The third group, that\u0027s the red light,"},{"Start":"00:53.180 ","End":"00:54.980","Text":"how many red lights do I have?"},{"Start":"00:54.980 ","End":"01:00.365","Text":"Again, that\u0027s 2. How many permutations or how many ways can the lights be ordered?"},{"Start":"01:00.365 ","End":"01:08.510","Text":"That\u0027ll be 6 factorial divided by the number of internal orderings for each group."},{"Start":"01:08.510 ","End":"01:13.736","Text":"That\u0027ll be 2 factorial times 2 factorial times 2 factorial."},{"Start":"01:13.736 ","End":"01:18.090","Text":"Now, that comes out to 90."}],"ID":12804},{"Watched":false,"Name":"Exrcise 4","Duration":"1m 35s","ChapterTopicVideoID":12326,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12326.jpeg","UploadDate":"2018-07-22T16:46:16.1970000","DurationForVideoObject":"PT1M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this question, we\u0027re asked how many numbers can be formed under the following digits,"},{"Start":"00:03.990 ","End":"00:06.585","Text":"1, 2, 2, 2, and 6."},{"Start":"00:06.585 ","End":"00:09.040","Text":"First of all, we notice that we have"},{"Start":"00:09.040 ","End":"00:15.345","Text":"5 numbers and we also noticed that we have 3 groups, r equals to 3."},{"Start":"00:15.345 ","End":"00:19.335","Text":"The first group is a group that has the digit 1,"},{"Start":"00:19.335 ","End":"00:22.425","Text":"so how many ones do I have, I have 1."},{"Start":"00:22.425 ","End":"00:26.640","Text":"The second group is the group that has the number 2."},{"Start":"00:26.640 ","End":"00:28.260","Text":"How many 2s do I have?"},{"Start":"00:28.260 ","End":"00:29.955","Text":"Well, I have 3 2s."},{"Start":"00:29.955 ","End":"00:34.130","Text":"The third group is the group that has the 6s."},{"Start":"00:34.130 ","End":"00:36.890","Text":"How many 6s do I have? Well, I have 1."},{"Start":"00:36.890 ","End":"00:41.420","Text":"So how many numbers can I form out of these digits?"},{"Start":"00:41.420 ","End":"00:47.090","Text":"Well, we know that if we have groups where we only have 1 object in the group,"},{"Start":"00:47.090 ","End":"00:49.985","Text":"then we can disregard it. Why is that?"},{"Start":"00:49.985 ","End":"00:55.055","Text":"Well, the number of possible combinations would be 5 factorial"},{"Start":"00:55.055 ","End":"01:01.189","Text":"divided by 1 factorial times 3 factorial times 1 factorial."},{"Start":"01:01.189 ","End":"01:04.310","Text":"As we said, 1 factorial equals to 1,"},{"Start":"01:04.310 ","End":"01:08.915","Text":"so that\u0027ll be 5 factorial divided by 3 factorial,"},{"Start":"01:08.915 ","End":"01:12.390","Text":"and that comes out to 20."},{"Start":"01:12.390 ","End":"01:17.600","Text":"So I can create 20 different numbers out of these digits."},{"Start":"01:17.600 ","End":"01:24.110","Text":"For example, 12,226, I can form the number"},{"Start":"01:24.110 ","End":"01:33.125","Text":"16,222 or 61,222 and so on and so forth,"},{"Start":"01:33.125 ","End":"01:35.550","Text":"20 of these numbers."}],"ID":12805},{"Watched":false,"Name":"Exrcise 5","Duration":"2m 6s","ChapterTopicVideoID":12327,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12327.jpeg","UploadDate":"2018-07-22T16:46:21.4530000","DurationForVideoObject":"PT2M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"In a game of chance, there are 2 players and 10 squares."},{"Start":"00:03.225 ","End":"00:06.480","Text":"Now, player number 1 colors in 4 squares and"},{"Start":"00:06.480 ","End":"00:09.675","Text":"player number 2 needs to guess which 4 squares were colored."},{"Start":"00:09.675 ","End":"00:11.970","Text":"Now, we\u0027re asked, what\u0027s the probability that"},{"Start":"00:11.970 ","End":"00:15.615","Text":"player number 2 will guess all 4 squares in 1 try?"},{"Start":"00:15.615 ","End":"00:20.235","Text":"Here are 10 squares and player number 1 colors in 4 of them."},{"Start":"00:20.235 ","End":"00:22.125","Text":"Let\u0027s say he colors in this 1,"},{"Start":"00:22.125 ","End":"00:26.115","Text":"this 1, this 1, and this 1."},{"Start":"00:26.115 ","End":"00:31.920","Text":"Player number 2 needs to guess which 4 squares were colored in in 1 try."},{"Start":"00:31.920 ","End":"00:33.420","Text":"We have 10 squares."},{"Start":"00:33.420 ","End":"00:35.785","Text":"Now, if all the squares would have been different,"},{"Start":"00:35.785 ","End":"00:42.530","Text":"then the number of permutations of ordering these 10 squares would be 10 factorial."},{"Start":"00:42.530 ","End":"00:44.510","Text":"But the squares are the same."},{"Start":"00:44.510 ","End":"00:49.595","Text":"But they\u0027re not exactly the same because 4 of them are colored."},{"Start":"00:49.595 ","End":"00:51.785","Text":"We have here 2 groups,"},{"Start":"00:51.785 ","End":"00:55.200","Text":"r here equals to 2."},{"Start":"00:55.940 ","End":"01:01.970","Text":"How many squares are in the first group where the first group are the colored squares?"},{"Start":"01:01.970 ","End":"01:03.950","Text":"Well, I have 4 of them."},{"Start":"01:03.950 ","End":"01:05.765","Text":"In the second group,"},{"Start":"01:05.765 ","End":"01:08.730","Text":"the squares that aren\u0027t colored I have 6."},{"Start":"01:08.730 ","End":"01:14.910","Text":"How many ways then can I color in 4 squares out of a total of 10?"},{"Start":"01:14.910 ","End":"01:19.600","Text":"Well, that\u0027ll be 10 factorial divided by, now here,"},{"Start":"01:19.600 ","End":"01:22.430","Text":"I have to divide that by the number of"},{"Start":"01:22.430 ","End":"01:26.840","Text":"the permutations or the internal orderings for each 1 of the groups."},{"Start":"01:26.840 ","End":"01:31.100","Text":"That\u0027ll be 4 factorial divided by 6 factorial,"},{"Start":"01:31.100 ","End":"01:35.200","Text":"and that comes out to 210."},{"Start":"01:35.200 ","End":"01:39.110","Text":"This is the total number of permutations or"},{"Start":"01:39.110 ","End":"01:42.710","Text":"combinations of 4 squares that I can color in,"},{"Start":"01:42.710 ","End":"01:45.235","Text":"out of a total of 10 squares."},{"Start":"01:45.235 ","End":"01:47.630","Text":"We\u0027re asked what\u0027s the probability that"},{"Start":"01:47.630 ","End":"01:51.080","Text":"player number 2 will guess all 4 squares in 1 try?"},{"Start":"01:51.080 ","End":"01:59.790","Text":"That means that I have 1 out of a possible 210 combinations to choose from."},{"Start":"01:59.790 ","End":"02:06.540","Text":"This then would be the probability of getting the 4 squares correct."}],"ID":12806},{"Watched":false,"Name":"Exrcise 6","Duration":"2m 14s","ChapterTopicVideoID":12328,"CourseChapterTopicPlaylistID":264988,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12328.jpeg","UploadDate":"2018-07-22T16:46:26.8970000","DurationForVideoObject":"PT2M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.140","Text":"In this question we\u0027re asked,"},{"Start":"00:01.140 ","End":"00:02.595","Text":"how many different signals,"},{"Start":"00:02.595 ","End":"00:05.825","Text":"each composed of 10 flags arranged in a row,"},{"Start":"00:05.825 ","End":"00:08.960","Text":"can be created if 4 of the flags are white,"},{"Start":"00:08.960 ","End":"00:11.555","Text":"3 are blue, 2 are red, and 1 is black?"},{"Start":"00:11.555 ","End":"00:16.545","Text":"We assume that flags with the same color are completely identical."},{"Start":"00:16.545 ","End":"00:20.355","Text":"Here we have our 10 flags."},{"Start":"00:20.355 ","End":"00:22.755","Text":"We have 1, 2, 3, 4 white,"},{"Start":"00:22.755 ","End":"00:25.695","Text":"3 blue, 2 red, and 1 black."},{"Start":"00:25.695 ","End":"00:29.685","Text":"N here, that equals to 10."},{"Start":"00:29.685 ","End":"00:37.070","Text":"We also know that these 10 flags are split up into how many groups?"},{"Start":"00:37.070 ","End":"00:38.990","Text":"Well, 1, 2, 3, 4,"},{"Start":"00:38.990 ","End":"00:42.725","Text":"so r here, that equals to 4 groups."},{"Start":"00:42.725 ","End":"00:47.340","Text":"N_1, that\u0027s the white flags that equals to 4."},{"Start":"00:47.340 ","End":"00:51.600","Text":"N_2, that\u0027s the blue flags that equals to 3."},{"Start":"00:51.600 ","End":"00:55.125","Text":"N_3, that\u0027s the red flags that equals to 2."},{"Start":"00:55.125 ","End":"00:59.340","Text":"N_4, that\u0027s the black flag that equals to 1."},{"Start":"00:59.340 ","End":"01:01.110","Text":"We have 1 black flag."},{"Start":"01:01.110 ","End":"01:06.980","Text":"If all of the flags would have been of different colors,"},{"Start":"01:06.980 ","End":"01:08.845","Text":"then we would have had"},{"Start":"01:08.845 ","End":"01:14.855","Text":"10 factorial permutations or combinations of putting them in order, but we don\u0027t."},{"Start":"01:14.855 ","End":"01:18.770","Text":"We have here 10 flags which are grouped into"},{"Start":"01:18.770 ","End":"01:23.620","Text":"the different colors where the flags in each color are completely identical."},{"Start":"01:23.620 ","End":"01:31.220","Text":"That means that when we\u0027re asked about how many ways of ordering these flags,"},{"Start":"01:31.220 ","End":"01:33.590","Text":"I have the 10 factorial,"},{"Start":"01:33.590 ","End":"01:38.450","Text":"but I have to divide that by the internal orderings of each group."},{"Start":"01:38.450 ","End":"01:46.140","Text":"That\u0027ll be 4 factorial times 3 factorial times 2 factorial times 1 factorial."},{"Start":"01:46.140 ","End":"01:52.190","Text":"We said that if we have a category where we have only 1 object in that category,"},{"Start":"01:52.190 ","End":"01:55.085","Text":"in 1 group then we can disregard this."},{"Start":"01:55.085 ","End":"02:00.580","Text":"In any case, this equals to 12,600."},{"Start":"02:00.650 ","End":"02:08.285","Text":"That means I have 12,600 ways of ordering these flags here."},{"Start":"02:08.285 ","End":"02:13.740","Text":"I have 12,600 different signals."}],"ID":12807}],"Thumbnail":null,"ID":264988},{"Name":"Permutations with and without Returns","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"5m 13s","ChapterTopicVideoID":12329,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12329.jpeg","UploadDate":"2018-07-22T16:50:36.0470000","DurationForVideoObject":"PT5M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this chapter, I\u0027ll be talking about combinatorics,"},{"Start":"00:02.280 ","End":"00:05.625","Text":"specifically permutations with and without returns."},{"Start":"00:05.625 ","End":"00:08.160","Text":"Now, let\u0027s first talk about permutations of"},{"Start":"00:08.160 ","End":"00:12.075","Text":"ordered samplings with returns. Now, what is that?"},{"Start":"00:12.075 ","End":"00:18.479","Text":"We\u0027re looking at the number of ways to sample k objects from n different objects,"},{"Start":"00:18.479 ","End":"00:22.125","Text":"where the sampled objects are returned each time and"},{"Start":"00:22.125 ","End":"00:26.385","Text":"the order in which the objects are sampled is important."},{"Start":"00:26.385 ","End":"00:30.760","Text":"Well, that equals to n^k."},{"Start":"00:30.760 ","End":"00:34.925","Text":"Let\u0027s just take a look at an example here to make things a little bit clearer."},{"Start":"00:34.925 ","End":"00:37.760","Text":"In this example, we have 3 students that are selected"},{"Start":"00:37.760 ","End":"00:40.745","Text":"from a group of 10 students to be in a class committee."},{"Start":"00:40.745 ","End":"00:43.610","Text":"That committee has 3 different functions."},{"Start":"00:43.610 ","End":"00:45.560","Text":"For example, a Chairman,"},{"Start":"00:45.560 ","End":"00:48.140","Text":"Secretary and Assistant and we\u0027re also"},{"Start":"00:48.140 ","End":"00:51.800","Text":"told that a student can serve in more than 1 function."},{"Start":"00:51.800 ","End":"00:55.925","Text":"We\u0027re asked, how many different committees can be assembled or"},{"Start":"00:55.925 ","End":"01:01.554","Text":"how many groups of 3 students can be selected out of a group of 10 students?"},{"Start":"01:01.554 ","End":"01:03.320","Text":"Well, here again,"},{"Start":"01:03.320 ","End":"01:07.610","Text":"we\u0027re asked for the number of permutations when"},{"Start":"01:07.610 ","End":"01:12.329","Text":"we\u0027re sampling with returns and we have ordered sampling."},{"Start":"01:12.329 ","End":"01:15.740","Text":"Now, let\u0027s just write down the information."},{"Start":"01:15.740 ","End":"01:18.680","Text":"First of all, n here that equals to 10."},{"Start":"01:18.680 ","End":"01:21.890","Text":"That\u0027s the total number of students that we have to work with."},{"Start":"01:21.890 ","End":"01:24.390","Text":"We\u0027re sampling 3 students,"},{"Start":"01:24.390 ","End":"01:26.805","Text":"so k here that equals to 3."},{"Start":"01:26.805 ","End":"01:32.540","Text":"So that means that if we want to sample 3 students out"},{"Start":"01:32.540 ","End":"01:37.985","Text":"of a possible of 10 students and we have sampling with returns,"},{"Start":"01:37.985 ","End":"01:40.325","Text":"and we have ordered sampling,"},{"Start":"01:40.325 ","End":"01:44.465","Text":"then the number of permutations would be n^k,"},{"Start":"01:44.465 ","End":"01:50.140","Text":"or in our case 10^3 and that equals to 1,000."},{"Start":"01:50.140 ","End":"01:54.028","Text":"Let\u0027s just try to understand why this is so."},{"Start":"01:54.028 ","End":"01:55.970","Text":"Well, first of all,"},{"Start":"01:55.970 ","End":"02:01.514","Text":"let\u0027s see how we can understand that this is sampling with returns."},{"Start":"02:01.514 ","End":"02:03.064","Text":"Well, it says so right here;"},{"Start":"02:03.064 ","End":"02:06.515","Text":"a student can serve in more than 1 function."},{"Start":"02:06.515 ","End":"02:09.290","Text":"When we see this, what does that mean?"},{"Start":"02:09.290 ","End":"02:12.649","Text":"That means that once a student has been sampled,"},{"Start":"02:12.649 ","End":"02:19.855","Text":"he has to be placed back into the pool in order to get a chance to be chosen again."},{"Start":"02:19.855 ","End":"02:26.570","Text":"That means that we\u0027re returning the object that has already been sampled."},{"Start":"02:26.570 ","End":"02:29.465","Text":"That means that we\u0027re sampling with returns."},{"Start":"02:29.465 ","End":"02:33.875","Text":"The other parameter here is ordered sampling."},{"Start":"02:33.875 ","End":"02:36.890","Text":"Now, how do we know that we have ordered sampling?"},{"Start":"02:36.890 ","End":"02:39.365","Text":"Well, again, we have to choose a Chairman,"},{"Start":"02:39.365 ","End":"02:40.940","Text":"a Secretary, and an Assistant."},{"Start":"02:40.940 ","End":"02:43.280","Text":"Now let\u0027s say we choose 3 students."},{"Start":"02:43.280 ","End":"02:46.830","Text":"We have Tom, we have Jane,"},{"Start":"02:46.830 ","End":"02:49.875","Text":"and we have Mike."},{"Start":"02:49.875 ","End":"02:52.355","Text":"Now, when we chose Tom,"},{"Start":"02:52.355 ","End":"02:54.995","Text":"he was chosen for the position of Chairman,"},{"Start":"02:54.995 ","End":"02:56.885","Text":"the first person to be chosen."},{"Start":"02:56.885 ","End":"03:01.130","Text":"The second person to be chosen was Jane for the position of Secretary."},{"Start":"03:01.130 ","End":"03:04.880","Text":"Mike was the third to be chosen for the position of Assistant."},{"Start":"03:04.880 ","End":"03:11.325","Text":"Now, if I would have chosen Jane first and then Mike,"},{"Start":"03:11.325 ","End":"03:13.425","Text":"and then Tom,"},{"Start":"03:13.425 ","End":"03:15.945","Text":"then Jane would be Chairman,"},{"Start":"03:15.945 ","End":"03:17.880","Text":"Mike would be Secretary and Tom,"},{"Start":"03:17.880 ","End":"03:19.470","Text":"would be an Assistant."},{"Start":"03:19.470 ","End":"03:25.235","Text":"Here, there\u0027s importance to the order in which the students have been sampled."},{"Start":"03:25.235 ","End":"03:30.710","Text":"Here we have sampling with returns or ordered sampling with returns."},{"Start":"03:30.710 ","End":"03:32.570","Text":"Now, if we take a look at this,"},{"Start":"03:32.570 ","End":"03:37.415","Text":"this is a special case of the multiplication principle."},{"Start":"03:37.415 ","End":"03:44.915","Text":"Now if we recall, the multiplication principle says that we have a trial with k stages,"},{"Start":"03:44.915 ","End":"03:48.800","Text":"and we want to multiply all the options in"},{"Start":"03:48.800 ","End":"03:54.584","Text":"each stage in order to see how many options do we have in total."},{"Start":"03:54.584 ","End":"03:56.300","Text":"Here, in our case, well,"},{"Start":"03:56.300 ","End":"03:59.960","Text":"how many possible students can we get for"},{"Start":"03:59.960 ","End":"04:04.735","Text":"the position of Chairman where we have 10 possible students to choose from?"},{"Start":"04:04.735 ","End":"04:09.020","Text":"Because a student can serve in more than 1 function,"},{"Start":"04:09.020 ","End":"04:13.805","Text":"then how many students do we have to choose from for the position of Secretary?"},{"Start":"04:13.805 ","End":"04:15.530","Text":"Well, again, that\u0027s 10."},{"Start":"04:15.530 ","End":"04:18.590","Text":"How many students do we have to choose from for"},{"Start":"04:18.590 ","End":"04:22.575","Text":"the position of an Assistant? Well, again, that\u0027s 10."},{"Start":"04:22.575 ","End":"04:25.380","Text":"That equals to 10^3."},{"Start":"04:25.380 ","End":"04:30.470","Text":"Now, as I said, this is a special case of the multiplication principle. Why is that?"},{"Start":"04:30.470 ","End":"04:33.350","Text":"Because if I have a trial with k stages,"},{"Start":"04:33.350 ","End":"04:36.875","Text":"then the total number of combinations would be"},{"Start":"04:36.875 ","End":"04:41.870","Text":"n_1 times n_2 times and so and so on 4th times n_k,"},{"Start":"04:41.870 ","End":"04:43.890","Text":"where n_1, n_2,"},{"Start":"04:43.890 ","End":"04:51.544","Text":"and up to n_k would be the number of possible objects to choose from in each stage."},{"Start":"04:51.544 ","End":"04:54.675","Text":"Now, here, n_1,"},{"Start":"04:54.675 ","End":"04:57.450","Text":"that\u0027s 10, that would be equal to n_2."},{"Start":"04:57.450 ","End":"04:59.325","Text":"That would be equal to n_3."},{"Start":"04:59.325 ","End":"05:01.550","Text":"That\u0027s what makes this a special case,"},{"Start":"05:01.550 ","End":"05:04.650","Text":"where all the n\u0027s are the same."},{"Start":"05:04.650 ","End":"05:13.310","Text":"There we go. This is the answer to how many different committees can be assembled. 1,000."}],"ID":12808},{"Watched":false,"Name":"Toturial 2","Duration":"5m 16s","ChapterTopicVideoID":12330,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12330.jpeg","UploadDate":"2018-07-22T16:50:50.5470000","DurationForVideoObject":"PT5M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"The next subject that we\u0027ll be talking about is permutations of"},{"Start":"00:02.745 ","End":"00:05.745","Text":"ordered sampling now without returns."},{"Start":"00:05.745 ","End":"00:07.020","Text":"Again, what does that mean?"},{"Start":"00:07.020 ","End":"00:10.140","Text":"We\u0027re looking for the number of permutations,"},{"Start":"00:10.140 ","End":"00:14.714","Text":"the number of ways to sample k objects from n different objects,"},{"Start":"00:14.714 ","End":"00:20.780","Text":"now here n has to be greater or equal to k. Where the sample is ordered where"},{"Start":"00:20.780 ","End":"00:24.470","Text":"the order in which we sample the objects is"},{"Start":"00:24.470 ","End":"00:28.580","Text":"important and the object sampled are not returned."},{"Start":"00:28.580 ","End":"00:30.650","Text":"Not like the previous subject."},{"Start":"00:30.650 ","End":"00:34.730","Text":"Right here, the sample is a sampled objects are not returned."},{"Start":"00:34.730 ","End":"00:37.700","Text":"Well, that equals to this thing right here."},{"Start":"00:37.700 ","End":"00:41.920","Text":"Now, this is denoted as n_k."},{"Start":"00:41.920 ","End":"00:49.085","Text":"Now that equals to n times n minus 1 times n minus 2,"},{"Start":"00:49.085 ","End":"00:54.425","Text":"and so on and so forth times n minus k minus 1."},{"Start":"00:54.425 ","End":"01:02.900","Text":"Now, that equals to n factorial divided by n minus K factorial."},{"Start":"01:02.900 ","End":"01:05.600","Text":"Now, let\u0027s just take a look at an example,"},{"Start":"01:05.600 ","End":"01:07.040","Text":"and then we\u0027ll come back here."},{"Start":"01:07.040 ","End":"01:10.910","Text":"Here will be using the same example that as in the last section."},{"Start":"01:10.910 ","End":"01:14.420","Text":"Now we have 3 students that are selected from a group of 10 students to"},{"Start":"01:14.420 ","End":"01:17.930","Text":"be in a class committee that has 3 different functions,"},{"Start":"01:17.930 ","End":"01:19.730","Text":"chairman, secretary, and assistant."},{"Start":"01:19.730 ","End":"01:21.140","Text":"But here\u0027s the difference."},{"Start":"01:21.140 ","End":"01:24.710","Text":"A student cannot serve more than 1 function."},{"Start":"01:24.710 ","End":"01:28.210","Text":"We\u0027re asked how many different committees can be assembled."},{"Start":"01:28.210 ","End":"01:31.160","Text":"Again, if a student is chosen for chairman,"},{"Start":"01:31.160 ","End":"01:34.475","Text":"he can\u0027t be a secretary or an assistant."},{"Start":"01:34.475 ","End":"01:37.745","Text":"He can only serve in 1 function."},{"Start":"01:37.745 ","End":"01:42.410","Text":"If that\u0027s the case, let\u0027s use the multiplication principle."},{"Start":"01:42.410 ","End":"01:44.390","Text":"For the position of chairman,"},{"Start":"01:44.390 ","End":"01:46.700","Text":"well, how many students do have to choose from?"},{"Start":"01:46.700 ","End":"01:49.055","Text":"I have 10 students to choose from."},{"Start":"01:49.055 ","End":"01:51.800","Text":"Now, that student is taken."},{"Start":"01:51.800 ","End":"01:53.900","Text":"For the position of secretary,"},{"Start":"01:53.900 ","End":"01:55.730","Text":"how many students do I have to choose from?"},{"Start":"01:55.730 ","End":"01:57.725","Text":"Well, 9 students."},{"Start":"01:57.725 ","End":"02:00.650","Text":"For the position of assistant well,"},{"Start":"02:00.650 ","End":"02:02.940","Text":"once someone is taken for chairman,"},{"Start":"02:02.940 ","End":"02:04.890","Text":"1 is then was taken for secretary,"},{"Start":"02:04.890 ","End":"02:06.495","Text":"so 10 minus 2,"},{"Start":"02:06.495 ","End":"02:08.340","Text":"that equals to 8."},{"Start":"02:08.340 ","End":"02:13.445","Text":"We have 10 times 9 times 8 and that equals to 720."},{"Start":"02:13.445 ","End":"02:17.510","Text":"This is using the multiplication principle to"},{"Start":"02:17.510 ","End":"02:22.670","Text":"calculate the number of different committees that can be assembled."},{"Start":"02:22.670 ","End":"02:28.580","Text":"Now let\u0027s take a look at our equation from above."},{"Start":"02:28.580 ","End":"02:31.560","Text":"We have 10_3,"},{"Start":"02:31.560 ","End":"02:32.825","Text":"now, what does that equal to?"},{"Start":"02:32.825 ","End":"02:35.179","Text":"Well, according to the equation,"},{"Start":"02:35.179 ","End":"02:44.315","Text":"n_k well that equals to n times n minus 1 times n minus 2 times,"},{"Start":"02:44.315 ","End":"02:48.830","Text":"and so on and so forth times n minus k minus 1."},{"Start":"02:48.830 ","End":"02:51.765","Text":"Now, what does this?"},{"Start":"02:51.765 ","End":"02:54.155","Text":"When do we stop? What is this equal to?"},{"Start":"02:54.155 ","End":"02:59.150","Text":"Well, n here that equals to 10 and k that equals to 3,"},{"Start":"02:59.150 ","End":"03:01.095","Text":"so 10 minus,"},{"Start":"03:01.095 ","End":"03:02.370","Text":"well k minus 1,"},{"Start":"03:02.370 ","End":"03:03.990","Text":"that\u0027s 3 minus 1, that\u0027s 2,"},{"Start":"03:03.990 ","End":"03:06.345","Text":"so 10 minus 2, that equals to 8."},{"Start":"03:06.345 ","End":"03:11.630","Text":"We have to stop multiplying backwards when we reach 8."},{"Start":"03:11.630 ","End":"03:13.100","Text":"We start with 10."},{"Start":"03:13.100 ","End":"03:17.765","Text":"We start with n, 10 times 9 times 8,"},{"Start":"03:17.765 ","End":"03:20.780","Text":"and that also equals to 720."},{"Start":"03:20.780 ","End":"03:22.805","Text":"We see we have the same results."},{"Start":"03:22.805 ","End":"03:31.465","Text":"Now, this also equals to n factorial divided by n minus k factorial."},{"Start":"03:31.465 ","End":"03:38.755","Text":"In our case, there\u0027ll be 10 factorial divided by 10 minus 3, that\u0027s 7 factorial."},{"Start":"03:38.755 ","End":"03:40.735","Text":"What does that equal to?"},{"Start":"03:40.735 ","End":"03:50.380","Text":"Well, 10 factorial divided by 7 factorial and that equals to 1 times 2 times 3, 4, 5,"},{"Start":"03:50.380 ","End":"03:52.855","Text":"6, 7, 8, 9,"},{"Start":"03:52.855 ","End":"03:59.410","Text":"10 divided by 7 factorial 1 times 2 times 3 times 4,"},{"Start":"03:59.410 ","End":"04:01.090","Text":"5, 6, and 7."},{"Start":"04:01.090 ","End":"04:05.200","Text":"We can see that this area right here cancels each other"},{"Start":"04:05.200 ","End":"04:10.310","Text":"out and we\u0027re left with 8 times 9 times 10."},{"Start":"04:10.310 ","End":"04:16.120","Text":"Excellent. We have the same results, 720."},{"Start":"04:16.120 ","End":"04:19.010","Text":"Now, just as a side note,"},{"Start":"04:19.010 ","End":"04:27.915","Text":"we\u0027re using this type of notation here we have 10_3."},{"Start":"04:27.915 ","End":"04:35.015","Text":"That means that we want to know the number of different permutations that we have"},{"Start":"04:35.015 ","End":"04:41.750","Text":"by selecting 3 students or 3 objects from 10 different objects without returns."},{"Start":"04:41.750 ","End":"04:45.530","Text":"Now sometimes you can see this notation,"},{"Start":"04:45.530 ","End":"04:49.385","Text":"p, k up here and n down here."},{"Start":"04:49.385 ","End":"04:52.955","Text":"Now, p is for permutation n would be 10."},{"Start":"04:52.955 ","End":"04:55.475","Text":"In our case it would be 3 would be 3."},{"Start":"04:55.475 ","End":"04:58.910","Text":"We\u0027re looking here for the number of"},{"Start":"04:58.910 ","End":"05:03.755","Text":"permutations of choosing 3 objects from 10 different objects."},{"Start":"05:03.755 ","End":"05:07.144","Text":"I will not be using this notation."},{"Start":"05:07.144 ","End":"05:09.155","Text":"I\u0027ll be using this notation."},{"Start":"05:09.155 ","End":"05:11.840","Text":"It\u0027s much easier, much cleaner, much simpler."},{"Start":"05:11.840 ","End":"05:16.320","Text":"But again, don\u0027t be surprised if you see this in the literature."}],"ID":12809},{"Watched":false,"Name":"Exercise 1 - Parts a-b","Duration":"5m 43s","ChapterTopicVideoID":12331,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12331.jpeg","UploadDate":"2018-07-22T16:51:04.4430000","DurationForVideoObject":"PT5M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.015","Text":"In this question, we have a political party with 20 members."},{"Start":"00:03.015 ","End":"00:07.650","Text":"Now it\u0027s necessary to select 3 members for 3 different positions."},{"Start":"00:07.650 ","End":"00:11.745","Text":"Now, we\u0027re told that a member can serve in more than 1 position."},{"Start":"00:11.745 ","End":"00:17.145","Text":"We\u0027re asked how many combinations are there for distributing the positions?"},{"Start":"00:17.145 ","End":"00:19.890","Text":"First of all, we have 20 members,"},{"Start":"00:19.890 ","End":"00:23.160","Text":"so n here, that equals to 20."},{"Start":"00:23.160 ","End":"00:25.650","Text":"We have to select 3 members,"},{"Start":"00:25.650 ","End":"00:28.875","Text":"so k that equals to 3."},{"Start":"00:28.875 ","End":"00:31.230","Text":"The next thing that we want to do is,"},{"Start":"00:31.230 ","End":"00:34.515","Text":"we want to know what type of sampling do we have?"},{"Start":"00:34.515 ","End":"00:40.285","Text":"Well, here, it\u0027s obvious that we have ordered sampling with returns."},{"Start":"00:40.285 ","End":"00:42.080","Text":"Now, why is that?"},{"Start":"00:42.080 ","End":"00:48.200","Text":"Well, it\u0027s ordered sampling because we have 3 different positions and we\u0027re"},{"Start":"00:48.200 ","End":"00:54.605","Text":"placing sampled members into the different positions."},{"Start":"00:54.605 ","End":"00:59.900","Text":"Now it\u0027s with returns because we\u0027re told that a member can serve in more than 1 position."},{"Start":"00:59.900 ","End":"01:02.090","Text":"That means that once chosen,"},{"Start":"01:02.090 ","End":"01:06.560","Text":"he has to be placed back into the pool in order to be chosen again."},{"Start":"01:06.560 ","End":"01:12.260","Text":"Now, let\u0027s just use the multiplication principle to see how we can calculate this."},{"Start":"01:12.260 ","End":"01:14.300","Text":"For the first position,"},{"Start":"01:14.300 ","End":"01:16.970","Text":"how many options do I have?"},{"Start":"01:16.970 ","End":"01:19.505","Text":"Well, I have 20 members to choose from."},{"Start":"01:19.505 ","End":"01:24.515","Text":"Now, since a member can serve in more than 1 position,"},{"Start":"01:24.515 ","End":"01:27.770","Text":"there 20 members to choose from for"},{"Start":"01:27.770 ","End":"01:32.405","Text":"the second position and 20 members to choose from for the third position."},{"Start":"01:32.405 ","End":"01:38.965","Text":"That means we\u0027re looking at 20^3 and that equals to 8,000."},{"Start":"01:38.965 ","End":"01:46.610","Text":"Now let\u0027s take a look at the equation for ordered sampling with returns."},{"Start":"01:46.610 ","End":"01:51.485","Text":"Well, the number of permutations here would be n^k."},{"Start":"01:51.485 ","End":"01:56.565","Text":"Well, we know that n equals 20 and k equals to 3."},{"Start":"01:56.565 ","End":"02:02.180","Text":"That will be 20^3 and that equals to 8,000."},{"Start":"02:02.180 ","End":"02:06.740","Text":"In this section, we\u0027re told that a member can serve in only 1 position."},{"Start":"02:06.740 ","End":"02:12.050","Text":"We\u0027re asked how many combinations are there for distributing the positions."},{"Start":"02:12.050 ","End":"02:16.985","Text":"Here we have ordered sampling without returns."},{"Start":"02:16.985 ","End":"02:18.755","Text":"Now, why do I know that?"},{"Start":"02:18.755 ","End":"02:22.550","Text":"Well, it says so, a member can serve in only 1 position,"},{"Start":"02:22.550 ","End":"02:26.330","Text":"that means once he\u0027s been placed in a specific position,"},{"Start":"02:26.330 ","End":"02:28.805","Text":"he can\u0027t serve in another position."},{"Start":"02:28.805 ","End":"02:32.850","Text":"That\u0027s why we have without returns."},{"Start":"02:32.850 ","End":"02:34.950","Text":"Now, what\u0027s the notation here?"},{"Start":"02:34.950 ","End":"02:37.690","Text":"Well, that will be n_k,"},{"Start":"02:37.790 ","End":"02:42.465","Text":"and in our case that will be 20 sub 3."},{"Start":"02:42.465 ","End":"02:48.485","Text":"N here that equals to 20 and k that equals to 3."},{"Start":"02:48.485 ","End":"02:50.935","Text":"Now, before we calculate that,"},{"Start":"02:50.935 ","End":"02:53.040","Text":"let\u0027s just see how we can calculate"},{"Start":"02:53.040 ","End":"02:58.865","Text":"the number of combinations by using the multiplication principle."},{"Start":"02:58.865 ","End":"03:00.665","Text":"For the first position,"},{"Start":"03:00.665 ","End":"03:02.720","Text":"how many members can we choose from?"},{"Start":"03:02.720 ","End":"03:06.030","Text":"Well, we can choose 20 members."},{"Start":"03:06.030 ","End":"03:09.260","Text":"For the second position we can choose"},{"Start":"03:09.260 ","End":"03:15.170","Text":"only 19 members because that 1 member was already chosen for position number 1."},{"Start":"03:15.170 ","End":"03:17.540","Text":"For the third position,"},{"Start":"03:17.540 ","End":"03:21.165","Text":"we have 18 members to choose from."},{"Start":"03:21.165 ","End":"03:26.415","Text":"Here that equals to 6,840."},{"Start":"03:26.415 ","End":"03:29.835","Text":"Now let\u0027s get back to our notation here,"},{"Start":"03:29.835 ","End":"03:32.010","Text":"n sub k or 20 sub 3."},{"Start":"03:32.010 ","End":"03:37.170","Text":"Well, the equation is that n sub k,"},{"Start":"03:37.170 ","End":"03:42.635","Text":"that equals to n times n minus 1 times,"},{"Start":"03:42.635 ","End":"03:49.715","Text":"and so on and so forth until n minus k minus 1."},{"Start":"03:49.715 ","End":"03:56.720","Text":"Well, that\u0027s when we stop multiplying backwards from n. K equals to 3,"},{"Start":"03:56.720 ","End":"04:00.020","Text":"so k minus 1 equals 2, n is 20."},{"Start":"04:00.020 ","End":"04:01.640","Text":"So 20 minus 2,"},{"Start":"04:01.640 ","End":"04:03.730","Text":"that equals to 18."},{"Start":"04:03.730 ","End":"04:10.050","Text":"Here that equals to 20 times 19 times, and when do we stop?"},{"Start":"04:10.050 ","End":"04:13.425","Text":"We stop at 18. There we go."},{"Start":"04:13.425 ","End":"04:18.300","Text":"That\u0027s the same multiplication here."},{"Start":"04:18.300 ","End":"04:21.490","Text":"That equals to 6,840."},{"Start":"04:21.490 ","End":"04:26.100","Text":"Now, let\u0027s take a look at it from a factorial perspective."},{"Start":"04:26.100 ","End":"04:33.785","Text":"That equals to n factorial divided by n minus k factorial."},{"Start":"04:33.785 ","End":"04:42.090","Text":"That means that we\u0027re looking at 20 factorial divided by 20 minus 3, that\u0027s 17 factorial."},{"Start":"04:42.090 ","End":"04:45.045","Text":"If we take a look at this,"},{"Start":"04:45.045 ","End":"04:47.265","Text":"what\u0027s 20 factorial,"},{"Start":"04:47.265 ","End":"04:50.180","Text":"and what\u0027s 17 factorial?"},{"Start":"04:50.180 ","End":"04:53.825","Text":"Well, that equals to 1 times 2 times 3,"},{"Start":"04:53.825 ","End":"04:55.175","Text":"and so on and so forth,"},{"Start":"04:55.175 ","End":"05:01.935","Text":"times 16 times 17 times 18, 19, 20."},{"Start":"05:01.935 ","End":"05:05.700","Text":"In the denominator, we have 17 factorial."},{"Start":"05:05.700 ","End":"05:07.760","Text":"That\u0027s 1 times 2 times 3,"},{"Start":"05:07.760 ","End":"05:09.005","Text":"and so on and so forth,"},{"Start":"05:09.005 ","End":"05:11.780","Text":"times 16 times 17."},{"Start":"05:11.780 ","End":"05:14.600","Text":"All these cancels each other out."},{"Start":"05:14.600 ","End":"05:19.125","Text":"We\u0027re left with 18 times 19 times 20."},{"Start":"05:19.125 ","End":"05:22.710","Text":"Again, that equals to 6,840."},{"Start":"05:22.710 ","End":"05:27.170","Text":"Here, where we have ordered sampling without returns,"},{"Start":"05:27.170 ","End":"05:34.910","Text":"the number of combinations for distributing the position is 6,840."},{"Start":"05:34.910 ","End":"05:39.755","Text":"Whereas before, where we had ordered sampling with returns,"},{"Start":"05:39.755 ","End":"05:43.740","Text":"the number of combinations was 8,000."}],"ID":12810},{"Watched":false,"Name":"Exercise 2 - Parts a-c","Duration":"5m 40s","ChapterTopicVideoID":12332,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12332.jpeg","UploadDate":"2018-07-22T16:51:23.4330000","DurationForVideoObject":"PT5M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"In a game of chance, there are 4 boxes marked A to"},{"Start":"00:02.790 ","End":"00:06.870","Text":"D. Each box must be filled with a digit from 0-9,"},{"Start":"00:06.870 ","End":"00:11.985","Text":"and the winner is the 1 who guesses all the digits in all the boxes correctly."},{"Start":"00:11.985 ","End":"00:16.380","Text":"And we\u0027re asked, what\u0027s the probability of winning the game?"},{"Start":"00:16.380 ","End":"00:18.930","Text":"Here are 4 boxes,"},{"Start":"00:18.930 ","End":"00:20.430","Text":"A, B, C, and D,"},{"Start":"00:20.430 ","End":"00:22.650","Text":"and we have to put into each box,"},{"Start":"00:22.650 ","End":"00:24.345","Text":"a digit from 0-9."},{"Start":"00:24.345 ","End":"00:27.180","Text":"An example could be 1, 2, 3,"},{"Start":"00:27.180 ","End":"00:29.565","Text":"and 4 or 5,"},{"Start":"00:29.565 ","End":"00:31.290","Text":"2, 7,"},{"Start":"00:31.290 ","End":"00:34.495","Text":"7, and so on and so forth."},{"Start":"00:34.495 ","End":"00:38.430","Text":"When we\u0027re asked what\u0027s the probability of winning the game of"},{"Start":"00:38.430 ","End":"00:42.330","Text":"guessing all the numbers in the 4 boxes,"},{"Start":"00:42.330 ","End":"00:48.590","Text":"then the first thing that I want to do is guess or calculate my sample size,"},{"Start":"00:48.590 ","End":"00:51.130","Text":"the size of my sample space."},{"Start":"00:51.130 ","End":"00:53.280","Text":"What does that equal to?"},{"Start":"00:53.280 ","End":"00:55.115","Text":"In order to answer that,"},{"Start":"00:55.115 ","End":"01:01.525","Text":"I need to think of this as ordered sampling with returns."},{"Start":"01:01.525 ","End":"01:04.130","Text":"Why is that? Well, first of all,"},{"Start":"01:04.130 ","End":"01:06.454","Text":"why is it ordered sampling?"},{"Start":"01:06.454 ","End":"01:13.700","Text":"Let\u0027s say I take the number 9,8,7,6."},{"Start":"01:13.700 ","End":"01:18.680","Text":"I can choose the same digits but in different order."},{"Start":"01:18.680 ","End":"01:22.595","Text":"I can have 8,6,7, and 9."},{"Start":"01:22.595 ","End":"01:25.745","Text":"The same digits but in the wrong order."},{"Start":"01:25.745 ","End":"01:27.830","Text":"So therefore, I won\u0027t be winning."},{"Start":"01:27.830 ","End":"01:33.120","Text":"I want to guess this number and not this number."},{"Start":"01:33.120 ","End":"01:35.930","Text":"That means that we have ordered sampling."},{"Start":"01:35.930 ","End":"01:37.835","Text":"The order here is important,"},{"Start":"01:37.835 ","End":"01:41.105","Text":"and its with returns. Why is that?"},{"Start":"01:41.105 ","End":"01:45.725","Text":"Because it can have the same number in the different boxes."},{"Start":"01:45.725 ","End":"01:51.290","Text":"What\u0027s the probability of winning?"},{"Start":"01:51.290 ","End":"01:55.640","Text":"As I said, we have to calculate the sample space,"},{"Start":"01:55.640 ","End":"01:57.925","Text":"the size of the sample space."},{"Start":"01:57.925 ","End":"02:01.610","Text":"When we have ordered sampling with returns,"},{"Start":"02:01.610 ","End":"02:09.930","Text":"then the total number of permutations here would be n to the power of k. Now,"},{"Start":"02:09.930 ","End":"02:14.610","Text":"n here, that equals to 10."},{"Start":"02:14.610 ","End":"02:18.785","Text":"We have here 10 numbers, and what\u0027s k?"},{"Start":"02:18.785 ","End":"02:22.160","Text":"K is the numbers that we have to choose."},{"Start":"02:22.160 ","End":"02:25.085","Text":"We\u0027re sampling 4 numbers."},{"Start":"02:25.085 ","End":"02:27.245","Text":"As we said in our case,"},{"Start":"02:27.245 ","End":"02:33.250","Text":"that\u0027s 10^4 and that equals to 10,000."},{"Start":"02:33.250 ","End":"02:35.600","Text":"What\u0027s the probability of winning?"},{"Start":"02:35.600 ","End":"02:40.430","Text":"I have to choose the exact series of numbers in"},{"Start":"02:40.430 ","End":"02:46.085","Text":"the same order and there\u0027s only 1 possibility to choose from."},{"Start":"02:46.085 ","End":"02:52.340","Text":"So the probability of winning would be 1 divided by 10^4,"},{"Start":"02:52.340 ","End":"02:56.510","Text":"or 1 divided by 10,000."},{"Start":"02:56.510 ","End":"03:00.080","Text":"In this section, we\u0027re asked what\u0027s the probability that the digit"},{"Start":"03:00.080 ","End":"03:03.909","Text":"3 does not appear in any box in the winning number,"},{"Start":"03:03.909 ","End":"03:11.715","Text":"so we\u0027re looking for the probability that there\u0027s no digit 3 in these 4 boxes."},{"Start":"03:11.715 ","End":"03:13.460","Text":"In the denominator,"},{"Start":"03:13.460 ","End":"03:18.140","Text":"we have the size of the sample space which is 10-the power of 4."},{"Start":"03:18.140 ","End":"03:20.155","Text":"What\u0027s in the numerator?"},{"Start":"03:20.155 ","End":"03:21.440","Text":"Well, in the numerator,"},{"Start":"03:21.440 ","End":"03:29.060","Text":"we have to see how many numbers I can create where I don\u0027t have the digit 3."},{"Start":"03:29.060 ","End":"03:34.009","Text":"We\u0027re still looking at ordered sampling with returns."},{"Start":"03:34.009 ","End":"03:37.760","Text":"But now n here that equals to 9,"},{"Start":"03:37.760 ","End":"03:42.425","Text":"because now I have only 9 digits to choose from instead of 10,"},{"Start":"03:42.425 ","End":"03:45.800","Text":"and k that equals to 4,"},{"Start":"03:45.800 ","End":"03:48.215","Text":"because I still have the 4 boxes."},{"Start":"03:48.215 ","End":"03:52.085","Text":"That would be 9 to the power of 4."},{"Start":"03:52.085 ","End":"03:55.850","Text":"This then would be the probability that I do"},{"Start":"03:55.850 ","End":"04:00.244","Text":"not have the digit 3 in any 1 of the 4 boxes."},{"Start":"04:00.244 ","End":"04:03.020","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"04:03.020 ","End":"04:06.140","Text":"the digit 4 appears at least once in the winning number."},{"Start":"04:06.140 ","End":"04:07.970","Text":"For example, I have 1,"},{"Start":"04:07.970 ","End":"04:10.415","Text":"4, 2, and 3."},{"Start":"04:10.415 ","End":"04:12.795","Text":"I have 4,"},{"Start":"04:12.795 ","End":"04:15.810","Text":"5, 4, 5."},{"Start":"04:15.810 ","End":"04:17.430","Text":"We have let\u0027s say 4,"},{"Start":"04:17.430 ","End":"04:18.630","Text":"4, 4,"},{"Start":"04:18.630 ","End":"04:21.360","Text":"1 and so on and so forth."},{"Start":"04:21.360 ","End":"04:24.945","Text":"We\u0027re looking for this probability."},{"Start":"04:24.945 ","End":"04:29.020","Text":"The probability that I have the digit 4 at least once."},{"Start":"04:29.020 ","End":"04:34.240","Text":"As I said, this is very difficult to calculate because we have a lot of these numbers."},{"Start":"04:34.240 ","End":"04:43.320","Text":"It\u0027ll be easy for us to calculate the probability of the complimentary set."},{"Start":"04:43.320 ","End":"04:49.135","Text":"The probability of having the digit 4 at least once in the number here."},{"Start":"04:49.135 ","End":"04:57.530","Text":"That would be equal to 1 minus the probability that there\u0027s no digit 4."},{"Start":"04:57.530 ","End":"04:59.370","Text":"What does that equal to?"},{"Start":"04:59.370 ","End":"05:02.290","Text":"Well, that\u0027s 1 minus this probability,"},{"Start":"05:02.290 ","End":"05:08.440","Text":"but this is the same probability as not having the digit 3 here."},{"Start":"05:09.620 ","End":"05:15.635","Text":"In the denominator, we have the size of our sample space that\u0027s 10^4,"},{"Start":"05:15.635 ","End":"05:19.270","Text":"and when I don\u0027t have any 4,"},{"Start":"05:19.270 ","End":"05:22.370","Text":"that means that I have 9 digits to choose from"},{"Start":"05:22.370 ","End":"05:26.090","Text":"instead of 10 digits to choose from for each 1 of the boxes."},{"Start":"05:26.090 ","End":"05:29.120","Text":"That means that will be 9^4,"},{"Start":"05:29.120 ","End":"05:34.580","Text":"so 1 minus 9^4 divided by 10^4."},{"Start":"05:34.580 ","End":"05:41.280","Text":"That\u0027s the probability that the digit 4 appears at least once."}],"ID":12811},{"Watched":false,"Name":"Exercise 3","Duration":"3m 27s","ChapterTopicVideoID":12333,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12333.jpeg","UploadDate":"2018-07-22T16:51:31.3630000","DurationForVideoObject":"PT3M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this question, we\u0027re given 22 people in a group,"},{"Start":"00:03.240 ","End":"00:08.445","Text":"and we\u0027re asked what\u0027s the probability that at least 2 of them have the same birthday?"},{"Start":"00:08.445 ","End":"00:11.910","Text":"We\u0027re looking for this probability right here."},{"Start":"00:11.910 ","End":"00:14.115","Text":"Now, what does that equal to?"},{"Start":"00:14.115 ","End":"00:19.890","Text":"Well, the size of this event that at least 2 people have the same birthday,"},{"Start":"00:19.890 ","End":"00:22.475","Text":"that\u0027s very complicated to calculate."},{"Start":"00:22.475 ","End":"00:24.670","Text":"It\u0027ll be better for us to go,"},{"Start":"00:24.670 ","End":"00:29.745","Text":"and calculate this probability through the complementary set."},{"Start":"00:29.745 ","End":"00:36.015","Text":"This probability is equal to 1 minus the probability of the complementary set,"},{"Start":"00:36.015 ","End":"00:43.060","Text":"and the complementary set is that all the people have different birthdays."},{"Start":"00:43.280 ","End":"00:46.339","Text":"This is what we want to calculate."},{"Start":"00:46.339 ","End":"00:51.365","Text":"Now in calculating this probability right here,"},{"Start":"00:51.365 ","End":"00:55.550","Text":"let\u0027s think of ordered sample with returns."},{"Start":"00:55.550 ","End":"01:00.655","Text":"We want to know the total number of permutations,"},{"Start":"01:00.655 ","End":"01:04.220","Text":"from an ordered sample with return. Now, why is that?"},{"Start":"01:04.220 ","End":"01:09.320","Text":"Well, we have 365 days to choose from,"},{"Start":"01:09.320 ","End":"01:14.690","Text":"so that\u0027ll be our n and we want to calculate 22 dates,"},{"Start":"01:14.690 ","End":"01:17.975","Text":"so k here would be 22 dates."},{"Start":"01:17.975 ","End":"01:23.060","Text":"What is the total number of permutations of calculating"},{"Start":"01:23.060 ","End":"01:28.920","Text":"or choosing 22 dates out of 365 days?"},{"Start":"01:28.920 ","End":"01:36.810","Text":"Well, we know that that\u0027s n^k and that equals to 365^22."},{"Start":"01:36.810 ","End":"01:39.380","Text":"Getting back to our probability,"},{"Start":"01:39.380 ","End":"01:41.525","Text":"that\u0027ll be equal to 1 minus,"},{"Start":"01:41.525 ","End":"01:43.400","Text":"and in the denominator,"},{"Start":"01:43.400 ","End":"01:46.670","Text":"we have the size of our sample space."},{"Start":"01:46.670 ","End":"01:49.430","Text":"Now our size of a sample space would be the total number of"},{"Start":"01:49.430 ","End":"01:52.685","Text":"permutations of choosing 22 dates"},{"Start":"01:52.685 ","End":"01:59.525","Text":"from 365 days, and that\u0027s 365^22."},{"Start":"01:59.525 ","End":"02:06.830","Text":"Now, let\u0027s take a look at the numerator that all the people now have different birthdays."},{"Start":"02:06.830 ","End":"02:12.850","Text":"If the first person has a birth date on 1 day,"},{"Start":"02:12.850 ","End":"02:16.130","Text":"there are 365 days to choose from,"},{"Start":"02:16.130 ","End":"02:22.130","Text":"but the second person has 364 days to choose from,"},{"Start":"02:22.130 ","End":"02:24.410","Text":"and the 3rd person,"},{"Start":"02:24.410 ","End":"02:29.270","Text":"his birthday has to be 1 out of 363 days,"},{"Start":"02:29.270 ","End":"02:30.898","Text":"and so on, and so forth."},{"Start":"02:30.898 ","End":"02:38.660","Text":"In all, we have 365 times 364 times 363, 22 times."},{"Start":"02:38.660 ","End":"02:44.660","Text":"I multiply the numbers from 365 backward, 22 times,"},{"Start":"02:44.660 ","End":"02:53.340","Text":"but that\u0027s exactly the definition of 365_22."},{"Start":"02:54.020 ","End":"02:59.705","Text":"This is how we choose different days without returns."},{"Start":"02:59.705 ","End":"03:05.630","Text":"This is the probability of at least 2 people that have the same birthday,"},{"Start":"03:05.630 ","End":"03:11.870","Text":"that\u0027s 1 minus the probability that all the people have different birthdays."},{"Start":"03:11.870 ","End":"03:13.910","Text":"Now, let\u0027s just calculate that."},{"Start":"03:13.910 ","End":"03:19.040","Text":"That comes out to 0.476 approximately."},{"Start":"03:19.040 ","End":"03:22.550","Text":"There we go, this is the probability that"},{"Start":"03:22.550 ","End":"03:28.200","Text":"at least 2 people have the same birthday in a group of 22 people."}],"ID":12812},{"Watched":false,"Name":"Exercise 4 - Parts a-b","Duration":"3m 33s","ChapterTopicVideoID":12334,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12334.jpeg","UploadDate":"2018-07-22T16:51:39.3330000","DurationForVideoObject":"PT3M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this question, we\u0027re given 3 people that arranged to meet"},{"Start":"00:02.670 ","End":"00:05.655","Text":"each other at the Hilton Hotel in Singapore."},{"Start":"00:05.655 ","End":"00:10.190","Text":"Now, unfortunately, there are 5 Hilton Hotels in Singapore."},{"Start":"00:10.190 ","End":"00:11.775","Text":"We\u0027re asked and section a,"},{"Start":"00:11.775 ","End":"00:16.425","Text":"what\u0027s the probability of all 3 people meeting at the same hotel?"},{"Start":"00:16.425 ","End":"00:18.920","Text":"Whenever I\u0027m asked about probabilities,"},{"Start":"00:18.920 ","End":"00:23.930","Text":"the first thing that I want to do is calculate the size of my sample space."},{"Start":"00:23.930 ","End":"00:28.235","Text":"I\u0027m looking for this quantity right here."},{"Start":"00:28.235 ","End":"00:30.170","Text":"Now, let\u0027s see what I have here."},{"Start":"00:30.170 ","End":"00:33.650","Text":"First of all, I have 5 Hilton Hotels,"},{"Start":"00:33.650 ","End":"00:35.780","Text":"so this will be the total number of"},{"Start":"00:35.780 ","End":"00:40.775","Text":"hotels that are in Singapore and I have to choose 3 hotels out of the 5."},{"Start":"00:40.775 ","End":"00:43.490","Text":"So k here that equals to 3, and why is that?"},{"Start":"00:43.490 ","End":"00:49.430","Text":"Because 3 people choose 3 hotels out of the possible 5 hotels."},{"Start":"00:49.430 ","End":"00:54.830","Text":"How many possible combinations of choosing 3 hotel side of 5?"},{"Start":"00:54.830 ","End":"00:58.745","Text":"Well, let\u0027s take a look at this as"},{"Start":"00:58.745 ","End":"01:05.239","Text":"calculating the number of permutations from an ordered sample with returns."},{"Start":"01:05.239 ","End":"01:06.680","Text":"Now, why is that?"},{"Start":"01:06.680 ","End":"01:10.490","Text":"Well, it\u0027s an ordered sample because it\u0027s important to know"},{"Start":"01:10.490 ","End":"01:15.215","Text":"who out of the 3 people shows which hotel,"},{"Start":"01:15.215 ","End":"01:20.030","Text":"and its with returns because 2 people can choose the same hotel."},{"Start":"01:20.030 ","End":"01:22.310","Text":"Now, since we have this,"},{"Start":"01:22.310 ","End":"01:26.960","Text":"then the number of permutations or possible combinations is n"},{"Start":"01:26.960 ","End":"01:32.085","Text":"to the power of k. Now in our case that\u0027s 5 to the power of 3."},{"Start":"01:32.085 ","End":"01:36.950","Text":"The size of the sample space is 5 to the power of 3."},{"Start":"01:36.950 ","End":"01:45.110","Text":"Now we\u0027re asked about the probability that all the people meet at the same hotel."},{"Start":"01:45.110 ","End":"01:49.565","Text":"Well, in the denominator we have 5 to the power of 3."},{"Start":"01:49.565 ","End":"01:55.205","Text":"We have the total number of combinations of choosing 3 hotels out of 5."},{"Start":"01:55.205 ","End":"01:57.530","Text":"Now, if all the people meet,"},{"Start":"01:57.530 ","End":"02:03.270","Text":"then let\u0027s say the first person chooses 1 out of the 5 hotels."},{"Start":"02:03.270 ","End":"02:07.310","Text":"He has 5 possibilities to choose from."},{"Start":"02:07.310 ","End":"02:09.320","Text":"Now, the second person,"},{"Start":"02:09.320 ","End":"02:11.420","Text":"he doesn\u0027t have any degrees of freedom."},{"Start":"02:11.420 ","End":"02:13.220","Text":"He has to choose that hotel,"},{"Start":"02:13.220 ","End":"02:16.450","Text":"and the same goes for the third person."},{"Start":"02:16.450 ","End":"02:19.865","Text":"We have 5 divided by 5 cubed,"},{"Start":"02:19.865 ","End":"02:25.670","Text":"and that equals to 1 divided by 5 squared or 1 divided by 25."},{"Start":"02:25.670 ","End":"02:31.265","Text":"So that\u0027s the probability that all 3 people will meet at the same hotel."},{"Start":"02:31.265 ","End":"02:34.025","Text":"In this section, we\u0027re asked what\u0027s the probability if"},{"Start":"02:34.025 ","End":"02:37.130","Text":"all 3 people going to different hotels?"},{"Start":"02:37.130 ","End":"02:41.580","Text":"We\u0027re looking at this probability right here."},{"Start":"02:43.120 ","End":"02:45.695","Text":"What\u0027s in the denominator?"},{"Start":"02:45.695 ","End":"02:49.310","Text":"Well, in the denominator we have the size of the sample space with"},{"Start":"02:49.310 ","End":"02:54.050","Text":"the total number of combinations of choosing 3 hotels at 5,"},{"Start":"02:54.050 ","End":"02:56.375","Text":"so that\u0027ll be 5 to the power of 3."},{"Start":"02:56.375 ","End":"02:59.525","Text":"Now, in the numerator, well,"},{"Start":"02:59.525 ","End":"03:05.315","Text":"this is just like an ordered sample without returns. Now why is that?"},{"Start":"03:05.315 ","End":"03:08.030","Text":"Well, if the first person chooses the hotel,"},{"Start":"03:08.030 ","End":"03:10.520","Text":"he has 5 hotels to choose from."},{"Start":"03:10.520 ","End":"03:14.615","Text":"But the second person has 4 hotels to choose from,"},{"Start":"03:14.615 ","End":"03:18.500","Text":"and the third person has 3 hotels to choose from."},{"Start":"03:18.500 ","End":"03:26.615","Text":"Well, that\u0027s exactly the definition of 5 sub 3 and that\u0027s divided by 5 to the power of 3."},{"Start":"03:26.615 ","End":"03:31.350","Text":"This equals to 0.48."}],"ID":12813},{"Watched":false,"Name":"Exercise 5 - Parts a-b","Duration":"2m 41s","ChapterTopicVideoID":12335,"CourseChapterTopicPlaylistID":264989,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12335.jpeg","UploadDate":"2018-07-22T16:51:45.5630000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this question, we\u0027re given that there are 40 students in a class."},{"Start":"00:03.210 ","End":"00:06.030","Text":"Now, 5 must be chosen for a class committee."},{"Start":"00:06.030 ","End":"00:11.145","Text":"Now, we\u0027re asked, in how many different ways can the committee be selected if"},{"Start":"00:11.145 ","End":"00:16.905","Text":"the committee has 5 different positions and a student can serve in more than 1 position?"},{"Start":"00:16.905 ","End":"00:23.560","Text":"Well, here we have an ordered sampling with returns."},{"Start":"00:23.560 ","End":"00:25.565","Text":"Now, why is that?"},{"Start":"00:25.565 ","End":"00:29.630","Text":"Well, first of all, it\u0027s with returns because we\u0027re told that"},{"Start":"00:29.630 ","End":"00:34.100","Text":"a student can serve in more than 1 position."},{"Start":"00:34.100 ","End":"00:36.994","Text":"This deals with the returns."},{"Start":"00:36.994 ","End":"00:39.005","Text":"Now, why is it an ordered sampling?"},{"Start":"00:39.005 ","End":"00:44.360","Text":"Well, we know it\u0027s an ordered sampling because we have 5 different positions."},{"Start":"00:44.360 ","End":"00:50.390","Text":"That means that the order in which the people are sampled, that\u0027s important."},{"Start":"00:50.390 ","End":"00:52.865","Text":"If Joe is sampled 1st,"},{"Start":"00:52.865 ","End":"00:57.170","Text":"then he goes to the 1st position and Tom goes to the 2nd position,"},{"Start":"00:57.170 ","End":"01:00.020","Text":"and it\u0027ll be completely different if Tom goes to"},{"Start":"01:00.020 ","End":"01:03.200","Text":"the 1st position and Joe goes to the 2nd position."},{"Start":"01:03.200 ","End":"01:06.335","Text":"So this is an ordered sampling with returns."},{"Start":"01:06.335 ","End":"01:10.280","Text":"Now, how many students are there in a class of 40?"},{"Start":"01:10.280 ","End":"01:13.460","Text":"So that means that n here that equals to 40,"},{"Start":"01:13.460 ","End":"01:16.790","Text":"and we\u0027re sampling 5 students,"},{"Start":"01:16.790 ","End":"01:18.170","Text":"so k here,"},{"Start":"01:18.170 ","End":"01:20.210","Text":"that equals to 5."},{"Start":"01:20.210 ","End":"01:24.530","Text":"So we know that when we have an ordered sampling with returns,"},{"Start":"01:24.530 ","End":"01:29.735","Text":"then the number of combinations of choosing 5 out of 40 students,"},{"Start":"01:29.735 ","End":"01:32.915","Text":"well that equals to n to the power of k,"},{"Start":"01:32.915 ","End":"01:36.515","Text":"or in our case 40 to the power of 5."},{"Start":"01:36.515 ","End":"01:40.519","Text":"In this section, we\u0027re told that the committee has 5 different positions,"},{"Start":"01:40.519 ","End":"01:44.795","Text":"and a student cannot serve in more than 1 position."},{"Start":"01:44.795 ","End":"01:51.290","Text":"Well, this is an ordered sampling without returns, and why is that?"},{"Start":"01:51.290 ","End":"01:58.670","Text":"Well because the student can\u0027t serve on more than 1 position."},{"Start":"01:58.670 ","End":"02:01.370","Text":"Now, that means that if he\u0027s chosen,"},{"Start":"02:01.370 ","End":"02:05.210","Text":"he\u0027s not put back into the pool in order to be chosen again."},{"Start":"02:05.210 ","End":"02:09.300","Text":"So again, we have n equals to 40,"},{"Start":"02:09.300 ","End":"02:11.795","Text":"that\u0027s the total number of students to choose from,"},{"Start":"02:11.795 ","End":"02:16.340","Text":"and k equals to 5 because we have 5 positions to fill,"},{"Start":"02:16.340 ","End":"02:24.655","Text":"and the total number of combinations is 40 sub 5."},{"Start":"02:24.655 ","End":"02:26.970","Text":"Now, this is a very big number."},{"Start":"02:26.970 ","End":"02:34.100","Text":"It comes out to 78,960,960."},{"Start":"02:34.100 ","End":"02:42.000","Text":"This is a total number of combinations of choosing 5 students out of 40 without returns."}],"ID":12814}],"Thumbnail":null,"ID":264989},{"Name":"Non-Sequential Sampling without Returns","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"1m 47s","ChapterTopicVideoID":12336,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12336.jpeg","UploadDate":"2018-07-22T16:54:12.3430000","DurationForVideoObject":"PT1M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this chapter, I\u0027ll be talking about combinatorics,"},{"Start":"00:02.565 ","End":"00:06.270","Text":"specifically, non-sequential sampling without returns."},{"Start":"00:06.270 ","End":"00:10.050","Text":"Or we can call this an unordered sample without returns."},{"Start":"00:10.050 ","End":"00:12.330","Text":"Now, what\u0027s the definition here?"},{"Start":"00:12.330 ","End":"00:15.390","Text":"Well, we\u0027re looking to calculate the number of"},{"Start":"00:15.390 ","End":"00:18.660","Text":"possibilities for sampling k different objects"},{"Start":"00:18.660 ","End":"00:21.210","Text":"from n different objects when the order of"},{"Start":"00:21.210 ","End":"00:25.305","Text":"the sampled objects is non-important and there are no returns."},{"Start":"00:25.305 ","End":"00:27.015","Text":"Now, if we recall,"},{"Start":"00:27.015 ","End":"00:28.380","Text":"in the last chapter,"},{"Start":"00:28.380 ","End":"00:31.970","Text":"we were looking at ordered sampling without returns and"},{"Start":"00:31.970 ","End":"00:36.800","Text":"the number of possibilities there was n_k."},{"Start":"00:36.800 ","End":"00:44.870","Text":"Now that equal to n factorial divided by n minus k factorial."},{"Start":"00:44.870 ","End":"00:48.450","Text":"Now, we have an unordered sampling."},{"Start":"00:48.450 ","End":"00:49.895","Text":"What do I need to do?"},{"Start":"00:49.895 ","End":"00:54.170","Text":"Well, I need to take this n_k."},{"Start":"00:54.170 ","End":"00:56.570","Text":"Now, I have to neutralize"},{"Start":"00:56.570 ","End":"01:01.520","Text":"the internal ordering of the sampled objects because they\u0027re not important now."},{"Start":"01:01.520 ","End":"01:07.100","Text":"I have to divide this by k factorial. What do I get?"},{"Start":"01:07.100 ","End":"01:16.185","Text":"Well, I get n factorial divided by n minus k factorial, that\u0027s the n_k."},{"Start":"01:16.185 ","End":"01:20.795","Text":"But then I have to add here the k factorial here."},{"Start":"01:20.795 ","End":"01:22.250","Text":"Now, as I said,"},{"Start":"01:22.250 ","End":"01:27.440","Text":"this is equal to n over k. This is the notation that I\u0027ll be using."},{"Start":"01:27.440 ","End":"01:34.640","Text":"This is how you represent the number of possibilities for sampling k different objects"},{"Start":"01:34.640 ","End":"01:37.340","Text":"from n different objects when the order of"},{"Start":"01:37.340 ","End":"01:42.480","Text":"the sampled objects is not important and there are no returns."},{"Start":"01:43.040 ","End":"01:47.970","Text":"Let\u0027s just look at an example to see how we can use this."}],"ID":12815},{"Watched":false,"Name":"Tutorial 2","Duration":"2m 20s","ChapterTopicVideoID":12337,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12337.jpeg","UploadDate":"2018-07-22T16:54:18.0330000","DurationForVideoObject":"PT2M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In this example, we\u0027re asked,"},{"Start":"00:01.260 ","End":"00:05.760","Text":"how many combinations do we have of choosing 3 out of 10 students for a committee,"},{"Start":"00:05.760 ","End":"00:09.135","Text":"where the committee has no defined position to fill?"},{"Start":"00:09.135 ","End":"00:10.890","Text":"Well, first of all,"},{"Start":"00:10.890 ","End":"00:14.250","Text":"we know that we have 10 students,"},{"Start":"00:14.250 ","End":"00:16.080","Text":"so n here equals to 10,"},{"Start":"00:16.080 ","End":"00:18.000","Text":"and we have to sample 3,"},{"Start":"00:18.000 ","End":"00:20.265","Text":"so k is 3."},{"Start":"00:20.265 ","End":"00:23.475","Text":"Now, we\u0027re in a situation where we have"},{"Start":"00:23.475 ","End":"00:28.710","Text":"non-ordered sampling without returns. Why is that?"},{"Start":"00:28.710 ","End":"00:30.960","Text":"Well, first of all,"},{"Start":"00:30.960 ","End":"00:34.320","Text":"it\u0027s non-ordered sampling because it really doesn\u0027t"},{"Start":"00:34.320 ","End":"00:38.780","Text":"matter in which order we choose the 3 students."},{"Start":"00:38.780 ","End":"00:42.450","Text":"We\u0027re not shooting them for a specific position."},{"Start":"00:42.450 ","End":"00:45.050","Text":"There\u0027s no predefined position to fill."},{"Start":"00:45.050 ","End":"00:47.090","Text":"We\u0027re not choosing the first one to be"},{"Start":"00:47.090 ","End":"00:49.700","Text":"the chairman and the second one to be the secretary,"},{"Start":"00:49.700 ","End":"00:52.160","Text":"and the third one to be the assistant."},{"Start":"00:52.160 ","End":"00:59.195","Text":"All we want to do is to choose 3 students for the committee and it\u0027s without returns."},{"Start":"00:59.195 ","End":"01:02.510","Text":"Well, that\u0027s because we\u0027re choosing 3 different people."},{"Start":"01:02.510 ","End":"01:06.605","Text":"We\u0027re not choosing 1 person to fill 3 positions."},{"Start":"01:06.605 ","End":"01:10.740","Text":"We have to choose 3 different students."},{"Start":"01:11.300 ","End":"01:14.885","Text":"What do we have here? How do we calculate this?"},{"Start":"01:14.885 ","End":"01:17.720","Text":"Well, according to the equation, well,"},{"Start":"01:17.720 ","End":"01:23.555","Text":"that equals to 10 over 3 and what is that equal to?"},{"Start":"01:23.555 ","End":"01:33.390","Text":"Well, that\u0027s 10 factorial divided by 7 factorial times 3 factorial."},{"Start":"01:33.390 ","End":"01:34.755","Text":"Now if we recall,"},{"Start":"01:34.755 ","End":"01:36.420","Text":"that will be n/k,"},{"Start":"01:36.420 ","End":"01:38.150","Text":"that\u0027s our equation,"},{"Start":"01:38.150 ","End":"01:46.865","Text":"that equals to n factorial divided by n minus k factorial times k factorial."},{"Start":"01:46.865 ","End":"01:48.995","Text":"This is exactly what we have here,"},{"Start":"01:48.995 ","End":"01:51.545","Text":"n factorial, that\u0027s 10 factorial,"},{"Start":"01:51.545 ","End":"01:53.600","Text":"n equals 10 and k equals 3."},{"Start":"01:53.600 ","End":"01:56.150","Text":"N minus k, that\u0027s 10 minus 3,"},{"Start":"01:56.150 ","End":"02:00.035","Text":"that\u0027s 7 factorial and then we have 3 factorial."},{"Start":"02:00.035 ","End":"02:02.075","Text":"So what does that equal to?"},{"Start":"02:02.075 ","End":"02:04.175","Text":"That equals to 120."},{"Start":"02:04.175 ","End":"02:08.210","Text":"So we have 120 different combinations of"},{"Start":"02:08.210 ","End":"02:12.200","Text":"choosing 3 out of 10 students for a committee where"},{"Start":"02:12.200 ","End":"02:15.395","Text":"the committee has no defined positions to fill"},{"Start":"02:15.395 ","End":"02:21.540","Text":"or where we have non-ordered samplings without returns."}],"ID":12816},{"Watched":false,"Name":"Tutorial 3","Duration":"4m 18s","ChapterTopicVideoID":12338,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12338.jpeg","UploadDate":"2018-07-22T16:54:27.7330000","DurationForVideoObject":"PT4M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"Before we go on to doing exercises,"},{"Start":"00:02.820 ","End":"00:05.130","Text":"let\u0027s consider several notes here."},{"Start":"00:05.130 ","End":"00:08.130","Text":"The 1st one is notation."},{"Start":"00:08.130 ","End":"00:14.850","Text":"In most cases, we\u0027ll be looking at this notation right here and over k. Now sometimes in"},{"Start":"00:14.850 ","End":"00:16.380","Text":"the literature you\u0027ll be seeing"},{"Start":"00:16.380 ","End":"00:22.560","Text":"this notation c with a superscript k and subscript n. Now,"},{"Start":"00:22.560 ","End":"00:27.225","Text":"this is the most common notation and I\u0027ll be using that."},{"Start":"00:27.225 ","End":"00:30.225","Text":"Secondly, n over k,"},{"Start":"00:30.225 ","End":"00:35.385","Text":"that equals to n over n minus k. Well,"},{"Start":"00:35.385 ","End":"00:36.855","Text":"let\u0027s take a look at this."},{"Start":"00:36.855 ","End":"00:38.820","Text":"Let\u0027s take a look at the example that we had."},{"Start":"00:38.820 ","End":"00:43.519","Text":"We had 10 over 3 and according to this,"},{"Start":"00:43.519 ","End":"00:46.490","Text":"that equals to 10 over 7."},{"Start":"00:46.490 ","End":"00:48.140","Text":"Now, why is that?"},{"Start":"00:48.140 ","End":"00:52.430","Text":"Well, choosing 3 students out of 10 to be in"},{"Start":"00:52.430 ","End":"00:58.415","Text":"a committee is just like choosing 7 students out of 10 not be in the committee."},{"Start":"00:58.415 ","End":"01:02.210","Text":"Now, let\u0027s take a look at the equations."},{"Start":"01:02.210 ","End":"01:04.660","Text":"Well, n over k,"},{"Start":"01:04.660 ","End":"01:13.370","Text":"that equals to n factorial divided by n minus k factorial, times k factorial."},{"Start":"01:13.370 ","End":"01:16.415","Text":"Now, what does that equal to?"},{"Start":"01:16.415 ","End":"01:20.555","Text":"Does that equal to n over n minus k?"},{"Start":"01:20.555 ","End":"01:22.925","Text":"Well, let\u0027s take a look."},{"Start":"01:22.925 ","End":"01:28.595","Text":"N over n minus k that equals to n factorial."},{"Start":"01:28.595 ","End":"01:34.215","Text":"Now, here we have n minus,"},{"Start":"01:34.215 ","End":"01:41.076","Text":"now this will be our k. It will be n minus k,"},{"Start":"01:41.076 ","End":"01:47.895","Text":"factorial times n minus k factorial."},{"Start":"01:47.895 ","End":"01:49.485","Text":"This will be our k now."},{"Start":"01:49.485 ","End":"01:51.115","Text":"Now, what does that equal to?"},{"Start":"01:51.115 ","End":"01:59.790","Text":"That equals to n factorial divided by k factorial times n minus k factorial."},{"Start":"01:59.790 ","End":"02:06.775","Text":"We see that there is an equality here. Note number 3."},{"Start":"02:06.775 ","End":"02:14.435","Text":"Note number 3 is basically a private or a special case of number 2."},{"Start":"02:14.435 ","End":"02:18.620","Text":"We have n over n minus 1,"},{"Start":"02:18.620 ","End":"02:22.790","Text":"and we want to know if that equals to n over 1."},{"Start":"02:22.790 ","End":"02:24.845","Text":"Well, n over n minus 1,"},{"Start":"02:24.845 ","End":"02:33.050","Text":"that equals to n factorial divided by n minus 1 factorial times 1 factorial."},{"Start":"02:33.050 ","End":"02:35.420","Text":"Now, what does that equal to?"},{"Start":"02:35.420 ","End":"02:38.240","Text":"What does n over 1 equal?"},{"Start":"02:38.240 ","End":"02:46.230","Text":"That equals to n factorial divided by 1 factorial times n minus 1 factorial."},{"Start":"02:46.230 ","End":"02:48.360","Text":"We see that that\u0027s the same thing."},{"Start":"02:48.360 ","End":"02:51.815","Text":"Now, what\u0027s the meaning of this?"},{"Start":"02:51.815 ","End":"02:58.070","Text":"Well, all this means is that we\u0027re choosing n minus 1 objects out of n objects."},{"Start":"02:58.070 ","End":"03:02.870","Text":"Right now, how many ways can we choose n minus 1 objects out of n objects?"},{"Start":"03:02.870 ","End":"03:06.920","Text":"Well we have n ways of doing that."},{"Start":"03:06.920 ","End":"03:10.255","Text":"If we have to choose 1 out of 10,"},{"Start":"03:10.255 ","End":"03:12.020","Text":"1 person out of 10, well,"},{"Start":"03:12.020 ","End":"03:14.855","Text":"how many ways can we choose 1 person out of 10?"},{"Start":"03:14.855 ","End":"03:18.470","Text":"Well, we have 10 ways of choosing 1 person out of 10."},{"Start":"03:18.470 ","End":"03:20.960","Text":"This is note number 3."},{"Start":"03:20.960 ","End":"03:23.225","Text":"What about note number 4?"},{"Start":"03:23.225 ","End":"03:30.215","Text":"N over n, that equals to n over 0 and that equals to 1."},{"Start":"03:30.215 ","End":"03:31.550","Text":"Well, n over n,"},{"Start":"03:31.550 ","End":"03:38.360","Text":"that\u0027s n factorial divided by n factorial times 0 factorial."},{"Start":"03:38.360 ","End":"03:43.925","Text":"Well, these cancel each other out so I have 1 divided by 0 factorial,"},{"Start":"03:43.925 ","End":"03:46.725","Text":"and 0 factorial, well,"},{"Start":"03:46.725 ","End":"03:48.090","Text":"that equals to 1."},{"Start":"03:48.090 ","End":"03:50.690","Text":"We have 1 over 1 and that equals to 1."},{"Start":"03:50.690 ","End":"03:54.350","Text":"Now what about n over 0?"},{"Start":"03:54.350 ","End":"04:02.040","Text":"Well, that equals to n factorial divided by 0 factorial times n factorial."},{"Start":"04:02.040 ","End":"04:07.370","Text":"Again, these cancel each other out and that equals to 1. There we go."},{"Start":"04:07.370 ","End":"04:10.970","Text":"We have to know these equalities"},{"Start":"04:10.970 ","End":"04:15.425","Text":"here in order to play around with them and do the exercises."},{"Start":"04:15.425 ","End":"04:18.120","Text":"Good luck with the exercises."}],"ID":12817},{"Watched":false,"Name":"Exercise 1 - Parts a-c","Duration":"3m 14s","ChapterTopicVideoID":12339,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12339.jpeg","UploadDate":"2018-07-22T16:54:34.5630000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this question, we\u0027re given that there are 15 girls and 10 boys and a class."},{"Start":"00:04.005 ","End":"00:06.950","Text":"5 different students are chosen for a committee."},{"Start":"00:06.950 ","End":"00:10.080","Text":"We\u0027re asked, how many possible ways can the committee be"},{"Start":"00:10.080 ","End":"00:15.430","Text":"composed if the choice of boys and girls in the committee is unrestricted?"},{"Start":"00:15.590 ","End":"00:19.170","Text":"What\u0027s the total number of boys and girls in the class?"},{"Start":"00:19.170 ","End":"00:21.390","Text":"Well, that\u0027s 15 plus 10,"},{"Start":"00:21.390 ","End":"00:23.250","Text":"and that equals to 25,"},{"Start":"00:23.250 ","End":"00:25.350","Text":"so n equals to 25."},{"Start":"00:25.350 ","End":"00:28.200","Text":"How many students are we sampling?"},{"Start":"00:28.200 ","End":"00:31.320","Text":"Well, 5 different students."},{"Start":"00:31.320 ","End":"00:38.100","Text":"Here we have a non-ordered sampling without returns. Why is that?"},{"Start":"00:38.100 ","End":"00:43.950","Text":"Well, it\u0027s not ordered because all we want is 5 students."},{"Start":"00:43.950 ","End":"00:47.525","Text":"We don\u0027t care in what order they\u0027ll be sampled."},{"Start":"00:47.525 ","End":"00:52.790","Text":"Secondly, it\u0027s without returns because we want 5 different students."},{"Start":"00:52.790 ","End":"00:56.825","Text":"We don\u0027t want the same student over and over again 5 times,"},{"Start":"00:56.825 ","End":"00:59.410","Text":"we want 5 different students."},{"Start":"00:59.410 ","End":"01:01.325","Text":"What do we have here?"},{"Start":"01:01.325 ","End":"01:05.870","Text":"How many possible ways can we choose 5 students out of 25?"},{"Start":"01:05.870 ","End":"01:08.150","Text":"Well, that\u0027s 25 over 5."},{"Start":"01:08.150 ","End":"01:18.690","Text":"That equals to 53,130 different ways of choosing 5 students out of 25 students."},{"Start":"01:18.690 ","End":"01:24.680","Text":"In this section we\u0027re asked how many possible ways can I choose 5 students out"},{"Start":"01:24.680 ","End":"01:30.680","Text":"of 25 students when the committee must consist of 3 girls and 2 boys."},{"Start":"01:30.680 ","End":"01:34.565","Text":"Well, we know that we have 15 girls in total,"},{"Start":"01:34.565 ","End":"01:37.400","Text":"and we have 10 boys in total."},{"Start":"01:37.400 ","End":"01:42.680","Text":"Out of the 15, we want to choose 3 girls and out of the 10 boys,"},{"Start":"01:42.680 ","End":"01:44.330","Text":"we want to choose 2 boys."},{"Start":"01:44.330 ","End":"01:47.300","Text":"Well, choosing 3 out of 15 girls,"},{"Start":"01:47.300 ","End":"01:49.160","Text":"that\u0027s 15 over 3,"},{"Start":"01:49.160 ","End":"01:51.635","Text":"and choosing 2 boys out of 10,"},{"Start":"01:51.635 ","End":"01:53.270","Text":"that\u0027s 10 over 2."},{"Start":"01:53.270 ","End":"01:57.980","Text":"We have to multiply these 2 expressions together. Why is that?"},{"Start":"01:57.980 ","End":"02:05.285","Text":"Well, for every combination of 3 girls that I choose out of the 15,"},{"Start":"02:05.285 ","End":"02:07.520","Text":"well, I have to multiply that by"},{"Start":"02:07.520 ","End":"02:12.835","Text":"all the possible combinations of choosing 2 boys out of 10."},{"Start":"02:12.835 ","End":"02:15.980","Text":"By multiplying both expressions right here,"},{"Start":"02:15.980 ","End":"02:18.365","Text":"I get the total number of combinations here,"},{"Start":"02:18.365 ","End":"02:25.145","Text":"and that comes out to 20,475."},{"Start":"02:25.145 ","End":"02:31.024","Text":"In this section, we still have to choose 5 students out of the 25 for the committee,"},{"Start":"02:31.024 ","End":"02:34.400","Text":"but there cannot be any boys in the committee."},{"Start":"02:34.400 ","End":"02:38.030","Text":"Again, we have 15 girls in total."},{"Start":"02:38.030 ","End":"02:40.925","Text":"We have 10 boys in total."},{"Start":"02:40.925 ","End":"02:42.950","Text":"Out of the 15 girls,"},{"Start":"02:42.950 ","End":"02:44.660","Text":"we have to choose 5 girls,"},{"Start":"02:44.660 ","End":"02:46.100","Text":"and out of the 10 points,"},{"Start":"02:46.100 ","End":"02:48.920","Text":"we have to choose 0 boys."},{"Start":"02:48.920 ","End":"02:51.950","Text":"Choosing 5 out of 15 girls,"},{"Start":"02:51.950 ","End":"02:54.425","Text":"so that\u0027s 15 over 5."},{"Start":"02:54.425 ","End":"02:59.195","Text":"We have to multiply that by 10 over 0."},{"Start":"02:59.195 ","End":"03:01.805","Text":"We know that 10 over 0,"},{"Start":"03:01.805 ","End":"03:03.800","Text":"that equals to 1."},{"Start":"03:03.800 ","End":"03:08.610","Text":"All this comes out to 15 over 5."},{"Start":"03:08.800 ","End":"03:15.490","Text":"That equals to 3,003."}],"ID":12818},{"Watched":false,"Name":"Exercise 2 - Parts a-d","Duration":"4m 47s","ChapterTopicVideoID":12340,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12340.jpeg","UploadDate":"2018-07-22T16:54:45.1670000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this question, we\u0027re told that the student wishes to"},{"Start":"00:02.310 ","End":"00:05.220","Text":"select 5 elective courses in the semester."},{"Start":"00:05.220 ","End":"00:07.140","Text":"There are 10 available courses,"},{"Start":"00:07.140 ","End":"00:10.019","Text":"5 humanities courses, 3 social sciences,"},{"Start":"00:10.019 ","End":"00:11.580","Text":"and 2 math courses."},{"Start":"00:11.580 ","End":"00:13.095","Text":"We\u0027re essence section a."},{"Start":"00:13.095 ","End":"00:16.315","Text":"How many different selections can he create from himself?"},{"Start":"00:16.315 ","End":"00:19.530","Text":"First of all, what\u0027s the total number of courses?"},{"Start":"00:19.530 ","End":"00:22.485","Text":"That would be our n, and that equals to 10."},{"Start":"00:22.485 ","End":"00:29.145","Text":"Now that equals to the 5 humanities plus 3 Social Sciences plus 2 math courses."},{"Start":"00:29.145 ","End":"00:33.060","Text":"How many courses does that student select?"},{"Start":"00:33.060 ","End":"00:34.905","Text":"he select 5 courses."},{"Start":"00:34.905 ","End":"00:38.295","Text":"k here, that equals to 5."},{"Start":"00:38.295 ","End":"00:43.930","Text":"Here we have a non ordered sampling without returned. Why is that?"},{"Start":"00:43.930 ","End":"00:47.150","Text":"It\u0027s non ordered because if I"},{"Start":"00:47.150 ","End":"00:51.250","Text":"choose the humanities course first and then a mathematics course"},{"Start":"00:51.250 ","End":"00:54.485","Text":"it really doesn\u0027t matter if I first choose"},{"Start":"00:54.485 ","End":"00:58.715","Text":"that or I choose the mathematics course and then the humanity course,"},{"Start":"00:58.715 ","End":"01:01.885","Text":"I still choose the 2 courses."},{"Start":"01:01.885 ","End":"01:05.690","Text":"It\u0027s without returns because if I choose"},{"Start":"01:05.690 ","End":"01:10.805","Text":"the same course over and over again I\u0027m still choosing 1 course."},{"Start":"01:10.805 ","End":"01:12.080","Text":"It\u0027s without returns."},{"Start":"01:12.080 ","End":"01:16.030","Text":"I need to choose 5 different courses."},{"Start":"01:16.030 ","End":"01:21.095","Text":"If that\u0027s the case, I have 10 courses to choose from,"},{"Start":"01:21.095 ","End":"01:23.995","Text":"and I have to choose 5 of them."},{"Start":"01:23.995 ","End":"01:28.085","Text":"How many possible ways can I choose 5 courses out of the 10?"},{"Start":"01:28.085 ","End":"01:35.614","Text":"That\u0027s 252 different ways of choosing 5 courses from 10 courses."},{"Start":"01:35.614 ","End":"01:41.615","Text":"In this section, we\u0027re asked how many selections include 3 humanities courses?"},{"Start":"01:41.615 ","End":"01:45.110","Text":"We have a total of 5 humanities courses."},{"Start":"01:45.110 ","End":"01:46.685","Text":"I have to choose 3 of them."},{"Start":"01:46.685 ","End":"01:50.185","Text":"I have 5 over 3."},{"Start":"01:50.185 ","End":"01:55.220","Text":"That\u0027s the number of humanities or the number of times"},{"Start":"01:55.220 ","End":"01:59.600","Text":"or combinations of choosing 3 humanities courses out of the 5."},{"Start":"01:59.600 ","End":"02:03.375","Text":"But I still need to choose 2 other courses."},{"Start":"02:03.375 ","End":"02:07.220","Text":"Out of the 5 remaining courses,"},{"Start":"02:07.220 ","End":"02:10.205","Text":"I have to choose 2 courses."},{"Start":"02:10.205 ","End":"02:13.610","Text":"That will be 5 over 3 with this 5."},{"Start":"02:13.610 ","End":"02:17.150","Text":"That means that the humanities,"},{"Start":"02:17.150 ","End":"02:20.500","Text":"that\u0027s the total humanities courses."},{"Start":"02:20.500 ","End":"02:27.375","Text":"This 5 represents the remaining courses."},{"Start":"02:27.375 ","End":"02:29.160","Text":"What does that equal to?"},{"Start":"02:29.160 ","End":"02:31.920","Text":"Well that equals to 100."},{"Start":"02:31.920 ","End":"02:35.300","Text":"There are a 100 possible combinations of choosing"},{"Start":"02:35.300 ","End":"02:40.174","Text":"5 courses where 3 of them are humanities courses."},{"Start":"02:40.174 ","End":"02:42.770","Text":"In this section we\u0027re asked how many selections have"},{"Start":"02:42.770 ","End":"02:46.775","Text":"2 courses that are not humanities courses."},{"Start":"02:46.775 ","End":"02:50.405","Text":"If they\u0027re not humanities courses,"},{"Start":"02:50.405 ","End":"02:55.935","Text":"then out of the 5 remaining courses,"},{"Start":"02:55.935 ","End":"02:59.100","Text":"I have to choose 2 of them."},{"Start":"02:59.100 ","End":"03:05.685","Text":"Out of the 5 humanities course I need to choose 3 courses."},{"Start":"03:05.685 ","End":"03:15.790","Text":"As we said, this 5 for presents the 5 courses that are not humanities."},{"Start":"03:16.460 ","End":"03:23.230","Text":"This 5 represents the courses that are humanities."},{"Start":"03:24.230 ","End":"03:27.900","Text":"That equals to 100."},{"Start":"03:27.900 ","End":"03:29.300","Text":"If we take a look,"},{"Start":"03:29.300 ","End":"03:34.070","Text":"this is exactly the same question as we had in the previous section."},{"Start":"03:34.070 ","End":"03:37.760","Text":"The previous sections, we had to calculate the number"},{"Start":"03:37.760 ","End":"03:41.735","Text":"of selections where we had 3 humanities courses."},{"Start":"03:41.735 ","End":"03:46.250","Text":"By asking how many selections have 2 courses that are not humanities,"},{"Start":"03:46.250 ","End":"03:49.535","Text":"that means the 3 courses are humanities."},{"Start":"03:49.535 ","End":"03:54.575","Text":"This is basically the same question that we had in the previous section."},{"Start":"03:54.575 ","End":"03:59.090","Text":"In this section we\u0027re asked how many selections have 2 humanities courses,"},{"Start":"03:59.090 ","End":"04:02.795","Text":"2 social science courses and 1 mathematics course."},{"Start":"04:02.795 ","End":"04:06.560","Text":"Out of the 5 humanities courses,"},{"Start":"04:06.560 ","End":"04:07.700","Text":"I need to choose 2."},{"Start":"04:07.700 ","End":"04:12.275","Text":"That\u0027ll be 5 over 2 times,"},{"Start":"04:12.275 ","End":"04:19.500","Text":"I need to choose 2 social science courses out of a total of 3 times 3 over 2."},{"Start":"04:19.500 ","End":"04:25.005","Text":"Then out of a total of 2 math courses I need to choose 1."},{"Start":"04:25.005 ","End":"04:31.110","Text":"This would be the number of selections that have 2 humanities courses,"},{"Start":"04:31.110 ","End":"04:33.030","Text":"2 social science courses,"},{"Start":"04:33.030 ","End":"04:34.815","Text":"and 1 math course."},{"Start":"04:34.815 ","End":"04:36.470","Text":"That equals to 10,"},{"Start":"04:36.470 ","End":"04:39.530","Text":"5 over 2 equals 10 times 3 over 2."},{"Start":"04:39.530 ","End":"04:42.050","Text":"That equals to 3 times 2 over 1,"},{"Start":"04:42.050 ","End":"04:43.925","Text":"that equals to 2,"},{"Start":"04:43.925 ","End":"04:47.100","Text":"and that equals to 60."}],"ID":12819},{"Watched":false,"Name":"Exercise 3 - Parts a-c","Duration":"4m 40s","ChapterTopicVideoID":12341,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12341.jpeg","UploadDate":"2018-07-22T16:54:58.0070000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this question, we\u0027re given 30 students in"},{"Start":"00:02.370 ","End":"00:05.430","Text":"a class of which 12 are male and 18 are female."},{"Start":"00:05.430 ","End":"00:09.045","Text":"Now, a committee of 4 students must be chosen from the class,"},{"Start":"00:09.045 ","End":"00:11.025","Text":"and the students are chosen randomly."},{"Start":"00:11.025 ","End":"00:16.615","Text":"We\u0027re asked what\u0027s the probability that the committee would consists solely of female?"},{"Start":"00:16.615 ","End":"00:23.475","Text":"First of all, we have here a non-ordered sample without returns. Now, why is that?"},{"Start":"00:23.475 ","End":"00:28.350","Text":"It\u0027s non ordered because it really doesn\u0027t matter in which order I choose the 4 students."},{"Start":"00:28.350 ","End":"00:31.410","Text":"All I need is 4 different people."},{"Start":"00:31.410 ","End":"00:33.705","Text":"Now, because they\u0027re different people,"},{"Start":"00:33.705 ","End":"00:35.685","Text":"then it\u0027s without returns."},{"Start":"00:35.685 ","End":"00:39.675","Text":"Secondly, let\u0027s just write out the information that we have here."},{"Start":"00:39.675 ","End":"00:42.189","Text":"We have 12 males,"},{"Start":"00:42.189 ","End":"00:44.615","Text":"we have 18 females."},{"Start":"00:44.615 ","End":"00:47.675","Text":"We have a total of 30 students,"},{"Start":"00:47.675 ","End":"00:50.825","Text":"and we need to choose 4 of them."},{"Start":"00:50.825 ","End":"00:55.400","Text":"Now, whenever we\u0027re asked about the probability of something,"},{"Start":"00:55.400 ","End":"01:00.055","Text":"the first thing that we want to do is calculate the size of the sample space."},{"Start":"01:00.055 ","End":"01:02.280","Text":"What would that be?"},{"Start":"01:02.280 ","End":"01:08.250","Text":"We\u0027re choosing 4 students randomly, from 30 students."},{"Start":"01:08.250 ","End":"01:14.390","Text":"How many possible combinations of choosing 4 students out of 30?"},{"Start":"01:14.390 ","End":"01:17.160","Text":"That\u0027s 30 over 4."},{"Start":"01:17.270 ","End":"01:22.880","Text":"Now we\u0027re looking for the probability that all the people on the committee,"},{"Start":"01:22.880 ","End":"01:26.225","Text":"all 4 students, are female."},{"Start":"01:26.225 ","End":"01:30.610","Text":"In the denominator we have the size of our sample space."},{"Start":"01:30.610 ","End":"01:32.625","Text":"That\u0027s 30 over 4."},{"Start":"01:32.625 ","End":"01:34.170","Text":"In the numerator,"},{"Start":"01:34.170 ","End":"01:41.525","Text":"we have to count all the possible combinations of getting 4 students that are female."},{"Start":"01:41.525 ","End":"01:49.395","Text":"Well, that means I have to choose 4 females out of the total number of females."},{"Start":"01:49.395 ","End":"01:51.530","Text":"Out of the total number of males,"},{"Start":"01:51.530 ","End":"01:52.730","Text":"I have to choose 0."},{"Start":"01:52.730 ","End":"01:54.425","Text":"Now 12 over 0,"},{"Start":"01:54.425 ","End":"01:56.135","Text":"that equals to 1."},{"Start":"01:56.135 ","End":"02:02.915","Text":"I can write this out as 18 over 4 divided by 30 over 4,"},{"Start":"02:02.915 ","End":"02:08.390","Text":"now that equals to 0.1117 approximately."},{"Start":"02:08.390 ","End":"02:12.455","Text":"This then would be the probability of having"},{"Start":"02:12.455 ","End":"02:16.910","Text":"all 4 students being female in the committee."},{"Start":"02:16.910 ","End":"02:19.700","Text":"In this section we\u0027re asked what\u0027s the probability"},{"Start":"02:19.700 ","End":"02:22.060","Text":"that the committee will have only 1 female?"},{"Start":"02:22.060 ","End":"02:27.845","Text":"We\u0027re looking for the probability of only 1 female."},{"Start":"02:27.845 ","End":"02:32.360","Text":"In the denominator, we have the size of the sample space,"},{"Start":"02:32.360 ","End":"02:34.280","Text":"which is 30 over 4."},{"Start":"02:34.280 ","End":"02:38.030","Text":"In the numerator, we\u0027re choosing only 1 female,"},{"Start":"02:38.030 ","End":"02:41.450","Text":"so we have to choose 1 out of a possible of 18 female."},{"Start":"02:41.450 ","End":"02:44.765","Text":"That\u0027ll be 18 over 1 times,"},{"Start":"02:44.765 ","End":"02:49.520","Text":"now we have to choose 3 males out of a total of 12 males."},{"Start":"02:49.520 ","End":"02:51.955","Text":"That\u0027ll be 12 over 3."},{"Start":"02:51.955 ","End":"02:58.710","Text":"That equals to 0.1445 approximately."},{"Start":"02:58.710 ","End":"03:03.260","Text":"This is the probability of having only 1 female."},{"Start":"03:03.260 ","End":"03:06.230","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"03:06.230 ","End":"03:08.809","Text":"the committee will have at least 1 females."},{"Start":"03:08.809 ","End":"03:15.240","Text":"We\u0027re looking for the probability that we have at least 1 female."},{"Start":"03:15.240 ","End":"03:18.885","Text":"Well, that means that I can have either 1 female or 2,"},{"Start":"03:18.885 ","End":"03:22.100","Text":"or 3 or 4 females in the committee."},{"Start":"03:22.100 ","End":"03:25.280","Text":"But that\u0027s pretty complicated to calculate."},{"Start":"03:25.280 ","End":"03:27.545","Text":"But if I use the complimentary set,"},{"Start":"03:27.545 ","End":"03:29.900","Text":"I think that\u0027ll be easier so let\u0027s do that."},{"Start":"03:29.900 ","End":"03:34.940","Text":"This probability is equal to 1 minus the probability."},{"Start":"03:34.940 ","End":"03:38.605","Text":"What\u0027s the complimentary set of at least 1 female?"},{"Start":"03:38.605 ","End":"03:41.745","Text":"That means that there\u0027s no females."},{"Start":"03:41.745 ","End":"03:43.875","Text":"No females."},{"Start":"03:43.875 ","End":"03:45.815","Text":"If there are no females,"},{"Start":"03:45.815 ","End":"03:51.085","Text":"then that means that we have all boys, all males."},{"Start":"03:51.085 ","End":"03:53.910","Text":"That equals to what?"},{"Start":"03:53.910 ","End":"03:59.975","Text":"That\u0027s 1 minus the probability of having all males when the denominator,"},{"Start":"03:59.975 ","End":"04:02.060","Text":"we have the size of our sample space,"},{"Start":"04:02.060 ","End":"04:04.100","Text":"that\u0027s 30 over 4."},{"Start":"04:04.100 ","End":"04:07.475","Text":"Because all of them and we have"},{"Start":"04:07.475 ","End":"04:11.780","Text":"an all-male committee then all 4 students have to be male."},{"Start":"04:11.780 ","End":"04:13.820","Text":"Out of the 12 students,"},{"Start":"04:13.820 ","End":"04:15.320","Text":"the 12 males that we have,"},{"Start":"04:15.320 ","End":"04:16.985","Text":"we have to choose 4."},{"Start":"04:16.985 ","End":"04:20.015","Text":"Now, I could write here 18 over 0,"},{"Start":"04:20.015 ","End":"04:22.190","Text":"but again, we know that that equals to 1."},{"Start":"04:22.190 ","End":"04:24.410","Text":"Really we don\u0027t have to write this out."},{"Start":"04:24.410 ","End":"04:29.160","Text":"This comes out to 0.9819."},{"Start":"04:32.470 ","End":"04:40.110","Text":"This then would be the probability that we have at least 1 female."}],"ID":12820},{"Watched":false,"Name":"Exercise 4 - Parts a-d","Duration":"6m 55s","ChapterTopicVideoID":12342,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12342.jpeg","UploadDate":"2018-07-22T16:55:13.2270000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this question, we have a lottery and we have to choose"},{"Start":"00:02.430 ","End":"00:05.690","Text":"5 numbers out of a total of 45 numbers."},{"Start":"00:05.690 ","End":"00:08.450","Text":"The numbers are between 1 and 45."},{"Start":"00:08.450 ","End":"00:13.110","Text":"We\u0027re asked what\u0027s the probability that the winning entry will have all even numbers?"},{"Start":"00:13.110 ","End":"00:16.470","Text":"First of all, we are in a situation we have"},{"Start":"00:16.470 ","End":"00:20.415","Text":"a non ordered sample without returns. Why is that?"},{"Start":"00:20.415 ","End":"00:25.725","Text":"It\u0027s without returns because we have to select 5 different numbers."},{"Start":"00:25.725 ","End":"00:31.930","Text":"It\u0027s non ordered because it really doesn\u0027t matter in which order we choose the 5 numbers."},{"Start":"00:31.930 ","End":"00:36.675","Text":"We\u0027re looking at the probability of something."},{"Start":"00:36.675 ","End":"00:40.220","Text":"Whenever we\u0027re looking at the probability of something then we have to"},{"Start":"00:40.220 ","End":"00:43.900","Text":"calculate the size of the sample space."},{"Start":"00:43.900 ","End":"00:48.435","Text":"Well, we have to choose 5 numbers out of 45."},{"Start":"00:48.435 ","End":"00:52.700","Text":"The size of the sample space would be 45 over 5,"},{"Start":"00:52.700 ","End":"00:58.040","Text":"and that\u0027ll be the total number of combinations of choosing 5 numbers out of 45."},{"Start":"00:58.040 ","End":"01:05.580","Text":"Now, we\u0027re looking at the probability that all the numbers that are chosen are even."},{"Start":"01:05.810 ","End":"01:12.810","Text":"In the denominator, we have the size of the sample space and that\u0027s 45 over 5,"},{"Start":"01:12.810 ","End":"01:14.985","Text":"and in the numerator,"},{"Start":"01:14.985 ","End":"01:17.640","Text":"we have to choose 5 numbers that are even."},{"Start":"01:17.640 ","End":"01:21.560","Text":"How many even numbers do I have between 1 and 45?"},{"Start":"01:21.560 ","End":"01:24.975","Text":"I have 22 numbers that are even."},{"Start":"01:24.975 ","End":"01:27.535","Text":"Out of the 22 even numbers,"},{"Start":"01:27.535 ","End":"01:30.085","Text":"I have to choose 5 numbers."},{"Start":"01:30.085 ","End":"01:38.665","Text":"This then would be my probability and that equals to 0.0216."},{"Start":"01:38.665 ","End":"01:41.410","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"01:41.410 ","End":"01:44.710","Text":"the winning entry will have at most 1 even number?"},{"Start":"01:44.710 ","End":"01:55.005","Text":"We\u0027re looking at the probability of the entry having at most 1 even number."},{"Start":"01:55.005 ","End":"02:01.190","Text":"When we\u0027re looking at the phrase \"at most\" what does that mean?"},{"Start":"02:01.190 ","End":"02:03.935","Text":"It means that the winning entry will either have"},{"Start":"02:03.935 ","End":"02:11.130","Text":"0 even numbers or the winning entry will have 1 even number and no more."},{"Start":"02:11.130 ","End":"02:15.525","Text":"These 2 events are mutually exclusive."},{"Start":"02:15.525 ","End":"02:18.980","Text":"Whenever we have 2 mutually exclusive events,"},{"Start":"02:18.980 ","End":"02:23.690","Text":"then the probability of both these events would be the sum of the events."},{"Start":"02:23.690 ","End":"02:26.070","Text":"Let\u0027s just write that."},{"Start":"02:26.090 ","End":"02:33.710","Text":"Here we go. We\u0027re looking at the probability that I have no even numbers in"},{"Start":"02:33.710 ","End":"02:42.730","Text":"the winning entry plus the probability that I have 1 even number in the winning entry."},{"Start":"02:42.730 ","End":"02:45.210","Text":"Let\u0027s just calculate that."},{"Start":"02:45.210 ","End":"02:51.945","Text":"What\u0027s the probability of having no even number in the winning entry?"},{"Start":"02:51.945 ","End":"02:58.855","Text":"In the denominator we have the size of the sample space which is 45 over 5."},{"Start":"02:58.855 ","End":"03:02.300","Text":"If I don\u0027t have any even number,"},{"Start":"03:02.300 ","End":"03:04.595","Text":"so then how many even numbers do I have?"},{"Start":"03:04.595 ","End":"03:09.395","Text":"I have 22 even numbers from 1 to 45."},{"Start":"03:09.395 ","End":"03:11.360","Text":"Now, I want no even number."},{"Start":"03:11.360 ","End":"03:13.130","Text":"I\u0027m not choosing any even number."},{"Start":"03:13.130 ","End":"03:21.180","Text":"I have 22 over 0 here times all 5 numbers are odd numbers."},{"Start":"03:21.180 ","End":"03:25.880","Text":"Out of the 23 odd numbers that I have between 1 and 45,"},{"Start":"03:25.880 ","End":"03:29.774","Text":"I have to choose 5 plus,"},{"Start":"03:29.774 ","End":"03:34.660","Text":"now, I\u0027m dealing with the probability that I have 1 even number."},{"Start":"03:34.660 ","End":"03:38.720","Text":"Again, that\u0027ll be 45 over 5 in the denominator."},{"Start":"03:38.720 ","End":"03:40.655","Text":"Now if I have 1 even number,"},{"Start":"03:40.655 ","End":"03:44.735","Text":"then out of the 22 even numbers I choose 1,"},{"Start":"03:44.735 ","End":"03:51.220","Text":"and out of the 23 odd numbers I choose 4."},{"Start":"03:51.230 ","End":"03:53.960","Text":"After doing all the math here,"},{"Start":"03:53.960 ","End":"03:57.620","Text":"that comes out to 0.187."},{"Start":"03:57.620 ","End":"04:03.530","Text":"This then would be the probability that I have at most 1 even number."},{"Start":"04:03.530 ","End":"04:06.290","Text":"This section, we\u0027re asked what\u0027s the probability that"},{"Start":"04:06.290 ","End":"04:09.230","Text":"the winning entry will have at least 1 even number?"},{"Start":"04:09.230 ","End":"04:16.160","Text":"We\u0027re looking for the probability that we have at least 1 even number,"},{"Start":"04:16.160 ","End":"04:18.440","Text":"that means that we can either have 1,"},{"Start":"04:18.440 ","End":"04:19.490","Text":"or 2, or 3,"},{"Start":"04:19.490 ","End":"04:21.575","Text":"or 4, or 5 even numbers."},{"Start":"04:21.575 ","End":"04:28.315","Text":"That\u0027s pretty complicated to calculate but we can use the complementary set here."},{"Start":"04:28.315 ","End":"04:33.285","Text":"This probability equals to 1 minus the probability,"},{"Start":"04:33.285 ","End":"04:38.300","Text":"what\u0027s the complimentary set of at least 1 number being even?"},{"Start":"04:38.300 ","End":"04:44.475","Text":"That means that there\u0027s no even numbers."},{"Start":"04:44.475 ","End":"04:53.040","Text":"That means that if there\u0027s no even numbers and we\u0027re looking at all the numbers are odd."},{"Start":"04:54.490 ","End":"04:58.130","Text":"What\u0027s the probability that all the numbers are odd here?"},{"Start":"04:58.130 ","End":"04:59.360","Text":"Well, in the denominator,"},{"Start":"04:59.360 ","End":"05:01.610","Text":"we have 45 over 5."},{"Start":"05:01.610 ","End":"05:04.450","Text":"That\u0027s the size of my sample space."},{"Start":"05:04.450 ","End":"05:06.810","Text":"In the numerator,"},{"Start":"05:06.810 ","End":"05:09.180","Text":"how many odd numbers do I have?"},{"Start":"05:09.180 ","End":"05:16.190","Text":"I have 23 odd numbers and I have to have all 5 numbers to be odd,"},{"Start":"05:16.190 ","End":"05:20.165","Text":"so I\u0027m choosing 5 out of the 23 odd numbers."},{"Start":"05:20.165 ","End":"05:27.780","Text":"This then would be the probability that I have at least 1 even number."},{"Start":"05:27.780 ","End":"05:32.585","Text":"That\u0027s 1 minus 23 over 5 divided by 45 over 5,"},{"Start":"05:32.585 ","End":"05:37.015","Text":"and that comes out to 0.972."},{"Start":"05:37.015 ","End":"05:40.100","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"05:40.100 ","End":"05:44.550","Text":"the winning entry will consist solely of numbers larger than 30?"},{"Start":"05:44.870 ","End":"05:50.090","Text":"We\u0027re looking at the probability of all numbers being greater than 30."},{"Start":"05:50.090 ","End":"05:53.875","Text":"How many numbers do I have that are greater than 30?"},{"Start":"05:53.875 ","End":"05:59.970","Text":"Well, that\u0027s 31, 32, 33, until 45."},{"Start":"05:59.970 ","End":"06:02.510","Text":"How many numbers do I have here?"},{"Start":"06:02.510 ","End":"06:05.660","Text":"That\u0027ll be b minus a plus 1."},{"Start":"06:05.660 ","End":"06:11.360","Text":"That\u0027s how to calculate how many numbers I have between 2 whole numbers,"},{"Start":"06:11.360 ","End":"06:15.965","Text":"b here would be 45 minus 31 plus 1,"},{"Start":"06:15.965 ","End":"06:18.155","Text":"and that equals to 15."},{"Start":"06:18.155 ","End":"06:23.140","Text":"I have 15 numbers that are greater than 30."},{"Start":"06:23.140 ","End":"06:26.929","Text":"The probability that all the numbers are greater than 31,"},{"Start":"06:26.929 ","End":"06:30.380","Text":"the denominator, that\u0027ll be the size of my sample space,"},{"Start":"06:30.380 ","End":"06:32.665","Text":"that will be 45 over 5."},{"Start":"06:32.665 ","End":"06:37.670","Text":"In the numerator, out of the 50 numbers that are greater than 30,"},{"Start":"06:37.670 ","End":"06:39.635","Text":"I have to choose 5,"},{"Start":"06:39.635 ","End":"06:41.270","Text":"all 5 of the numbers."},{"Start":"06:41.270 ","End":"06:47.930","Text":"Now, that equals to 0.00246 approximately."},{"Start":"06:47.930 ","End":"06:56.160","Text":"That\u0027s the probability of choosing 5 numbers that all of them would be greater than 30."}],"ID":12821},{"Watched":false,"Name":"Exercise 5 - Parts a-b","Duration":"6m 56s","ChapterTopicVideoID":12343,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12343.jpeg","UploadDate":"2018-07-22T16:55:29.2630000","DurationForVideoObject":"PT6M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this question, we have 52 cards in a pack of cards."},{"Start":"00:03.270 ","End":"00:04.680","Text":"We have 13 spades,"},{"Start":"00:04.680 ","End":"00:07.890","Text":"13 hearts, 13 diamonds, and 13 clubs."},{"Start":"00:07.890 ","End":"00:11.250","Text":"Now, each of the 4 suits has 9 cards,"},{"Start":"00:11.250 ","End":"00:13.950","Text":"number 2-10, and the remaining cards are a jack,"},{"Start":"00:13.950 ","End":"00:15.480","Text":"a queen, a king, and an ace."},{"Start":"00:15.480 ","End":"00:18.900","Text":"The deck is an ordinary deck with no jokers."},{"Start":"00:18.900 ","End":"00:20.925","Text":"Now, 2 people are playing poker."},{"Start":"00:20.925 ","End":"00:24.750","Text":"Each of them is dealt 5 cards randomly with no returns."},{"Start":"00:24.750 ","End":"00:27.660","Text":"We\u0027re asked, what\u0027s the probability that"},{"Start":"00:27.660 ","End":"00:33.280","Text":"the first player would get all the kings and the second player would get all the queens?"},{"Start":"00:33.290 ","End":"00:38.250","Text":"The first thing that we need to do when we\u0027re talking about probability is to"},{"Start":"00:38.250 ","End":"00:42.364","Text":"calculate the size of a sample space."},{"Start":"00:42.364 ","End":"00:44.000","Text":"Now, what would that be?"},{"Start":"00:44.000 ","End":"00:48.150","Text":"Let\u0027s say we deal 5 cards to the first player."},{"Start":"00:48.150 ","End":"00:52.715","Text":"Well, 5 cards needs to be chosen out of a total of 52 cards,"},{"Start":"00:52.715 ","End":"00:56.420","Text":"so we have 52/5,"},{"Start":"00:56.420 ","End":"01:02.840","Text":"that\u0027s the number of combinations of choosing 5 cards out of 52 cards."},{"Start":"01:02.840 ","End":"01:05.330","Text":"Now, for the second player, again,"},{"Start":"01:05.330 ","End":"01:09.095","Text":"the second player gets 5 cards as well, but now,"},{"Start":"01:09.095 ","End":"01:11.630","Text":"out of a total of 47."},{"Start":"01:11.630 ","End":"01:13.130","Text":"Why is that?"},{"Start":"01:13.130 ","End":"01:16.310","Text":"We\u0027ve already given 5 cards to the first player,"},{"Start":"01:16.310 ","End":"01:19.670","Text":"so 52 minus 5, that\u0027s 47."},{"Start":"01:19.670 ","End":"01:25.325","Text":"The second player gets 5 cards out of a total of 47 cards."},{"Start":"01:25.325 ","End":"01:29.940","Text":"That would be the size of the sample space."},{"Start":"01:29.940 ","End":"01:33.320","Text":"Let\u0027s calculate the probability now that the first player will"},{"Start":"01:33.320 ","End":"01:37.160","Text":"get all the kings and the second will get all the queens."},{"Start":"01:37.160 ","End":"01:43.070","Text":"In the denominator, we have the size of our sample space"},{"Start":"01:43.070 ","End":"01:49.195","Text":"which is 52/5 times 47/5."},{"Start":"01:49.195 ","End":"01:50.720","Text":"Now, for the first player,"},{"Start":"01:50.720 ","End":"01:52.700","Text":"the first player gets the 4 kings."},{"Start":"01:52.700 ","End":"01:55.190","Text":"How many kings are there in the deck?"},{"Start":"01:55.190 ","End":"02:00.280","Text":"There are 4 kings in the deck and the first player needs to get all 4 of them."},{"Start":"02:00.280 ","End":"02:03.840","Text":"Now, he\u0027s dealt the 4 kings,"},{"Start":"02:03.840 ","End":"02:06.120","Text":"but actually, he\u0027s dealt 5 cards,"},{"Start":"02:06.120 ","End":"02:08.220","Text":"so he needs to get 1 more card."},{"Start":"02:08.220 ","End":"02:10.415","Text":"Now 1 more card out of how many?"},{"Start":"02:10.415 ","End":"02:17.933","Text":"Well, we\u0027re taking all the 4 kings and all the 4 queens away from 52"},{"Start":"02:17.933 ","End":"02:25.520","Text":"because he needs to get 1 card out of all the cards which are not kings and not queens."},{"Start":"02:25.520 ","End":"02:28.970","Text":"All the kings were calculated here."},{"Start":"02:28.970 ","End":"02:31.070","Text":"How many cards are left?"},{"Start":"02:31.070 ","End":"02:33.980","Text":"4 kings and 4 queens, that\u0027s 8,"},{"Start":"02:33.980 ","End":"02:38.225","Text":"so 52 minus 8, that\u0027s 44."},{"Start":"02:38.225 ","End":"02:45.515","Text":"The first player needs to get 1 more card out of the 44 remaining cards."},{"Start":"02:45.515 ","End":"02:48.170","Text":"Now, what about the second player?"},{"Start":"02:48.170 ","End":"02:52.290","Text":"The second player gets the 4 queens,"},{"Start":"02:53.060 ","End":"02:55.610","Text":"how many queens are there in the deck?"},{"Start":"02:55.610 ","End":"03:00.660","Text":"There are 4 queens in the deck and the second player gets all 4 of them."},{"Start":"03:01.000 ","End":"03:06.805","Text":"Up till now, 9 cards have been dealt,"},{"Start":"03:06.805 ","End":"03:13.005","Text":"that means that 52 minus 9, that\u0027s 43."},{"Start":"03:13.005 ","End":"03:16.940","Text":"So there are 43 cards that are remaining and"},{"Start":"03:16.940 ","End":"03:23.265","Text":"the second player needs 1 more card right out of the 43."},{"Start":"03:23.265 ","End":"03:28.040","Text":"This then would be the probability that"},{"Start":"03:28.040 ","End":"03:32.855","Text":"the first player gets 4 kings and the second player gets 4 queens."},{"Start":"03:32.855 ","End":"03:39.094","Text":"Now, this probability is approximately 0."},{"Start":"03:39.094 ","End":"03:41.960","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"03:41.960 ","End":"03:44.555","Text":"that 1 of the players would get the ace of hearts?"},{"Start":"03:44.555 ","End":"03:47.480","Text":"Well, whenever we\u0027re asked this type of question,"},{"Start":"03:47.480 ","End":"03:54.890","Text":"we need to recognize that we need to split this event into 2 mutually exclusive events."},{"Start":"03:54.890 ","End":"03:58.670","Text":"We have 1 ace of hearts and that ace of"},{"Start":"03:58.670 ","End":"04:03.785","Text":"hearts can belong either to the first player or to the second player."},{"Start":"04:03.785 ","End":"04:08.515","Text":"Let\u0027s just write this probability out."},{"Start":"04:08.515 ","End":"04:14.000","Text":"We have here the probability that 1 of the players gets the ace of hearts,"},{"Start":"04:14.000 ","End":"04:18.155","Text":"that equals to the probability that the first player gets the ace of hearts,"},{"Start":"04:18.155 ","End":"04:23.540","Text":"or the probability that the second player gets the ace of hearts."},{"Start":"04:23.540 ","End":"04:26.615","Text":"That means that we have to add up these probabilities."},{"Start":"04:26.615 ","End":"04:29.240","Text":"Let\u0027s first calculate this probability"},{"Start":"04:29.240 ","End":"04:31.880","Text":"right here that the first player gets the ace of hearts."},{"Start":"04:31.880 ","End":"04:33.740","Text":"Well, in the denominator,"},{"Start":"04:33.740 ","End":"04:36.425","Text":"we have the size of the sample space,"},{"Start":"04:36.425 ","End":"04:42.680","Text":"and that\u0027s 52/5 times 47/5."},{"Start":"04:42.680 ","End":"04:44.450","Text":"Now in the numerator, what do we have?"},{"Start":"04:44.450 ","End":"04:48.780","Text":"We have only 1 ace of hearts and the first player gets it,"},{"Start":"04:48.780 ","End":"04:51.800","Text":"so out of 1 ace of hearts,"},{"Start":"04:51.800 ","End":"04:54.785","Text":"the first person gets that ace of hearts."},{"Start":"04:54.785 ","End":"04:58.880","Text":"Now, how many cards are left for the first player?"},{"Start":"04:58.880 ","End":"05:02.840","Text":"51 cards are left for the first player and he needs to"},{"Start":"05:02.840 ","End":"05:07.955","Text":"get 4 cards out of the remaining 51."},{"Start":"05:07.955 ","End":"05:11.720","Text":"Now, since 5 cards have been dealt to the first player, well,"},{"Start":"05:11.720 ","End":"05:20.350","Text":"the second player gets 5 cards out of the remaining 47 cards."},{"Start":"05:20.350 ","End":"05:24.815","Text":"That\u0027s the probability that the first player gets the ace of hearts."},{"Start":"05:24.815 ","End":"05:26.690","Text":"Now what about the second player,"},{"Start":"05:26.690 ","End":"05:30.642","Text":"the probability that the second player will get the ace of hearts?"},{"Start":"05:30.642 ","End":"05:32.260","Text":"Again, in the denominator,"},{"Start":"05:32.260 ","End":"05:34.535","Text":"we have the size of the sample space,"},{"Start":"05:34.535 ","End":"05:40.375","Text":"and that\u0027s 52/5 times 47/5."},{"Start":"05:40.375 ","End":"05:44.105","Text":"Here, let\u0027s start with the first player."},{"Start":"05:44.105 ","End":"05:48.770","Text":"The first player gets 5 cards out of 51,"},{"Start":"05:48.770 ","End":"05:55.590","Text":"not 52 because we have the ace of hearts given to the second player."},{"Start":"05:55.590 ","End":"05:59.810","Text":"The first player gets 5 out of the remaining 51."},{"Start":"05:59.810 ","End":"06:02.360","Text":"Now, for the second player,"},{"Start":"06:02.360 ","End":"06:04.265","Text":"he gets 1 card,"},{"Start":"06:04.265 ","End":"06:06.455","Text":"which is the ace of hearts,"},{"Start":"06:06.455 ","End":"06:09.155","Text":"so there\u0027s 1 ace of hearts and he gets it."},{"Start":"06:09.155 ","End":"06:12.720","Text":"Now, he needs another 4 cards."},{"Start":"06:12.720 ","End":"06:14.735","Text":"How many cards are left?"},{"Start":"06:14.735 ","End":"06:24.010","Text":"We have 51 cards and 5 of the cards were given to the first player."},{"Start":"06:24.010 ","End":"06:27.975","Text":"That means that 51 minus 5, that\u0027s 46,"},{"Start":"06:27.975 ","End":"06:30.560","Text":"so out of the remaining 46 cards,"},{"Start":"06:30.560 ","End":"06:35.050","Text":"the second player gets the remaining 4 cards."},{"Start":"06:35.050 ","End":"06:41.540","Text":"This then is the calculation for the probability that the first player gets the ace of"},{"Start":"06:41.540 ","End":"06:44.180","Text":"hearts and this is the calculation for"},{"Start":"06:44.180 ","End":"06:47.930","Text":"the probability that the second player gets the ace of hearts."},{"Start":"06:47.930 ","End":"06:50.855","Text":"Now, the sum of these probabilities,"},{"Start":"06:50.855 ","End":"06:56.310","Text":"that equals to 0.1923."}],"ID":12822},{"Watched":false,"Name":"Exercise 5 - Parts c-d","Duration":"4m 30s","ChapterTopicVideoID":12344,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12344.jpeg","UploadDate":"2018-07-22T16:55:40.2800000","DurationForVideoObject":"PT4M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this section we\u0027re asked what\u0027s the probability that the first player will get"},{"Start":"00:03.630 ","End":"00:08.010","Text":"all black cards and the second player will get exactly 2 black cards."},{"Start":"00:08.010 ","End":"00:12.450","Text":"Let\u0027s just write down the probability here. Here we go."},{"Start":"00:12.450 ","End":"00:15.570","Text":"We\u0027re looking at the probability that the first player will get"},{"Start":"00:15.570 ","End":"00:19.770","Text":"all black cards and the second player will get exactly 2 black cards."},{"Start":"00:19.770 ","End":"00:23.520","Text":"Well, how many black cards are there in a deck?"},{"Start":"00:23.520 ","End":"00:29.160","Text":"Well, there are 26 black cards in a deck and 26 red cards in a deck."},{"Start":"00:29.160 ","End":"00:32.069","Text":"Now, let\u0027s get back to our probability."},{"Start":"00:32.069 ","End":"00:33.300","Text":"In the denominator,"},{"Start":"00:33.300 ","End":"00:36.434","Text":"we have here the size of the sample space,"},{"Start":"00:36.434 ","End":"00:42.235","Text":"which is 52 over 5 times 47 over 5."},{"Start":"00:42.235 ","End":"00:46.715","Text":"Now, what about this event right here?"},{"Start":"00:46.715 ","End":"00:52.025","Text":"Well, let\u0027s take a look. The first player well all 5 cards need to be black."},{"Start":"00:52.025 ","End":"00:54.260","Text":"Out of the 26 black cards,"},{"Start":"00:54.260 ","End":"00:55.430","Text":"we need to choose 5,"},{"Start":"00:55.430 ","End":"00:57.635","Text":"so that\u0027s 26 over 5."},{"Start":"00:57.635 ","End":"00:59.540","Text":"Now what about the second player well,"},{"Start":"00:59.540 ","End":"01:01.430","Text":"he gets exactly 2 black?"},{"Start":"01:01.430 ","End":"01:05.915","Text":"That means that he needs to get 3 red cards."},{"Start":"01:05.915 ","End":"01:08.960","Text":"Out of the 26 red cards,"},{"Start":"01:08.960 ","End":"01:10.730","Text":"he needs to get 3."},{"Start":"01:10.730 ","End":"01:15.110","Text":"Now, out of the remaining black cards,"},{"Start":"01:15.110 ","End":"01:17.420","Text":"he needs to get 2."},{"Start":"01:17.420 ","End":"01:20.840","Text":"How many black cards remain?"},{"Start":"01:20.840 ","End":"01:24.560","Text":"Well, 5 cards out of the 26 would give into the first there,"},{"Start":"01:24.560 ","End":"01:28.070","Text":"so 26 minus 5 is 21."},{"Start":"01:28.070 ","End":"01:30.920","Text":"Out of the remaining 21 cards,"},{"Start":"01:30.920 ","End":"01:33.800","Text":"the second player needs to get 2 cards."},{"Start":"01:33.800 ","End":"01:36.230","Text":"This then would be the probability,"},{"Start":"01:36.230 ","End":"01:41.675","Text":"and that comes out to approximately 0.009."},{"Start":"01:41.675 ","End":"01:45.680","Text":"In this section, we\u0027re asked what\u0027s the probability that the first player will get"},{"Start":"01:45.680 ","End":"01:50.045","Text":"at least 3 number cards where we\u0027re told that ace is not a number card."},{"Start":"01:50.045 ","End":"01:53.060","Text":"Well, first of all, how many number cards do we have in a suit?"},{"Start":"01:53.060 ","End":"01:55.670","Text":"But the number cards are from 2-10,"},{"Start":"01:55.670 ","End":"02:01.085","Text":"that means that we have 9 cards in a suit,"},{"Start":"02:01.085 ","End":"02:04.819","Text":"9 number cards, and we have 4 suits."},{"Start":"02:04.819 ","End":"02:08.930","Text":"That means that we have a total of 9 times 4,"},{"Start":"02:08.930 ","End":"02:14.195","Text":"that\u0027s 36 number cards in total in a deck."},{"Start":"02:14.195 ","End":"02:20.735","Text":"That means that we have 16 face cards for a total of 52 cards."},{"Start":"02:20.735 ","End":"02:24.995","Text":"Now let\u0027s calculate the probability here."},{"Start":"02:24.995 ","End":"02:30.590","Text":"This is the probability where the first player gets at least 3 number cards."},{"Start":"02:30.590 ","End":"02:35.330","Text":"Well, this event can be split up into 3 mutually exclusive events."},{"Start":"02:35.330 ","End":"02:42.365","Text":"The first mutually exclusive event is that the player gets 3 number cards in the hand."},{"Start":"02:42.365 ","End":"02:48.570","Text":"The next mutually exclusive event is if the player gets 4 number cards in a hand,"},{"Start":"02:48.570 ","End":"02:55.785","Text":"and the last event is when the player gets 5 number cards in a hand."},{"Start":"02:55.785 ","End":"02:58.790","Text":"Let\u0027s calculate this probability right here."},{"Start":"02:58.790 ","End":"03:03.110","Text":"In the denominator, we have the size of the sample space,"},{"Start":"03:03.110 ","End":"03:08.350","Text":"that\u0027s 52 over 5 times 47 over 5."},{"Start":"03:08.350 ","End":"03:10.430","Text":"Now, let\u0027s take a look at"},{"Start":"03:10.430 ","End":"03:16.985","Text":"the first mutually exclusive event where the player gets 3 number of cards in a hand."},{"Start":"03:16.985 ","End":"03:19.700","Text":"Well, how many number of cards are there in a deck?"},{"Start":"03:19.700 ","End":"03:23.330","Text":"Well, that\u0027s 36, and he needs 3 number of cards."},{"Start":"03:23.330 ","End":"03:30.980","Text":"Now, that means that he\u0027s dealt 2 cards that are face cards out of the 16."},{"Start":"03:30.980 ","End":"03:34.340","Text":"That\u0027s the first mutually exclusive event."},{"Start":"03:34.340 ","End":"03:40.400","Text":"Now the second event is that if he gets 4 number cards, out of the 36,"},{"Start":"03:40.400 ","End":"03:48.095","Text":"he gets 4, that means that out of the 16 face cards he gets 1 card."},{"Start":"03:48.095 ","End":"03:54.200","Text":"Plus, well, the last mutually exclusive event is if he gets 5 number of cards."},{"Start":"03:54.200 ","End":"03:58.130","Text":"So out of the 36 number of cards,"},{"Start":"03:58.130 ","End":"03:59.870","Text":"he gets 5 cards."},{"Start":"03:59.870 ","End":"04:02.150","Text":"That means that out of the 16 he gets 0,"},{"Start":"04:02.150 ","End":"04:04.475","Text":"so 16 over 0, that\u0027s 1,"},{"Start":"04:04.475 ","End":"04:06.440","Text":"and we disregard that."},{"Start":"04:06.440 ","End":"04:09.860","Text":"Now, all of this has to be multiplied by"},{"Start":"04:09.860 ","End":"04:14.105","Text":"the number of cards that are dealt to the second player."},{"Start":"04:14.105 ","End":"04:19.505","Text":"The second player gets 5 cards out of the remaining 47."},{"Start":"04:19.505 ","End":"04:21.845","Text":"We have 47 over 5."},{"Start":"04:21.845 ","End":"04:30.270","Text":"This probability here is equal to 0.837."}],"ID":12823},{"Watched":false,"Name":"Exercise 6 - Part a-c","Duration":"5m 22s","ChapterTopicVideoID":12345,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12345.jpeg","UploadDate":"2018-07-22T16:55:55.5300000","DurationForVideoObject":"PT5M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.239","Text":"In this question, we\u0027re given 4 study tracks at a college,"},{"Start":"00:03.239 ","End":"00:06.455","Text":"and we have 5 secretaries for each study track."},{"Start":"00:06.455 ","End":"00:08.910","Text":"Now, a 5 member committee is selected"},{"Start":"00:08.910 ","End":"00:12.270","Text":"randomly from all of the secretaries at the college,"},{"Start":"00:12.270 ","End":"00:14.820","Text":"and we\u0027re asked to calculate the following probabilities."},{"Start":"00:14.820 ","End":"00:16.230","Text":"Now, before we do that,"},{"Start":"00:16.230 ","End":"00:18.960","Text":"the first thing that we need to do is to calculate the size"},{"Start":"00:18.960 ","End":"00:21.635","Text":"of my sample space. How do we do that?"},{"Start":"00:21.635 ","End":"00:24.580","Text":"Well, we have 4 tracks, T_1,"},{"Start":"00:24.580 ","End":"00:28.470","Text":"T_2, T_3, and T_4."},{"Start":"00:28.470 ","End":"00:31.325","Text":"Now, for each track we have 5 secretaries."},{"Start":"00:31.325 ","End":"00:34.670","Text":"How many secretaries do we have in total?"},{"Start":"00:34.670 ","End":"00:36.830","Text":"Well n here that equals to 20."},{"Start":"00:36.830 ","End":"00:38.975","Text":"5 plus 5 plus 5 plus 5."},{"Start":"00:38.975 ","End":"00:42.890","Text":"How many secretaries do we select for the committee?"},{"Start":"00:42.890 ","End":"00:46.235","Text":"Well, it\u0027s a 5 member committee."},{"Start":"00:46.235 ","End":"00:50.074","Text":"That means that the size of my sample space,"},{"Start":"00:50.074 ","End":"00:55.175","Text":"well, I have to choose 5 secretaries from a total of 20."},{"Start":"00:55.175 ","End":"00:58.955","Text":"This would be the size of my sample space."},{"Start":"00:58.955 ","End":"01:03.500","Text":"Now, we can go ahead and calculate the probability in the first 1"},{"Start":"01:03.500 ","End":"01:08.635","Text":"is that all secretaries and the committee are from 1 specific study track."},{"Start":"01:08.635 ","End":"01:11.120","Text":"This is the probability."},{"Start":"01:11.120 ","End":"01:15.800","Text":"Now, in the denominator we have the size of the sample space,"},{"Start":"01:15.800 ","End":"01:17.570","Text":"which is 20 over 5."},{"Start":"01:17.570 ","End":"01:22.970","Text":"Now, if we want all the secretaries to be from 1 specific track,"},{"Start":"01:22.970 ","End":"01:25.693","Text":"well, how many secretaries do we have in a track?"},{"Start":"01:25.693 ","End":"01:27.475","Text":"We have 5."},{"Start":"01:27.475 ","End":"01:31.760","Text":"Now, how many secretaries are on the committee?"},{"Start":"01:31.760 ","End":"01:38.415","Text":"Were 5, so we have to choose 5 out of the 5 secretaries."},{"Start":"01:38.415 ","End":"01:43.670","Text":"5 secretaries out of the 5 that are in this 1 specific track."},{"Start":"01:43.670 ","End":"01:47.240","Text":"Now, that equals, well, 5 over 5,"},{"Start":"01:47.240 ","End":"01:50.114","Text":"that equals to 1 and 20 over 5,"},{"Start":"01:50.114 ","End":"01:55.070","Text":"that equals to 15,504."},{"Start":"01:55.070 ","End":"01:57.199","Text":"Now, if you use your calculator,"},{"Start":"01:57.199 ","End":"02:04.210","Text":"you might see this 6.45 times 10 to the power of minus 5,"},{"Start":"02:04.210 ","End":"02:12.960","Text":"or in decimal notation that\u0027ll be 0.0000645."},{"Start":"02:12.960 ","End":"02:18.815","Text":"Any 1 of these representations would be valid."},{"Start":"02:18.815 ","End":"02:21.830","Text":"In this section, we\u0027re asked for the probability that"},{"Start":"02:21.830 ","End":"02:24.845","Text":"all the secretaries and the committee are from the same track."},{"Start":"02:24.845 ","End":"02:26.735","Text":"Not 1 specific track,"},{"Start":"02:26.735 ","End":"02:28.340","Text":"but from the same track."},{"Start":"02:28.340 ","End":"02:33.980","Text":"That means that they could be all from track number 1 or all from track number 2,"},{"Start":"02:33.980 ","End":"02:35.360","Text":"and so on and so forth."},{"Start":"02:35.360 ","End":"02:38.075","Text":"We\u0027re looking for this probability."},{"Start":"02:38.075 ","End":"02:40.745","Text":"Well, that equals to the following."},{"Start":"02:40.745 ","End":"02:44.780","Text":"In the denominator, we have the size of the sample space,"},{"Start":"02:44.780 ","End":"02:46.640","Text":"which is 20 over 5."},{"Start":"02:46.640 ","End":"02:51.650","Text":"Now, because we didn\u0027t say which track,"},{"Start":"02:51.650 ","End":"02:54.455","Text":"so we have to first choose a track."},{"Start":"02:54.455 ","End":"02:58.270","Text":"We have 4 tracks and we choose 1 out of the 4."},{"Start":"02:58.270 ","End":"03:00.470","Text":"Now, having chosen that track,"},{"Start":"03:00.470 ","End":"03:06.470","Text":"we need to choose 5 secretaries out of the 5 in that track."},{"Start":"03:06.470 ","End":"03:14.780","Text":"That comes out to 4 over 15,504."},{"Start":"03:14.780 ","End":"03:17.960","Text":"In this section, we\u0027re asked for the probability that"},{"Start":"03:17.960 ","End":"03:21.530","Text":"at least 1 secretary selected from each study track."},{"Start":"03:21.530 ","End":"03:25.075","Text":"Let\u0027s write down this probability."},{"Start":"03:25.075 ","End":"03:31.370","Text":"We\u0027re looking for the probability that at least 1 secretary is from each track."},{"Start":"03:31.370 ","End":"03:34.100","Text":"Now, let\u0027s just look at this for a bit."},{"Start":"03:34.100 ","End":"03:36.575","Text":"We have 4 tracks, Track number 1,"},{"Start":"03:36.575 ","End":"03:38.000","Text":"Track number 2,"},{"Start":"03:38.000 ","End":"03:41.140","Text":"Track number 3, and Track number 4."},{"Start":"03:41.140 ","End":"03:44.360","Text":"Now, we want at least 1 secretary from each study tracks."},{"Start":"03:44.360 ","End":"03:46.145","Text":"We have a secretary from here,"},{"Start":"03:46.145 ","End":"03:47.735","Text":"a secretary from here,"},{"Start":"03:47.735 ","End":"03:49.160","Text":"from here, and from here,"},{"Start":"03:49.160 ","End":"03:51.740","Text":"and now we have 1 more secretary to choose."},{"Start":"03:51.740 ","End":"03:58.820","Text":"Now, that secretary could come from any 1 of the 4 tracks. Let\u0027s choose 1."},{"Start":"03:58.820 ","End":"04:04.435","Text":"Let\u0027s arbitrarily choose that the secretary comes from Track number 2."},{"Start":"04:04.435 ","End":"04:06.965","Text":"Now, that we understand that,"},{"Start":"04:06.965 ","End":"04:13.085","Text":"let\u0027s go back to our probability in the denominator we have the size of the sample space,"},{"Start":"04:13.085 ","End":"04:18.035","Text":"and that comes out to 15,504, that\u0027s 20 over 5."},{"Start":"04:18.035 ","End":"04:19.730","Text":"Now, what about the numerator?"},{"Start":"04:19.730 ","End":"04:23.015","Text":"Well first of all,"},{"Start":"04:23.015 ","End":"04:28.855","Text":"let\u0027s look at the track that has 2 secretaries."},{"Start":"04:28.855 ","End":"04:34.220","Text":"That means that I need to choose 1 out of the 4 tracks that has"},{"Start":"04:34.220 ","End":"04:39.920","Text":"the 2 secretaries and then out of the 5 secretaries in that track,"},{"Start":"04:39.920 ","End":"04:42.305","Text":"I have to choose 2."},{"Start":"04:42.305 ","End":"04:50.655","Text":"Now, for all the others I need 1 out of the 5 for each track."},{"Start":"04:50.655 ","End":"04:53.745","Text":"I have this is lets say for Track number 2,"},{"Start":"04:53.745 ","End":"04:55.740","Text":"this is for Track number 1,"},{"Start":"04:55.740 ","End":"05:00.945","Text":"this is for Track number 3 and this is for Track number 4."},{"Start":"05:00.945 ","End":"05:02.625","Text":"Track number 2,"},{"Start":"05:02.625 ","End":"05:04.875","Text":"I chose 2 secretaries,"},{"Start":"05:04.875 ","End":"05:06.690","Text":"in Track number 1,"},{"Start":"05:06.690 ","End":"05:09.330","Text":"I chose 1 secretary, Track number 3,"},{"Start":"05:09.330 ","End":"05:11.438","Text":"I chose 1 secretary out of 5,"},{"Start":"05:11.438 ","End":"05:12.525","Text":"and Track number 4,"},{"Start":"05:12.525 ","End":"05:14.610","Text":"I chose 1 secretary out of 5."},{"Start":"05:14.610 ","End":"05:19.770","Text":"Now, that comes out to 0.3225."}],"ID":12824},{"Watched":false,"Name":"Exercise 7 - Parts a","Duration":"7m 46s","ChapterTopicVideoID":12346,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12346.jpeg","UploadDate":"2018-07-22T16:56:23.9370000","DurationForVideoObject":"PT7M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this question, we\u0027re asked to prove that this expression"},{"Start":"00:03.300 ","End":"00:07.440","Text":"right here on the left is equal to this expression right here on the right."},{"Start":"00:07.440 ","End":"00:12.840","Text":"Let\u0027s call this expression on the right A and this expression on the left,"},{"Start":"00:12.840 ","End":"00:14.835","Text":"we\u0027ll call that B."},{"Start":"00:14.835 ","End":"00:18.370","Text":"Let\u0027s start with A."},{"Start":"00:18.470 ","End":"00:20.910","Text":"What does A equal to?"},{"Start":"00:20.910 ","End":"00:27.030","Text":"A, well, that equals to n plus 1 over k plus 1. What am I doing?"},{"Start":"00:27.030 ","End":"00:31.694","Text":"I\u0027m choosing k plus 1 objects from n plus 1 objects."},{"Start":"00:31.694 ","End":"00:39.770","Text":"Now, that\u0027s defined as n plus 1 factorial divided by k"},{"Start":"00:39.770 ","End":"00:44.960","Text":"plus 1 factorial times n plus"},{"Start":"00:44.960 ","End":"00:51.285","Text":"1 minus k plus 1 factorial."},{"Start":"00:51.285 ","End":"00:52.965","Text":"What does that equal to?"},{"Start":"00:52.965 ","End":"00:59.520","Text":"That equals to n plus 1 factorial divided by k"},{"Start":"00:59.520 ","End":"01:06.350","Text":"plus 1 factorial times n plus 1 minus k plus 1."},{"Start":"01:06.350 ","End":"01:12.905","Text":"Well, the 1s cancel each other out and we have n minus k factorial."},{"Start":"01:12.905 ","End":"01:16.470","Text":"We\u0027ll call this A."},{"Start":"01:16.780 ","End":"01:19.895","Text":"Now, what about B?"},{"Start":"01:19.895 ","End":"01:22.010","Text":"Well, what does that equal to?"},{"Start":"01:22.010 ","End":"01:30.270","Text":"Well, that\u0027s n over k plus n over k plus 1."},{"Start":"01:30.520 ","End":"01:39.485","Text":"What are we doing? We\u0027ll show that this thing is equal to this thing right here."},{"Start":"01:39.485 ","End":"01:41.420","Text":"What\u0027s this thing right here?"},{"Start":"01:41.420 ","End":"01:43.310","Text":"Well, that equals to the following."},{"Start":"01:43.310 ","End":"01:47.675","Text":"Well, n over k is n factorial divided by"},{"Start":"01:47.675 ","End":"01:54.550","Text":"k factorial times n minus k factorial plus,"},{"Start":"01:54.550 ","End":"01:56.675","Text":"n over k plus 1,"},{"Start":"01:56.675 ","End":"02:02.270","Text":"that\u0027ll be n factorial divided by k"},{"Start":"02:02.270 ","End":"02:10.995","Text":"plus 1 factorial times n minus k plus 1 factorial."},{"Start":"02:10.995 ","End":"02:17.165","Text":"That equals to n factorial divided by k factorial times"},{"Start":"02:17.165 ","End":"02:27.050","Text":"n minus k factorial plus n factorial divided by k plus 1 factorial times,"},{"Start":"02:27.050 ","End":"02:29.240","Text":"well, let\u0027s just simplify that,"},{"Start":"02:29.240 ","End":"02:34.550","Text":"that\u0027ll be n minus k minus 1 factorial."},{"Start":"02:34.550 ","End":"02:39.110","Text":"We want to create a common denominator here."},{"Start":"02:39.110 ","End":"02:40.910","Text":"In order to do that,"},{"Start":"02:40.910 ","End":"02:45.395","Text":"we need to understand the following principle."},{"Start":"02:45.395 ","End":"02:48.275","Text":"Whenever we have a factorial,"},{"Start":"02:48.275 ","End":"02:55.040","Text":"n factorial, and we want to multiply that by n plus 1."},{"Start":"02:55.040 ","End":"03:01.080","Text":"Well, that equals to n plus 1 factorial."},{"Start":"03:01.080 ","End":"03:03.380","Text":"Why is that? Well, what\u0027s n factorial?"},{"Start":"03:03.380 ","End":"03:05.660","Text":"That\u0027s 1 times 2 times 3,"},{"Start":"03:05.660 ","End":"03:06.935","Text":"and so on and so forth,"},{"Start":"03:06.935 ","End":"03:11.875","Text":"times n times n plus 1."},{"Start":"03:11.875 ","End":"03:16.205","Text":"Well, whenever we have this series right here is continuous 1, 2,"},{"Start":"03:16.205 ","End":"03:22.380","Text":"3 and so on and so forth up to n. Then we multiply that by n plus 1,"},{"Start":"03:22.380 ","End":"03:27.640","Text":"well, that\u0027s the definition of n plus 1 factorial."},{"Start":"03:29.060 ","End":"03:32.245","Text":"Once we understand that,"},{"Start":"03:32.245 ","End":"03:36.760","Text":"then let\u0027s get back here to calculate the common denominator."},{"Start":"03:36.760 ","End":"03:38.380","Text":"How do we do that?"},{"Start":"03:38.380 ","End":"03:42.370","Text":"Well, then this sum equals to the following."},{"Start":"03:42.370 ","End":"03:45.190","Text":"That\u0027ll be n factorial right here."},{"Start":"03:45.190 ","End":"03:47.275","Text":"This is this expression right here,"},{"Start":"03:47.275 ","End":"03:51.615","Text":"divided by k factorial."},{"Start":"03:51.615 ","End":"03:57.605","Text":"Now I want to multiply this by k plus 1."},{"Start":"03:57.605 ","End":"04:00.530","Text":"If I multiply this right here,"},{"Start":"04:00.530 ","End":"04:02.120","Text":"the denominator by k plus 1,"},{"Start":"04:02.120 ","End":"04:03.305","Text":"then in the numerator,"},{"Start":"04:03.305 ","End":"04:10.760","Text":"I also have to multiply this by k plus 1 times what\u0027s left."},{"Start":"04:10.760 ","End":"04:20.975","Text":"Well, that\u0027s n minus k factorial plus n factorial divided by,"},{"Start":"04:20.975 ","End":"04:30.290","Text":"well, here I have k plus 1 factorial times n minus k minus 1 factorial."},{"Start":"04:30.290 ","End":"04:37.880","Text":"Here I want to multiply this by n minus k. If I multiply this by n minus k,"},{"Start":"04:37.880 ","End":"04:39.830","Text":"then up here in the numerator,"},{"Start":"04:39.830 ","End":"04:43.190","Text":"I have to multiply this by n minus k as well."},{"Start":"04:43.190 ","End":"04:45.380","Text":"Why am I doing all this?"},{"Start":"04:45.380 ","End":"04:48.860","Text":"Well, let\u0027s take a look."},{"Start":"04:48.860 ","End":"04:51.080","Text":"What\u0027s this expression now?"},{"Start":"04:51.080 ","End":"04:56.915","Text":"That\u0027ll be n factorial times k plus 1 divided by,"},{"Start":"04:56.915 ","End":"04:59.900","Text":"what\u0027s k factorial times k plus 1?"},{"Start":"04:59.900 ","End":"05:02.810","Text":"Well, that\u0027s this expression right here."},{"Start":"05:02.810 ","End":"05:05.900","Text":"We have n factorial times n plus 1,"},{"Start":"05:05.900 ","End":"05:08.150","Text":"that equals to n plus 1 factorial."},{"Start":"05:08.150 ","End":"05:11.840","Text":"So k factorial times k plus 1,"},{"Start":"05:11.840 ","End":"05:20.480","Text":"that\u0027s k plus 1 factorial times n minus k plus, well,"},{"Start":"05:20.480 ","End":"05:27.530","Text":"here we have n factorial times n minus k divided by, well,"},{"Start":"05:27.530 ","End":"05:31.160","Text":"here we have k plus 1 factorial,"},{"Start":"05:31.160 ","End":"05:38.390","Text":"and here again here we have n minus k minus 1 times n minus k. Well,"},{"Start":"05:38.390 ","End":"05:47.265","Text":"that means that this thing right here becomes n minus k factorial."},{"Start":"05:47.265 ","End":"05:50.535","Text":"Now we have a common denominator."},{"Start":"05:50.535 ","End":"05:52.160","Text":"What does that equal to?"},{"Start":"05:52.160 ","End":"05:54.425","Text":"Well, here in the denominator,"},{"Start":"05:54.425 ","End":"06:01.070","Text":"we have k plus 1 factorial times n minus k factorial."},{"Start":"06:01.070 ","End":"06:07.744","Text":"Here we have n factorial times k plus 1"},{"Start":"06:07.744 ","End":"06:16.835","Text":"plus n factorial times n minus k. I hope you\u0027re not tired."},{"Start":"06:16.835 ","End":"06:18.275","Text":"Let\u0027s continue."},{"Start":"06:18.275 ","End":"06:20.240","Text":"What is this equal to?"},{"Start":"06:20.240 ","End":"06:29.150","Text":"Well, here again we have k plus 1 factorial times n minus k factorial."},{"Start":"06:29.150 ","End":"06:32.960","Text":"Let\u0027s take n factorial out. What are we left with?"},{"Start":"06:32.960 ","End":"06:39.715","Text":"Well, we\u0027re left with k plus 1 here plus n minus k,"},{"Start":"06:39.715 ","End":"06:44.060","Text":"that equals the k\u0027s cancel each other out,"},{"Start":"06:44.060 ","End":"06:52.340","Text":"k plus 1 plus n minus k. That\u0027ll be n factorial times n plus"},{"Start":"06:52.340 ","End":"07:02.400","Text":"1 divided by k plus 1 factorial times n minus k factorial."},{"Start":"07:02.400 ","End":"07:05.385","Text":"What\u0027s n factorial times n plus 1?"},{"Start":"07:05.385 ","End":"07:11.675","Text":"That\u0027s n plus 1 factorial divided by the denominator,"},{"Start":"07:11.675 ","End":"07:17.930","Text":"k plus 1 factorial times n minus k factorial."},{"Start":"07:17.930 ","End":"07:25.595","Text":"Well here we stop because now we\u0027ve reached the exact expression that we had an A."},{"Start":"07:25.595 ","End":"07:33.470","Text":"We have n plus 1 factorial divided by k plus 1 factorial times n minus k factorial."},{"Start":"07:33.470 ","End":"07:37.345","Text":"Well, this is exactly what we have here."},{"Start":"07:37.345 ","End":"07:41.475","Text":"This proves that this expression,"},{"Start":"07:41.475 ","End":"07:46.290","Text":"expression A, is equal to expression B."}],"ID":12825},{"Watched":false,"Name":"Exercise 7 - Parts b","Duration":"3m 16s","ChapterTopicVideoID":12347,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12347.jpeg","UploadDate":"2018-07-22T16:56:32.6600000","DurationForVideoObject":"PT3M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.165","Text":"Let\u0027s now take a look at this equation from a different perspective."},{"Start":"00:03.165 ","End":"00:06.495","Text":"Up to now, we\u0027ve proved this equation mathematically,"},{"Start":"00:06.495 ","End":"00:11.595","Text":"but let\u0027s take a look at this from a combinatorial perspective."},{"Start":"00:11.595 ","End":"00:14.220","Text":"We\u0027ll use some combinatorial logic here."},{"Start":"00:14.220 ","End":"00:18.525","Text":"We\u0027ll still call this A and this we\u0027ll call B."},{"Start":"00:18.525 ","End":"00:20.445","Text":"What\u0027s A right here?"},{"Start":"00:20.445 ","End":"00:28.020","Text":"Well, A is choosing k plus 1 people out of a group of n plus 1."},{"Start":"00:28.020 ","End":"00:30.740","Text":"Now, what does that mean?"},{"Start":"00:30.740 ","End":"00:36.020","Text":"Well, let\u0027s just make this a down to earth example here."},{"Start":"00:36.020 ","End":"00:39.935","Text":"Let\u0027s take a group of people here and I have n plus 1,"},{"Start":"00:39.935 ","End":"00:42.965","Text":"in the group, n plus 1 people in the group."},{"Start":"00:42.965 ","End":"00:45.410","Text":"I want to choose k plus 1 people."},{"Start":"00:45.410 ","End":"00:48.350","Text":"Now this 1 person right here,"},{"Start":"00:48.350 ","End":"00:49.715","Text":"well, let\u0027s point him out."},{"Start":"00:49.715 ","End":"00:51.460","Text":"Let\u0027s call him Joe."},{"Start":"00:51.460 ","End":"00:54.570","Text":"I have n people plus Joe,"},{"Start":"00:54.570 ","End":"00:56.190","Text":"n plus 1 people,"},{"Start":"00:56.190 ","End":"00:58.290","Text":"and out of this group,"},{"Start":"00:58.290 ","End":"01:02.985","Text":"I want to select k plus 1 people."},{"Start":"01:02.985 ","End":"01:06.170","Text":"This is what A means right here."},{"Start":"01:06.170 ","End":"01:10.230","Text":"Now, let\u0027s take a look at B."},{"Start":"01:10.570 ","End":"01:14.845","Text":"Well, when choosing k plus 1 people."},{"Start":"01:14.845 ","End":"01:16.770","Text":"Well, I have 2 scenarios,"},{"Start":"01:16.770 ","End":"01:19.450","Text":"2 mutually exclusive events."},{"Start":"01:19.450 ","End":"01:24.250","Text":"Joe can be in this group of k plus 1 that I chose."},{"Start":"01:24.250 ","End":"01:28.075","Text":"Or he cannot be in the k plus 1 people."},{"Start":"01:28.075 ","End":"01:32.205","Text":"Now, let\u0027s just calculate these 2 events right here."},{"Start":"01:32.205 ","End":"01:35.275","Text":"Now, if he\u0027s in the group,"},{"Start":"01:35.275 ","End":"01:38.410","Text":"well, how can we calculate this?"},{"Start":"01:38.410 ","End":"01:43.165","Text":"Well, that\u0027ll be n over k times 1 over 1."},{"Start":"01:43.165 ","End":"01:44.875","Text":"What do I mean by that?"},{"Start":"01:44.875 ","End":"01:49.645","Text":"Well, n over k means that Joe is not in the group and I"},{"Start":"01:49.645 ","End":"01:55.170","Text":"chose k people out of the n. Then I chose Joe,"},{"Start":"01:55.170 ","End":"01:57.690","Text":"and there\u0027s only 1 special person,"},{"Start":"01:57.690 ","End":"01:59.460","Text":"and I chose him."},{"Start":"01:59.460 ","End":"02:05.465","Text":"This would be the multiplication of n over k times 1 over 1."},{"Start":"02:05.465 ","End":"02:08.700","Text":"This is the first event."},{"Start":"02:09.340 ","End":"02:14.480","Text":"That\u0027s the number of combinations of choosing k plus 1 where"},{"Start":"02:14.480 ","End":"02:18.965","Text":"Joe is in the k plus 1 people right here."},{"Start":"02:18.965 ","End":"02:21.050","Text":"But there\u0027s another scenario."},{"Start":"02:21.050 ","End":"02:26.150","Text":"The scenario is that I choose k plus 1 people,"},{"Start":"02:26.150 ","End":"02:28.295","Text":"but Joe was not in the group,"},{"Start":"02:28.295 ","End":"02:33.440","Text":"that means that I\u0027m choosing out of a group of n people right here."},{"Start":"02:33.440 ","End":"02:42.245","Text":"In both cases, I choose k plus 1 people out of a group of n plus 1."},{"Start":"02:42.245 ","End":"02:45.560","Text":"As we can see in both cases,"},{"Start":"02:45.560 ","End":"02:49.675","Text":"we chose k plus 1 people,"},{"Start":"02:49.675 ","End":"02:52.260","Text":"out of a group of n plus 1,"},{"Start":"02:52.260 ","End":"02:54.435","Text":"that\u0027s what we meant in A."},{"Start":"02:54.435 ","End":"03:00.733","Text":"In B, we just split that up where Joe was included in the k plus 1 people,"},{"Start":"03:00.733 ","End":"03:03.960","Text":"and Joe was not included in the k plus 1 people."},{"Start":"03:03.960 ","End":"03:05.510","Text":"In both cases here,"},{"Start":"03:05.510 ","End":"03:10.655","Text":"we chose k plus 1 people out of a group of n plus 1."},{"Start":"03:10.655 ","End":"03:16.050","Text":"As we can see, A and B are equal to each other."}],"ID":12826},{"Watched":false,"Name":"Exercise 8 - Parts a-b","Duration":"6m 51s","ChapterTopicVideoID":12348,"CourseChapterTopicPlaylistID":264990,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12348.jpeg","UploadDate":"2018-07-22T16:57:00.9730000","DurationForVideoObject":"PT6M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.260","Text":"In this question, we\u0027re given 2n boys and 2n girls and they\u0027re divided into 2 groups."},{"Start":"00:07.260 ","End":"00:12.420","Text":"We\u0027re asked in how many different ways can the groups be selected if the 2 groups must be"},{"Start":"00:12.420 ","End":"00:18.030","Text":"the same size and in each group has the same number of boys and girls?"},{"Start":"00:18.030 ","End":"00:19.380","Text":"Before we answer that,"},{"Start":"00:19.380 ","End":"00:21.900","Text":"let\u0027s just understand where we\u0027re at."},{"Start":"00:21.900 ","End":"00:28.769","Text":"We have 2n boys and we have 2n girls,"},{"Start":"00:28.769 ","End":"00:32.685","Text":"a total of 4n people."},{"Start":"00:32.685 ","End":"00:37.560","Text":"Here, out of these 4n people,"},{"Start":"00:37.560 ","End":"00:39.810","Text":"we have to create 2 groups."},{"Start":"00:39.810 ","End":"00:45.750","Text":"We\u0027ll call this group number 1 and group number 2."},{"Start":"00:45.750 ","End":"00:50.150","Text":"Now, we\u0027re given that the groups are the same size."},{"Start":"00:50.150 ","End":"00:55.940","Text":"That means that if you have 4n people that\u0027s divided into 2 groups of the same size,"},{"Start":"00:55.940 ","End":"01:00.300","Text":"then group number 1 has 2n people and"},{"Start":"01:00.300 ","End":"01:05.710","Text":"group number 2 has also 2n people now that they\u0027re the same size."},{"Start":"01:05.710 ","End":"01:10.720","Text":"We\u0027re also given that each group has the same number of girls and boys,"},{"Start":"01:10.720 ","End":"01:14.665","Text":"so out of the 2n people in group number 1,"},{"Start":"01:14.665 ","End":"01:21.210","Text":"we have n boys and we have n girls."},{"Start":"01:21.210 ","End":"01:23.840","Text":"The same thing for group number 2,"},{"Start":"01:23.840 ","End":"01:30.465","Text":"we have n boys and n girls."},{"Start":"01:30.465 ","End":"01:33.135","Text":"Now that we\u0027ve understood this,"},{"Start":"01:33.135 ","End":"01:38.670","Text":"let\u0027s see how many different ways can the groups be selected."},{"Start":"01:39.680 ","End":"01:42.660","Text":"Let\u0027s take a look at group number 1."},{"Start":"01:42.660 ","End":"01:47.970","Text":"We have to select n boys out of total of 2n boys."},{"Start":"01:47.970 ","End":"01:51.390","Text":"In group number 1, out of 2n boys,"},{"Start":"01:51.390 ","End":"01:55.785","Text":"we have to select n of them and the same thing for the girls."},{"Start":"01:55.785 ","End":"01:58.050","Text":"Out of the 2n girls,"},{"Start":"01:58.050 ","End":"02:03.300","Text":"we have to select n. What happens to group number 2?"},{"Start":"02:03.300 ","End":"02:08.760","Text":"Well, if I\u0027ve selected n boys for group number 1, well,"},{"Start":"02:08.760 ","End":"02:15.200","Text":"I\u0027m left with n boys for group number 2 and I have to select n boys for group number 2,"},{"Start":"02:15.200 ","End":"02:18.950","Text":"so it\u0027ll be n over n. The same thing for the girls,"},{"Start":"02:18.950 ","End":"02:21.125","Text":"that will be n over n. I have"},{"Start":"02:21.125 ","End":"02:24.430","Text":"n girls remaining and I have to choose them for group number 2."},{"Start":"02:24.430 ","End":"02:30.839","Text":"Now, this comes out to 2n over n times"},{"Start":"02:30.839 ","End":"02:39.210","Text":"2n over n. That equals to 2n over n and we have to square that."},{"Start":"02:39.210 ","End":"02:46.900","Text":"This is the number of different ways that the groups can be selected."},{"Start":"02:46.900 ","End":"02:50.860","Text":"In this section, we\u0027re asked in how many different ways can the groups"},{"Start":"02:50.860 ","End":"02:54.685","Text":"be selected if each group has an equal number of girls and boys,"},{"Start":"02:54.685 ","End":"02:58.450","Text":"but the 2 groups are not necessarily the same size?"},{"Start":"02:58.450 ","End":"03:02.020","Text":"Well, the difficulty in answering these questions"},{"Start":"03:02.020 ","End":"03:06.580","Text":"is trying to understand what\u0027s really required of us."},{"Start":"03:06.580 ","End":"03:09.655","Text":"Let\u0027s just take a specific example here."},{"Start":"03:09.655 ","End":"03:14.100","Text":"Let\u0027s say I have 10 boys and 10 girls,"},{"Start":"03:14.100 ","End":"03:20.195","Text":"and I have group number 1 and group number 2."},{"Start":"03:20.195 ","End":"03:24.875","Text":"Well, let\u0027s assume that I choose 1 boy for group number 1."},{"Start":"03:24.875 ","End":"03:28.550","Text":"That means I have 9 boys for group number 2."},{"Start":"03:28.550 ","End":"03:30.980","Text":"If I choose 1 boy for group number 1,"},{"Start":"03:30.980 ","End":"03:34.250","Text":"I have to choose 1 girl for group number 1 because I need"},{"Start":"03:34.250 ","End":"03:37.960","Text":"an equal number of girls and boys in each group."},{"Start":"03:37.960 ","End":"03:42.875","Text":"If that\u0027s the case, then here I have to have 9 girls."},{"Start":"03:42.875 ","End":"03:48.050","Text":"A total of 2 people in group number 1,"},{"Start":"03:48.050 ","End":"03:51.030","Text":"a total of 18 people in group number 2."},{"Start":"03:51.030 ","End":"03:55.685","Text":"We see that we have an equal number of boys and girls in each group,"},{"Start":"03:55.685 ","End":"03:58.865","Text":"but the sizes of the group are different."},{"Start":"03:58.865 ","End":"04:00.920","Text":"Let\u0027s just take another example here."},{"Start":"04:00.920 ","End":"04:03.710","Text":"Let\u0027s say I have 2 boys in group number 1."},{"Start":"04:03.710 ","End":"04:06.965","Text":"That means I need 2 girls in group number 1,"},{"Start":"04:06.965 ","End":"04:08.810","Text":"a total of 4 people."},{"Start":"04:08.810 ","End":"04:11.180","Text":"If I have 2 boys in group number 1,"},{"Start":"04:11.180 ","End":"04:14.840","Text":"I have 8 boys in group number 2,"},{"Start":"04:14.840 ","End":"04:16.430","Text":"8 girls in group number 2,"},{"Start":"04:16.430 ","End":"04:18.875","Text":"a total of 16 people."},{"Start":"04:18.875 ","End":"04:23.240","Text":"Again, we see that we have the same number of boys and girls in each group,"},{"Start":"04:23.240 ","End":"04:25.310","Text":"but the group sizes are different."},{"Start":"04:25.310 ","End":"04:27.872","Text":"These are specific cases,"},{"Start":"04:27.872 ","End":"04:30.785","Text":"let\u0027s do this in a general way."},{"Start":"04:30.785 ","End":"04:38.400","Text":"Now we have 2n boys and 2n girls."},{"Start":"04:39.220 ","End":"04:41.330","Text":"What do we need to do?"},{"Start":"04:41.330 ","End":"04:47.270","Text":"Well, we need to choose boys and girls for group number 1 and group number 2."},{"Start":"04:47.270 ","End":"04:52.400","Text":"Again, each group has to have an equal number of girls and boys,"},{"Start":"04:52.400 ","End":"04:57.400","Text":"but the 2 groups do not necessarily have to be the same size."},{"Start":"04:57.400 ","End":"05:00.405","Text":"Let\u0027s take a look at group number 1."},{"Start":"05:00.405 ","End":"05:01.890","Text":"In group number 1,"},{"Start":"05:01.890 ","End":"05:04.335","Text":"out of the 2n boys,"},{"Start":"05:04.335 ","End":"05:07.360","Text":"I choose i boys."},{"Start":"05:07.880 ","End":"05:10.485","Text":"In group number 1,"},{"Start":"05:10.485 ","End":"05:12.140","Text":"out of the 2n girls,"},{"Start":"05:12.140 ","End":"05:15.365","Text":"I need to choose the same number of girls."},{"Start":"05:15.365 ","End":"05:22.850","Text":"Having chosen a specific number of boys and girls for group number 1,"},{"Start":"05:22.850 ","End":"05:29.270","Text":"well, I automatically choose the number of boys and girls for group number 2."},{"Start":"05:29.270 ","End":"05:36.080","Text":"For example, if I chose 1 boy and 1 girl in group number 1, then I have,"},{"Start":"05:36.080 ","End":"05:40.380","Text":"let\u0027s say 9 boys and 9 girls in group number 2,"},{"Start":"05:40.380 ","End":"05:43.590","Text":"if we have 10 boys and 10 girls in total"},{"Start":"05:43.590 ","End":"05:47.220","Text":"or if I choose 2 boys and 2 girls in group number 1,"},{"Start":"05:47.220 ","End":"05:50.985","Text":"then I have 8 boys and 8 girls in group number 2."},{"Start":"05:50.985 ","End":"05:54.600","Text":"Automatically, the second group is"},{"Start":"05:54.600 ","End":"06:00.720","Text":"determined once I determine the number of boys and girls in group number 1."},{"Start":"06:01.510 ","End":"06:05.750","Text":"The thing here is that I have many combinations."},{"Start":"06:05.750 ","End":"06:09.590","Text":"I can choose 1 boy and 1 girl for group number 1"},{"Start":"06:09.590 ","End":"06:14.600","Text":"or 2 boys and 2 girls and so on and so forth. What do I need to do?"},{"Start":"06:14.600 ","End":"06:19.670","Text":"I need to sum this product over all values of i,"},{"Start":"06:19.670 ","End":"06:22.790","Text":"where i goes from 1 to n. Why n?"},{"Start":"06:22.790 ","End":"06:26.555","Text":"Well, that\u0027s half of the number of boys and girls,"},{"Start":"06:26.555 ","End":"06:30.550","Text":"in order to make the 2 groups."},{"Start":"06:30.550 ","End":"06:37.070","Text":"Basically this then, well that equals to the sum where i goes"},{"Start":"06:37.070 ","End":"06:45.075","Text":"from 1 to n of 2n over i squared."},{"Start":"06:45.075 ","End":"06:51.869","Text":"This then would be the number of different ways that the groups can be selected."}],"ID":12827}],"Thumbnail":null,"ID":264990},{"Name":"Non-Sequential Sampling with Returns","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"1m 12s","ChapterTopicVideoID":12349,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12349.jpeg","UploadDate":"2018-07-22T17:03:11.7500000","DurationForVideoObject":"PT1M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this chapter, I\u0027ll be talking about combinatorics,"},{"Start":"00:02.220 ","End":"00:05.040","Text":"specifically non-sequential sampling with returns."},{"Start":"00:05.040 ","End":"00:08.670","Text":"This isn\u0027t a very common topic to encounter,"},{"Start":"00:08.670 ","End":"00:10.825","Text":"but we do have to talk about it."},{"Start":"00:10.825 ","End":"00:16.065","Text":"The number of ways for selecting k objects not necessarily different,"},{"Start":"00:16.065 ","End":"00:19.410","Text":"that means that we can choose the same object more than once,"},{"Start":"00:19.410 ","End":"00:24.150","Text":"from n objects, where the order of the sampled object is not important"},{"Start":"00:24.150 ","End":"00:29.115","Text":"and the object can be selected more than once is this expression right here."},{"Start":"00:29.115 ","End":"00:36.660","Text":"That\u0027s n plus k minus 1 over k. That equals to n plus k minus 1 over n minus 1."},{"Start":"00:36.660 ","End":"00:38.960","Text":"This equality is very simple to prove."},{"Start":"00:38.960 ","End":"00:40.640","Text":"I\u0027ll leave that up to you."},{"Start":"00:40.640 ","End":"00:44.300","Text":"We can also see this notation."},{"Start":"00:44.300 ","End":"00:50.660","Text":"We can see C superscript k subscript n plus k minus 1,"},{"Start":"00:50.660 ","End":"00:58.100","Text":"or C superscript n minus 1 subscript n plus k minus 1."},{"Start":"00:58.100 ","End":"01:01.580","Text":"I won\u0027t be using this notation right here."},{"Start":"01:01.580 ","End":"01:03.785","Text":"I\u0027ll be using this notation,"},{"Start":"01:03.785 ","End":"01:08.785","Text":"but I mention it because we do see this in the literature sometimes."},{"Start":"01:08.785 ","End":"01:12.270","Text":"Let\u0027s take a look at an example right now."}],"ID":12828},{"Watched":false,"Name":"Tutorial 2","Duration":"3m 48s","ChapterTopicVideoID":12350,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12350.jpeg","UploadDate":"2018-07-22T17:03:21.9230000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.395","Text":"In this example we\u0027re asked,"},{"Start":"00:01.395 ","End":"00:06.600","Text":"in how many different ways can 4 identical balls be distributed among 3 boxes,"},{"Start":"00:06.600 ","End":"00:10.427","Text":"when each box can contain more than 1 ball?"},{"Start":"00:10.427 ","End":"00:15.060","Text":"Here we have our 4 balls and we have our 3 boxes numbered 1, 2,"},{"Start":"00:15.060 ","End":"00:18.345","Text":"and 3, and what we want to do is we want to take each ball,"},{"Start":"00:18.345 ","End":"00:20.610","Text":"put them in 1 of the 3 boxes here."},{"Start":"00:20.610 ","End":"00:23.865","Text":"An example that would be something like this,"},{"Start":"00:23.865 ","End":"00:25.545","Text":"where we have a ball."},{"Start":"00:25.545 ","End":"00:28.785","Text":"In box number 1, we have 2 balls and box number 2,"},{"Start":"00:28.785 ","End":"00:31.680","Text":"and we have another ball in box number 3."},{"Start":"00:31.680 ","End":"00:34.860","Text":"Now, we don\u0027t know which of the 4 balls goes into"},{"Start":"00:34.860 ","End":"00:39.075","Text":"which of the boxes because the balls are identical."},{"Start":"00:39.075 ","End":"00:47.135","Text":"Here we have a classic case of non-sequential sampling with returns. Now why is that?"},{"Start":"00:47.135 ","End":"00:54.500","Text":"Well, let\u0027s start with the fact that we have sampling with returns. What do we do?"},{"Start":"00:54.500 ","End":"00:56.945","Text":"We choose a ball,"},{"Start":"00:56.945 ","End":"01:04.440","Text":"and then we have to sample 1 of the 3 boxes in which to put the ball."},{"Start":"01:05.600 ","End":"01:09.510","Text":"We do that for all the 4 balls right here."},{"Start":"01:09.510 ","End":"01:12.430","Text":"When we choose a ball,"},{"Start":"01:13.400 ","End":"01:16.485","Text":"how many options do I have?"},{"Start":"01:16.485 ","End":"01:20.250","Text":"Well, I sample 1 out of 3 boxes right here."},{"Start":"01:20.250 ","End":"01:25.440","Text":"So n here, that equals to 3 as the number of boxes."},{"Start":"01:25.440 ","End":"01:29.220","Text":"I do this for all 4 balls here."},{"Start":"01:29.220 ","End":"01:31.245","Text":"That means that k,"},{"Start":"01:31.245 ","End":"01:35.089","Text":"that equals to 4 because I sample 4 times."},{"Start":"01:35.089 ","End":"01:39.175","Text":"Now, why is it non-sequential?"},{"Start":"01:39.175 ","End":"01:45.740","Text":"Well, it\u0027s non-sequential because it really doesn\u0027t matter which ball I choose first."},{"Start":"01:45.740 ","End":"01:51.380","Text":"Now, I can identify or recognize the fact that this is"},{"Start":"01:51.380 ","End":"01:56.840","Text":"non-sequential sampling with returns by what\u0027s given to me."},{"Start":"01:56.840 ","End":"02:00.110","Text":"Whenever I see that I have identical objects,"},{"Start":"02:00.110 ","End":"02:02.045","Text":"in this case identical balls,"},{"Start":"02:02.045 ","End":"02:05.990","Text":"then I know that it\u0027s non-sequential because it really doesn\u0027t"},{"Start":"02:05.990 ","End":"02:10.990","Text":"matter which ball I choose when I sample the boxes."},{"Start":"02:10.990 ","End":"02:18.420","Text":"It\u0027s with returns because each box can contain more than 1 balls."},{"Start":"02:18.420 ","End":"02:20.145","Text":"Whenever I see something I guess,"},{"Start":"02:20.145 ","End":"02:23.220","Text":"then I know that its with returns."},{"Start":"02:23.220 ","End":"02:26.955","Text":"Now that I know my n and my k,"},{"Start":"02:26.955 ","End":"02:32.140","Text":"let\u0027s just plug in the values here in the expressions."},{"Start":"02:32.140 ","End":"02:34.009","Text":"What\u0027s the expression?"},{"Start":"02:34.009 ","End":"02:40.655","Text":"The expression is k plus n minus 1 over k,"},{"Start":"02:40.655 ","End":"02:48.080","Text":"and that equals to k plus n minus 1 over n minus 1."},{"Start":"02:48.080 ","End":"02:50.480","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"02:50.480 ","End":"02:56.930","Text":"K equals 4 plus n is 3 minus 1 over k,"},{"Start":"02:56.930 ","End":"02:58.540","Text":"where k is 4."},{"Start":"02:58.540 ","End":"03:00.375","Text":"Now, what does that equal to?"},{"Start":"03:00.375 ","End":"03:01.700","Text":"Well, that\u0027s 4 plus 3,"},{"Start":"03:01.700 ","End":"03:03.095","Text":"that\u0027s 7 minus 1,"},{"Start":"03:03.095 ","End":"03:06.320","Text":"that equals to 6 over 4."},{"Start":"03:06.320 ","End":"03:09.515","Text":"Now, that equals to 15."},{"Start":"03:09.515 ","End":"03:11.780","Text":"Now if you use this expression right here,"},{"Start":"03:11.780 ","End":"03:16.780","Text":"well that\u0027ll be k. That\u0027s 4 plus 3."},{"Start":"03:16.780 ","End":"03:18.480","Text":"That\u0027s 7 minus 1,"},{"Start":"03:18.480 ","End":"03:21.120","Text":"that\u0027s 6 over n minus 1,"},{"Start":"03:21.120 ","End":"03:25.695","Text":"where n is 3, so 3 minus 1, that\u0027ll be 2."},{"Start":"03:25.695 ","End":"03:28.680","Text":"That equals to 6 over 2,"},{"Start":"03:28.680 ","End":"03:37.440","Text":"and we know that 6 over 2 is equal to 6 over 4 and that equals to 15."},{"Start":"03:37.440 ","End":"03:41.299","Text":"Now that we\u0027ve solved this exercise,"},{"Start":"03:41.299 ","End":"03:48.240","Text":"let\u0027s just see how people got to this expression in the first place."}],"ID":12829},{"Watched":false,"Name":"Tutorial 3","Duration":"4m 10s","ChapterTopicVideoID":12351,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12351.jpeg","UploadDate":"2018-07-22T17:03:32.3600000","DurationForVideoObject":"PT4M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.035","Text":"Let\u0027s take a look at the example where we had 1 ball in box number 1,"},{"Start":"00:04.035 ","End":"00:07.155","Text":"2 balls in box number 2 and 1 ball in box number 3."},{"Start":"00:07.155 ","End":"00:10.500","Text":"Now, instead of putting them in boxes,"},{"Start":"00:10.500 ","End":"00:14.160","Text":"let\u0027s put all the 4 balls in a row."},{"Start":"00:14.160 ","End":"00:17.190","Text":"But we\u0027ll split them up into partitions,"},{"Start":"00:17.190 ","End":"00:19.920","Text":"and each partition has a divider."},{"Start":"00:19.920 ","End":"00:26.960","Text":"For example, this would be placing the 4 balls in a row with 3 partitions."},{"Start":"00:26.960 ","End":"00:30.050","Text":"Now, the partitions have 2 dividers. These are the lines."},{"Start":"00:30.050 ","End":"00:32.545","Text":"Again, in partition number 1,"},{"Start":"00:32.545 ","End":"00:35.510","Text":"we have 1 ball, just like we did in the box here."},{"Start":"00:35.510 ","End":"00:36.950","Text":"In partition number 2,"},{"Start":"00:36.950 ","End":"00:40.760","Text":"we have 2 balls just like here and partition number 3,"},{"Start":"00:40.760 ","End":"00:42.980","Text":"we have 1 ball just like here."},{"Start":"00:42.980 ","End":"00:46.570","Text":"Now, let\u0027s take a look at another example."},{"Start":"00:46.570 ","End":"00:50.685","Text":"Here we have the 3 partitions,"},{"Start":"00:50.685 ","End":"00:54.260","Text":"but we have the 2 dividers here and we have the 4 balls."},{"Start":"00:54.260 ","End":"00:57.505","Text":"But in partition number 1,"},{"Start":"00:57.505 ","End":"00:58.730","Text":"we have no balls."},{"Start":"00:58.730 ","End":"01:01.205","Text":"In partition number 2 we have 1 ball,"},{"Start":"01:01.205 ","End":"01:04.415","Text":"and in partition number 3 we have 3 balls."},{"Start":"01:04.415 ","End":"01:10.850","Text":"Now, instead of thinking about this as balls and partitions,"},{"Start":"01:10.850 ","End":"01:14.540","Text":"let\u0027s just take a look at this as balls and dividers."},{"Start":"01:14.540 ","End":"01:16.640","Text":"Now, how many dividers do I have?"},{"Start":"01:16.640 ","End":"01:19.295","Text":"Well, I have 2 dividers that are the same,"},{"Start":"01:19.295 ","End":"01:21.575","Text":"and I have 4 balls that are the same."},{"Start":"01:21.575 ","End":"01:23.435","Text":"How many objects do I have?"},{"Start":"01:23.435 ","End":"01:26.220","Text":"Well, I have 6 objects, 1, 2,"},{"Start":"01:26.220 ","End":"01:29.325","Text":"3, 4, 5 and 6."},{"Start":"01:29.325 ","End":"01:31.560","Text":"1 would be the first divider,"},{"Start":"01:31.560 ","End":"01:32.760","Text":"2 would be this ball,"},{"Start":"01:32.760 ","End":"01:34.530","Text":"3 would be this divider."},{"Start":"01:34.530 ","End":"01:37.590","Text":"4, 5, and 6 would be these 3 balls."},{"Start":"01:37.590 ","End":"01:42.525","Text":"Now, how many ways can I order this in a row?"},{"Start":"01:42.525 ","End":"01:44.295","Text":"Well, I have 6 objects."},{"Start":"01:44.295 ","End":"01:46.770","Text":"Well, if they were all different objects,"},{"Start":"01:46.770 ","End":"01:49.425","Text":"then that would be 6 factorial."},{"Start":"01:49.425 ","End":"01:59.290","Text":"But I have 4 balls that are identical and I have 2 dividers that are identical."},{"Start":"01:59.290 ","End":"02:06.950","Text":"I have to divide this 6 factorial by 4 factorial times 2 factorial."},{"Start":"02:06.950 ","End":"02:10.370","Text":"I have to take the 6 factorial and I have to"},{"Start":"02:10.370 ","End":"02:17.980","Text":"neutralize the combinations of the balls."},{"Start":"02:17.980 ","End":"02:20.240","Text":"Because now the balls are identical."},{"Start":"02:20.240 ","End":"02:24.335","Text":"There\u0027s no meaning to the order in which the balls are placed."},{"Start":"02:24.335 ","End":"02:29.575","Text":"I have to neutralize that by dividing 6 factorial by 4 factorial."},{"Start":"02:29.575 ","End":"02:37.770","Text":"Then I have to neutralize the number of combinations of ordering the 2 dividers here."},{"Start":"02:38.530 ","End":"02:41.555","Text":"Let\u0027s take a general case here,"},{"Start":"02:41.555 ","End":"02:48.935","Text":"if we had n boxes and we have k balls,"},{"Start":"02:48.935 ","End":"02:55.755","Text":"well then we would have n minus 1 dividers,"},{"Start":"02:55.755 ","End":"02:57.250","Text":"just like we did here."},{"Start":"02:57.250 ","End":"03:01.060","Text":"We have 3 partitions, but 2 dividers."},{"Start":"03:01.060 ","End":"03:07.540","Text":"Here instead of 6 factorial divided by 4 factorial times 2 factorial,"},{"Start":"03:07.540 ","End":"03:12.585","Text":"we would have k plus n minus 1."},{"Start":"03:12.585 ","End":"03:20.415","Text":"We have k which would be 4 plus n,"},{"Start":"03:20.415 ","End":"03:24.195","Text":"which would be 3 partitions minus 1,"},{"Start":"03:24.195 ","End":"03:28.835","Text":"that\u0027s 2 dividers factorial"},{"Start":"03:28.835 ","End":"03:35.825","Text":"divided by k factorial times n minus 1 factorial."},{"Start":"03:35.825 ","End":"03:44.665","Text":"That comes out to k plus n minus 1 over k factorial."},{"Start":"03:44.665 ","End":"03:52.340","Text":"We know that also equals to k plus n minus 1 over"},{"Start":"03:52.340 ","End":"04:00.050","Text":"n minus 1 so these 2 expressions are the same."},{"Start":"04:00.050 ","End":"04:04.370","Text":"They originate from this expression right"},{"Start":"04:04.370 ","End":"04:10.110","Text":"here and this is a logic behind this expression."}],"ID":12830},{"Watched":false,"Name":"Tutorial 4","Duration":"1m 1s","ChapterTopicVideoID":12352,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12352.jpeg","UploadDate":"2018-07-22T17:03:35.4230000","DurationForVideoObject":"PT1M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this table I\u0027ve summarized all the possible situations for sampling."},{"Start":"00:04.170 ","End":"00:07.020","Text":"I have sampling with returns and without returns,"},{"Start":"00:07.020 ","End":"00:10.980","Text":"and I have sequential sampling and non-sequential sampling,"},{"Start":"00:10.980 ","End":"00:15.230","Text":"where sequential sampling the order does matter when we\u0027re talking about permutations,"},{"Start":"00:15.230 ","End":"00:17.810","Text":"and in non-sequential sampling the order"},{"Start":"00:17.810 ","End":"00:20.340","Text":"doesn\u0027t matter when we\u0027re talking here about combinations."},{"Start":"00:20.340 ","End":"00:24.840","Text":"For example, here, this would be the number of"},{"Start":"00:24.840 ","End":"00:31.725","Text":"permutations where I have sequential sampling with returns,"},{"Start":"00:31.725 ","End":"00:37.280","Text":"and this would be the number of permutations of sequential sampling without returns."},{"Start":"00:37.280 ","End":"00:43.400","Text":"This would be the number of combinations of non-sequential sampling with returns"},{"Start":"00:43.400 ","End":"00:49.610","Text":"and this would be the number of combinations of non-sequential sampling without returns."},{"Start":"00:49.610 ","End":"00:52.820","Text":"Let\u0027s do the exercise here."},{"Start":"00:52.820 ","End":"00:58.295","Text":"They\u0027ll emphasize the differences for each 1 of the 4 cells here,"},{"Start":"00:58.295 ","End":"01:01.410","Text":"so good luck with the exercises."}],"ID":12831},{"Watched":false,"Name":"Exercise 1","Duration":"2m 30s","ChapterTopicVideoID":12353,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12353.jpeg","UploadDate":"2018-07-22T17:03:40.7200000","DurationForVideoObject":"PT2M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In this question, we\u0027re asked;"},{"Start":"00:01.260 ","End":"00:04.530","Text":"in how many ways can 8 identical balls be put into"},{"Start":"00:04.530 ","End":"00:08.550","Text":"5 boxes when a box can contain more than 1 ball?"},{"Start":"00:08.550 ","End":"00:14.460","Text":"Well, this is an iconic question for non-sequential sampling with returns."},{"Start":"00:14.460 ","End":"00:16.845","Text":"But how do I know that this is so?"},{"Start":"00:16.845 ","End":"00:19.275","Text":"Well, let\u0027s start with returns."},{"Start":"00:19.275 ","End":"00:20.685","Text":"What are we doing here?"},{"Start":"00:20.685 ","End":"00:23.115","Text":"Well, we\u0027re sampling boxes."},{"Start":"00:23.115 ","End":"00:29.520","Text":"For each ball, we\u0027re sampling a box in order to put the ball in it."},{"Start":"00:29.520 ","End":"00:33.120","Text":"A box can contain more than 1 ball."},{"Start":"00:33.120 ","End":"00:36.780","Text":"Now, n, that\u0027ll be the number of boxes,"},{"Start":"00:36.780 ","End":"00:39.740","Text":"and I have 5 boxes here."},{"Start":"00:39.740 ","End":"00:43.055","Text":"Now, what about the non-sequential part?"},{"Start":"00:43.055 ","End":"00:45.665","Text":"Well, the balls are identical."},{"Start":"00:45.665 ","End":"00:48.610","Text":"We\u0027re given that right here."},{"Start":"00:48.610 ","End":"00:50.210","Text":"If 2 balls, let\u0027s say,"},{"Start":"00:50.210 ","End":"00:51.740","Text":"are placed in the same box,"},{"Start":"00:51.740 ","End":"00:53.780","Text":"we can\u0027t differentiate between them."},{"Start":"00:53.780 ","End":"00:58.100","Text":"The internal ordering the balls in the box is impossible."},{"Start":"00:58.100 ","End":"01:02.000","Text":"Because, again, the balls are identical."},{"Start":"01:02.000 ","End":"01:04.535","Text":"How many balls do I have here?"},{"Start":"01:04.535 ","End":"01:09.825","Text":"Well, I have 8 identical balls, that\u0027s right here."},{"Start":"01:09.825 ","End":"01:15.285","Text":"Let\u0027s see what is the number of combinations that I have?"},{"Start":"01:15.285 ","End":"01:18.270","Text":"I\u0027m putting 8 identical balls into 5 boxes."},{"Start":"01:18.270 ","End":"01:23.445","Text":"Well, the expression would be k plus"},{"Start":"01:23.445 ","End":"01:30.150","Text":"n minus 1 over k. Or we can write it like this,"},{"Start":"01:30.150 ","End":"01:32.835","Text":"k plus n minus 1,"},{"Start":"01:32.835 ","End":"01:35.865","Text":"over n minus 1."},{"Start":"01:35.865 ","End":"01:38.510","Text":"Let\u0027s use this expression right here,"},{"Start":"01:38.510 ","End":"01:41.360","Text":"where k is 8, plus n,"},{"Start":"01:41.360 ","End":"01:46.915","Text":"that\u0027s 5, minus 1 over k. K is 8."},{"Start":"01:46.915 ","End":"01:49.310","Text":"Now, what does that equal to?"},{"Start":"01:49.310 ","End":"01:52.745","Text":"Well, that equals to 8 plus 5,"},{"Start":"01:52.745 ","End":"01:54.080","Text":"that\u0027s 13, minus 1,"},{"Start":"01:54.080 ","End":"01:57.280","Text":"that\u0027s 12, over 8."},{"Start":"01:57.280 ","End":"02:02.000","Text":"Now, that comes out to 495."},{"Start":"02:02.000 ","End":"02:04.610","Text":"Now if we use this expression right here,"},{"Start":"02:04.610 ","End":"02:06.890","Text":"well, that would be k plus n minus 1."},{"Start":"02:06.890 ","End":"02:08.110","Text":"Well, that\u0027s the same thing right here."},{"Start":"02:08.110 ","End":"02:12.410","Text":"That\u0027ll be 12 over n minus 1."},{"Start":"02:12.410 ","End":"02:15.215","Text":"Where n minus 1 is 4."},{"Start":"02:15.215 ","End":"02:20.670","Text":"Well, we know that 12 over 4 with that equals to 12 over 8,"},{"Start":"02:20.670 ","End":"02:23.265","Text":"and we\u0027ve calculated that right here."},{"Start":"02:23.265 ","End":"02:27.430","Text":"That, again, equals to 495."},{"Start":"02:27.430 ","End":"02:30.590","Text":"That\u0027s the answer to this question."}],"ID":12832},{"Watched":false,"Name":"Exercise 2","Duration":"2m 1s","ChapterTopicVideoID":12354,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12354.jpeg","UploadDate":"2018-07-22T17:03:44.9930000","DurationForVideoObject":"PT2M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this question, we\u0027re asked in how many ways can"},{"Start":"00:02.220 ","End":"00:05.910","Text":"5 identical notebooks be put into 3 different files?"},{"Start":"00:05.910 ","End":"00:09.165","Text":"This is pretty much the same questions that we had previously."},{"Start":"00:09.165 ","End":"00:11.010","Text":"Instead of notebooks,"},{"Start":"00:11.010 ","End":"00:12.885","Text":"you could have used balls,"},{"Start":"00:12.885 ","End":"00:14.490","Text":"and instead of files,"},{"Start":"00:14.490 ","End":"00:16.900","Text":"we could have used boxes."},{"Start":"00:17.270 ","End":"00:23.100","Text":"Why is this non-sequential sampling with returns?"},{"Start":"00:23.100 ","End":"00:28.860","Text":"It\u0027s not sequential because the notebooks here are identical."},{"Start":"00:28.860 ","End":"00:34.920","Text":"Whenever we choose a notebook and we have to sample a file in which to put the notebook,"},{"Start":"00:34.920 ","End":"00:36.830","Text":"a file can have"},{"Start":"00:36.830 ","End":"00:42.260","Text":"more than 1 notebook because we have 5 notebooks and we have only 3 files."},{"Start":"00:42.260 ","End":"00:47.345","Text":"It really doesn\u0027t matter the order of the notebooks within that specific file."},{"Start":"00:47.345 ","End":"00:49.250","Text":"This is not sequential."},{"Start":"00:49.250 ","End":"00:54.920","Text":"Why is this with returns?"},{"Start":"00:54.920 ","End":"01:02.030","Text":"Again because I can choose more than 1 file in which to put a notebook."},{"Start":"01:02.030 ","End":"01:06.600","Text":"A file can have more than 1 notebook."},{"Start":"01:07.010 ","End":"01:11.390","Text":"Again, how many boxes do I have?"},{"Start":"01:11.390 ","End":"01:12.575","Text":"That will be my n,"},{"Start":"01:12.575 ","End":"01:14.675","Text":"and that equals to 3."},{"Start":"01:14.675 ","End":"01:16.700","Text":"How many notebooks do I have?"},{"Start":"01:16.700 ","End":"01:18.170","Text":"I have 5."},{"Start":"01:18.170 ","End":"01:19.970","Text":"They\u0027re identical notebooks."},{"Start":"01:19.970 ","End":"01:24.604","Text":"Whenever I have non-sequential sampling with returns,"},{"Start":"01:24.604 ","End":"01:27.815","Text":"I want to know the number of combinations of putting"},{"Start":"01:27.815 ","End":"01:31.820","Text":"5 identical notebooks into 3 different files."},{"Start":"01:31.820 ","End":"01:39.520","Text":"That\u0027ll be equal to k plus n minus 1 over k. We said,"},{"Start":"01:39.520 ","End":"01:43.820","Text":"that that equals to k plus n minus 1 over n minus 1."},{"Start":"01:43.820 ","End":"01:46.985","Text":"I\u0027ll be using this expression right here."},{"Start":"01:46.985 ","End":"01:53.155","Text":"K here is 5 plus n is 3 minus 1 over 5."},{"Start":"01:53.155 ","End":"02:01.920","Text":"That equals to 7 over 5 and that equals to 21."}],"ID":12833},{"Watched":false,"Name":"Exercise 3","Duration":"3m 19s","ChapterTopicVideoID":12355,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12355.jpeg","UploadDate":"2018-07-22T17:03:52.4730000","DurationForVideoObject":"PT3M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.170","Text":"In this question we\u0027re asked,"},{"Start":"00:01.170 ","End":"00:05.280","Text":"in how many ways can 8 balls be put into 3 different boxes when a,"},{"Start":"00:05.280 ","End":"00:07.610","Text":"the balls are identical and b,"},{"Start":"00:07.610 ","End":"00:09.870","Text":"the balls are different from each other."},{"Start":"00:09.870 ","End":"00:11.790","Text":"Well, first of all,"},{"Start":"00:11.790 ","End":"00:16.125","Text":"in both cases we have sampling with returns. Why is that?"},{"Start":"00:16.125 ","End":"00:20.550","Text":"Well, we have 8 balls and we need to put them into 3 different boxes."},{"Start":"00:20.550 ","End":"00:22.470","Text":"When we choose a ball,"},{"Start":"00:22.470 ","End":"00:25.499","Text":"we have to sample 1 of 3 boxes."},{"Start":"00:25.499 ","End":"00:27.630","Text":"Then when we choose another ball,"},{"Start":"00:27.630 ","End":"00:31.485","Text":"we still have to choose 1 of 3 boxes."},{"Start":"00:31.485 ","End":"00:34.620","Text":"We are sampling with returns."},{"Start":"00:34.620 ","End":"00:37.240","Text":"We\u0027re sampling the boxes with returns."},{"Start":"00:37.240 ","End":"00:39.710","Text":"Now in section a,"},{"Start":"00:39.710 ","End":"00:41.900","Text":"we have the balls are identical."},{"Start":"00:41.900 ","End":"00:45.865","Text":"This means that we have non-sequential sampling."},{"Start":"00:45.865 ","End":"00:48.270","Text":"In b, the balls are different."},{"Start":"00:48.270 ","End":"00:51.080","Text":"Here we have sequential sampling. Now, why is that?"},{"Start":"00:51.080 ","End":"00:53.300","Text":"Well, in non-sequential sampling,"},{"Start":"00:53.300 ","End":"00:54.670","Text":"if we have, let\u0027s say,"},{"Start":"00:54.670 ","End":"00:58.880","Text":"2 balls, we put 1 in the first box and 1 in the second box."},{"Start":"00:58.880 ","End":"01:01.175","Text":"Well, we can\u0027t differentiate between the balls."},{"Start":"01:01.175 ","End":"01:04.445","Text":"There\u0027s no meaning to the order of the balls."},{"Start":"01:04.445 ","End":"01:07.250","Text":"But when the balls are different,"},{"Start":"01:07.250 ","End":"01:10.700","Text":"let\u0027s say we have 1 green ball and 1 orange ball."},{"Start":"01:10.700 ","End":"01:12.730","Text":"Well, in the first case,"},{"Start":"01:12.730 ","End":"01:17.960","Text":"the orange ball can be in the first box and the green 1 can be in the second box."},{"Start":"01:17.960 ","End":"01:21.410","Text":"That\u0027s totally different than the case where"},{"Start":"01:21.410 ","End":"01:25.600","Text":"the green ball is in the first box and the orange is in the second box."},{"Start":"01:25.600 ","End":"01:30.150","Text":"That\u0027s the difference between a and b here."},{"Start":"01:30.850 ","End":"01:33.620","Text":"Now that we understand the difference,"},{"Start":"01:33.620 ","End":"01:37.760","Text":"let\u0027s go to the summary table to see how we can"},{"Start":"01:37.760 ","End":"01:42.815","Text":"see or calculate the number of combinations in each case."},{"Start":"01:42.815 ","End":"01:46.610","Text":"Here\u0027s our summary table and we\u0027re in this line right here,"},{"Start":"01:46.610 ","End":"01:49.100","Text":"we have sampling with returns."},{"Start":"01:49.100 ","End":"01:51.755","Text":"Here we have sequential sampling."},{"Start":"01:51.755 ","End":"01:54.200","Text":"Here we have non-sequential sampling."},{"Start":"01:54.200 ","End":"01:56.330","Text":"For sequential sampling, if we recall,"},{"Start":"01:56.330 ","End":"02:00.920","Text":"that\u0027s n to the power of k. For non-sequential sampling,"},{"Start":"02:00.920 ","End":"02:05.210","Text":"that\u0027ll be n plus k minus 1 over k. Again,"},{"Start":"02:05.210 ","End":"02:09.090","Text":"that\u0027s only when we have sampling with returns."},{"Start":"02:09.430 ","End":"02:15.215","Text":"Again, we have here k plus n minus 1 over"},{"Start":"02:15.215 ","End":"02:21.845","Text":"k and here we\u0027ll have n to the power of k. Now n here,"},{"Start":"02:21.845 ","End":"02:24.900","Text":"well, that\u0027s the number of the boxes that we have,"},{"Start":"02:24.900 ","End":"02:27.645","Text":"we have 3. k,"},{"Start":"02:27.645 ","End":"02:29.430","Text":"well that\u0027s the number of balls."},{"Start":"02:29.430 ","End":"02:31.110","Text":"We have here 8."},{"Start":"02:31.110 ","End":"02:32.930","Text":"In section a,"},{"Start":"02:32.930 ","End":"02:34.610","Text":"where the balls are identical,"},{"Start":"02:34.610 ","End":"02:35.960","Text":"well, what do we have here?"},{"Start":"02:35.960 ","End":"02:46.430","Text":"We have 8 plus 3 minus 1 over 8."},{"Start":"02:46.430 ","End":"02:54.365","Text":"That means that we have 10 over 8 and that equals to 45."},{"Start":"02:54.365 ","End":"02:57.720","Text":"Here, n to the power of k, well,"},{"Start":"02:57.720 ","End":"03:02.655","Text":"we have here 3 to the power of 8."},{"Start":"03:02.655 ","End":"03:07.390","Text":"Now that equals to 6,561."},{"Start":"03:08.300 ","End":"03:11.210","Text":"These are the number of combinations."},{"Start":"03:11.210 ","End":"03:12.890","Text":"When the balls are identical,"},{"Start":"03:12.890 ","End":"03:14.945","Text":"we have 45 combinations,"},{"Start":"03:14.945 ","End":"03:19.530","Text":"and when the balls are different, we have 6,561."}],"ID":12834},{"Watched":false,"Name":"Exercise 4","Duration":"2m 48s","ChapterTopicVideoID":12356,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12356.jpeg","UploadDate":"2019-01-20T11:52:14.7600000","DurationForVideoObject":"PT2M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.170","Text":"In this question we\u0027re asked,"},{"Start":"00:01.170 ","End":"00:05.790","Text":"what\u0027s the number of non-negative whole numbers solutions to the following equation,"},{"Start":"00:05.790 ","End":"00:09.780","Text":"X_1 plus X_2, well, that equals to 3."},{"Start":"00:09.780 ","End":"00:13.995","Text":"This is very similar to the previous questions"},{"Start":"00:13.995 ","End":"00:18.180","Text":"where we had non-sequential sampling with returns."},{"Start":"00:18.180 ","End":"00:22.440","Text":"We\u0027re thinking of X_1 and X_2 just like bins,"},{"Start":"00:22.440 ","End":"00:24.300","Text":"just like the boxes."},{"Start":"00:24.300 ","End":"00:26.870","Text":"How many boxes do I have here?"},{"Start":"00:26.870 ","End":"00:28.140","Text":"That\u0027ll be my n, well,"},{"Start":"00:28.140 ","End":"00:29.325","Text":"that equals to 2."},{"Start":"00:29.325 ","End":"00:31.425","Text":"I have X_1 and X_2."},{"Start":"00:31.425 ","End":"00:35.284","Text":"Since I\u0027m dealing with non-negative whole numbers,"},{"Start":"00:35.284 ","End":"00:38.080","Text":"well then, how many balls do I have here?"},{"Start":"00:38.080 ","End":"00:40.810","Text":"Well, I have basically 3 balls."},{"Start":"00:40.810 ","End":"00:49.730","Text":"Now, I want to put the 3 balls into each 1 of the 2 bins here."},{"Start":"00:49.730 ","End":"00:53.545","Text":"For example, we can look at it like this."},{"Start":"00:53.545 ","End":"00:56.580","Text":"Here, I have my X\u0027s,"},{"Start":"00:56.580 ","End":"00:58.020","Text":"that\u0027s the green lines right here,"},{"Start":"00:58.020 ","End":"00:59.715","Text":"that\u0027s X_1 and X_2,"},{"Start":"00:59.715 ","End":"01:02.580","Text":"and here I have 3 balls."},{"Start":"01:02.580 ","End":"01:06.660","Text":"I have, let\u0027s say, X_1 that equals to 2,"},{"Start":"01:06.660 ","End":"01:09.960","Text":"because I have 2 balls in X_1 and X_2,"},{"Start":"01:09.960 ","End":"01:11.985","Text":"that equals to 1."},{"Start":"01:11.985 ","End":"01:15.850","Text":"Another example will be something like this."},{"Start":"01:15.850 ","End":"01:22.790","Text":"Here, we see the example where X_1 equals to 3 and X_2,"},{"Start":"01:22.790 ","End":"01:25.165","Text":"well, that equals to 0."},{"Start":"01:25.165 ","End":"01:31.205","Text":"Again, if we have non-sequential sampling with returns,"},{"Start":"01:31.205 ","End":"01:38.955","Text":"then how many combinations can we have when we have 2 boxes and 3 balls?"},{"Start":"01:38.955 ","End":"01:46.385","Text":"Well, that will be k plus n minus 1 over k. What does that equal to?"},{"Start":"01:46.385 ","End":"01:52.969","Text":"k equals to 3 plus 2 minus 1 over 3."},{"Start":"01:52.969 ","End":"01:57.125","Text":"Well, that equals to 3 plus 2,"},{"Start":"01:57.125 ","End":"01:58.670","Text":"that equals to 5 minus 1,"},{"Start":"01:58.670 ","End":"02:01.010","Text":"that\u0027s 4 over 3."},{"Start":"02:01.010 ","End":"02:03.650","Text":"That equals to 4,"},{"Start":"02:03.650 ","End":"02:06.065","Text":"there aren\u0027t many combinations here."},{"Start":"02:06.065 ","End":"02:11.620","Text":"Let\u0027s just take a look at all the other combinations that we have."},{"Start":"02:11.620 ","End":"02:14.655","Text":"Here we have the other 2 combinations."},{"Start":"02:14.655 ","End":"02:21.435","Text":"We have a total of 4 combinations of putting 3 balls into 2 bins,"},{"Start":"02:21.435 ","End":"02:30.320","Text":"or looking at non-negative whole numbers and placing them in the variables X_1 and X_2."},{"Start":"02:30.320 ","End":"02:34.920","Text":"Well here, X_1 that equals to 1 and X_2 that equals to."},{"Start":"02:34.920 ","End":"02:41.190","Text":"Here, X_1 equals to 0 and X_2 that equals to 3."},{"Start":"02:41.190 ","End":"02:49.050","Text":"This is the way we think of solving these types of problems."}],"ID":12835},{"Watched":false,"Name":"Exercise 5","Duration":"1m 37s","ChapterTopicVideoID":12357,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12357.jpeg","UploadDate":"2019-01-20T11:50:06.5900000","DurationForVideoObject":"PT1M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"In this question, we\u0027re asked what\u0027s the number of"},{"Start":"00:01.920 ","End":"00:05.280","Text":"nonnegative whole number solutions to the following equations."},{"Start":"00:05.280 ","End":"00:08.205","Text":"X_1 plus X_2 plus X_3 plus X_4,"},{"Start":"00:08.205 ","End":"00:10.140","Text":"and that equals to 20."},{"Start":"00:10.140 ","End":"00:13.470","Text":"This is exactly like the previous question,"},{"Start":"00:13.470 ","End":"00:17.535","Text":"we had non sequential sampling with returns."},{"Start":"00:17.535 ","End":"00:22.170","Text":"We thought of the X\u0027s as our boxes."},{"Start":"00:22.170 ","End":"00:24.705","Text":"How many boxes do I have here?"},{"Start":"00:24.705 ","End":"00:26.625","Text":"I have 4 boxes,"},{"Start":"00:26.625 ","End":"00:30.465","Text":"and this number right here would be the number of balls."},{"Start":"00:30.465 ","End":"00:35.110","Text":"I\u0027ll have 20 balls as an example."},{"Start":"00:35.110 ","End":"00:38.165","Text":"I can have 4 balls here,"},{"Start":"00:38.165 ","End":"00:41.450","Text":"4 balls here, 2 balls here."},{"Start":"00:41.450 ","End":"00:47.000","Text":"That\u0027ll be the non-negative whole numbers that I\u0027ll put into X_1,"},{"Start":"00:47.000 ","End":"00:48.230","Text":"X_2, and X_3,"},{"Start":"00:48.230 ","End":"00:52.650","Text":"so that means what\u0027s the number to be put into X_4,"},{"Start":"00:52.650 ","End":"00:54.495","Text":"well, 4 plus 4 plus 2,"},{"Start":"00:54.495 ","End":"00:56.100","Text":"that equals to 10."},{"Start":"00:56.100 ","End":"01:02.780","Text":"I need this to be 10 in order for the sum of all these numbers to be equal to 20."},{"Start":"01:02.780 ","End":"01:09.035","Text":"Well again, because this is non sequential sampling with returns,"},{"Start":"01:09.035 ","End":"01:15.380","Text":"then the number of combinations of placing 20 balls into 4 bins will then"},{"Start":"01:15.380 ","End":"01:21.725","Text":"be k plus n minus 1 over k. In our case,"},{"Start":"01:21.725 ","End":"01:26.630","Text":"that\u0027ll be 20 plus 4 minus 1 over 20."},{"Start":"01:26.630 ","End":"01:31.895","Text":"Now, that equals to 23 over 20,"},{"Start":"01:31.895 ","End":"01:37.740","Text":"and that equals to 1771."}],"ID":12836},{"Watched":false,"Name":"Exercise 6","Duration":"2m 22s","ChapterTopicVideoID":12358,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12358.jpeg","UploadDate":"2019-01-20T11:50:23.7370000","DurationForVideoObject":"PT2M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this question, we have an auction and we have"},{"Start":"00:02.340 ","End":"00:05.310","Text":"4 identical gold candlesticks that are displayed."},{"Start":"00:05.310 ","End":"00:08.265","Text":"Now, 3 collectors competed to purchase them."},{"Start":"00:08.265 ","End":"00:11.655","Text":"A collector can purchase more than 1 candlestick."},{"Start":"00:11.655 ","End":"00:14.115","Text":"Assuming that all the candlesticks were sold,"},{"Start":"00:14.115 ","End":"00:17.890","Text":"how many possible ways could they be sold to the collectors?"},{"Start":"00:17.890 ","End":"00:22.560","Text":"Guys, this is the same thing as we had in previous questions."},{"Start":"00:22.560 ","End":"00:27.330","Text":"The trick here is to identify that all the questions are the same."},{"Start":"00:27.330 ","End":"00:31.185","Text":"We have non-sequential sampling with returns."},{"Start":"00:31.185 ","End":"00:34.920","Text":"Why is that? Let\u0027s take a look."},{"Start":"00:34.920 ","End":"00:38.430","Text":"We have 4 identical gold candlesticks."},{"Start":"00:38.430 ","End":"00:40.070","Text":"That\u0027s just like saying,"},{"Start":"00:40.070 ","End":"00:42.404","Text":"we have 4 balls."},{"Start":"00:42.404 ","End":"00:44.565","Text":"K here, k,"},{"Start":"00:44.565 ","End":"00:46.800","Text":"that\u0027ll be equal to 4."},{"Start":"00:46.800 ","End":"00:49.470","Text":"We have 3 collectors."},{"Start":"00:49.470 ","End":"00:53.503","Text":"Well, 3 collectors is just like having 3 boxes,"},{"Start":"00:53.503 ","End":"00:54.710","Text":"so n here,"},{"Start":"00:54.710 ","End":"00:57.685","Text":"that equals to 3."},{"Start":"00:57.685 ","End":"01:01.250","Text":"Again, why is it non-sequential?"},{"Start":"01:01.250 ","End":"01:04.939","Text":"It\u0027s non-sequential because we have identical gold candlesticks."},{"Start":"01:04.939 ","End":"01:07.910","Text":"It really doesn\u0027t matter which candlestick goes to"},{"Start":"01:07.910 ","End":"01:11.480","Text":"which collector and why is it with returns?"},{"Start":"01:11.480 ","End":"01:13.895","Text":"Because we\u0027re sampling collectors."},{"Start":"01:13.895 ","End":"01:17.780","Text":"We\u0027re choosing the candlestick and it can either go to the first,"},{"Start":"01:17.780 ","End":"01:19.640","Text":"or second, or third collector."},{"Start":"01:19.640 ","End":"01:23.480","Text":"Then the second candlestick can again go to the first,"},{"Start":"01:23.480 ","End":"01:25.340","Text":"or second, or third collector."},{"Start":"01:25.340 ","End":"01:31.605","Text":"That means that we have sampling with returns here."},{"Start":"01:31.605 ","End":"01:35.705","Text":"We have here this expression,"},{"Start":"01:35.705 ","End":"01:45.410","Text":"k plus 1 minus k plus n minus 1 over k. That\u0027ll be the expression that calculates"},{"Start":"01:45.410 ","End":"01:50.060","Text":"the number of possible combinations of distributing"},{"Start":"01:50.060 ","End":"01:56.275","Text":"the 4 identical gold candlesticks to the 3 collectors."},{"Start":"01:56.275 ","End":"02:00.555","Text":"That\u0027ll be equal to k here, that equals to 4,"},{"Start":"02:00.555 ","End":"02:06.890","Text":"n that equals to 3 minus 1 over k. k is 4 now"},{"Start":"02:06.890 ","End":"02:13.220","Text":"that equals to 6 over 4 and that equals to 15."},{"Start":"02:13.220 ","End":"02:17.540","Text":"There are 15 possible combinations of distributing"},{"Start":"02:17.540 ","End":"02:23.280","Text":"the 4 identical gold candlesticks to the 3 collectors."}],"ID":12837},{"Watched":false,"Name":"Exercise 7","Duration":"1m 55s","ChapterTopicVideoID":12359,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12359.jpeg","UploadDate":"2019-01-20T11:50:38.2500000","DurationForVideoObject":"PT1M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this question, we have 2 letters that are selected from a set of 4 letters, A, B, C,"},{"Start":"00:04.770 ","End":"00:08.370","Text":"and D. A letter can be selected more than once,"},{"Start":"00:08.370 ","End":"00:11.040","Text":"but the order of the selected letter doesn\u0027t matter."},{"Start":"00:11.040 ","End":"00:15.660","Text":"We\u0027re asked how many ways are there of choosing the letters?"},{"Start":"00:15.660 ","End":"00:20.825","Text":"Let\u0027s just see an example of the sampling here."},{"Start":"00:20.825 ","End":"00:23.720","Text":"We can sample A and B."},{"Start":"00:23.720 ","End":"00:28.620","Text":"These are 2 letters and they\u0027re from the 4 letters here."},{"Start":"00:28.620 ","End":"00:34.485","Text":"I won\u0027t write B and A because I\u0027ve already sampled an A and a B."},{"Start":"00:34.485 ","End":"00:37.785","Text":"We\u0027re told that the order doesn\u0027t matter."},{"Start":"00:37.785 ","End":"00:42.070","Text":"I can also sample A and A. Why is that?"},{"Start":"00:42.070 ","End":"00:47.145","Text":"Well, because we\u0027re told a letter can be selected more than once."},{"Start":"00:47.145 ","End":"00:50.115","Text":"Other examples would be C and D,"},{"Start":"00:50.115 ","End":"00:54.515","Text":"or D and D. What do we have here?"},{"Start":"00:54.515 ","End":"01:01.675","Text":"We have k objects or k letters that I can select out of n letters."},{"Start":"01:01.675 ","End":"01:08.355","Text":"This is non-sequential sampling with returns. Why\u0027s that?"},{"Start":"01:08.355 ","End":"01:12.600","Text":"Well, it\u0027s non-sequential because the order doesn\u0027t matter and"},{"Start":"01:12.600 ","End":"01:19.085","Text":"it\u0027s with returns because I can sample the same letter over and over again."},{"Start":"01:19.085 ","End":"01:22.170","Text":"How many letters are we sampling?"},{"Start":"01:22.170 ","End":"01:24.105","Text":"We\u0027re sampling 2 letters."},{"Start":"01:24.105 ","End":"01:27.945","Text":"All in all we have 4 letters to choose from."},{"Start":"01:27.945 ","End":"01:31.520","Text":"Whenever we have non-sequential sampling with returns,"},{"Start":"01:31.520 ","End":"01:39.470","Text":"then the number of combinations would be k plus n minus 1 over k. In our case,"},{"Start":"01:39.470 ","End":"01:44.480","Text":"k is 2, n is 4 over 2,"},{"Start":"01:44.480 ","End":"01:47.750","Text":"so that\u0027ll be 2 plus 4 minus 1 over 2."},{"Start":"01:47.750 ","End":"01:52.460","Text":"That equals to 5 over 2,"},{"Start":"01:52.460 ","End":"01:55.860","Text":"and that equals to 10."}],"ID":12838},{"Watched":false,"Name":"Exercise 8","Duration":"3m 20s","ChapterTopicVideoID":12360,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12360.jpeg","UploadDate":"2019-01-20T11:50:59.3630000","DurationForVideoObject":"PT3M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this question, we have a lottery and"},{"Start":"00:02.220 ","End":"00:05.910","Text":"4 numbers must be selected from the numbers 1- 20."},{"Start":"00:05.910 ","End":"00:09.120","Text":"Now, the order in which the numbers are selected is not important."},{"Start":"00:09.120 ","End":"00:12.675","Text":"We\u0027re asked what is the chances of guessing all 4 numbers"},{"Start":"00:12.675 ","End":"00:17.100","Text":"if in a the same number cannot be selected more than once,"},{"Start":"00:17.100 ","End":"00:21.420","Text":"and in b the same number can be selected more than once."},{"Start":"00:21.420 ","End":"00:28.500","Text":"In a we have sampling without returns and in b we have sampling with returns,"},{"Start":"00:28.500 ","End":"00:34.779","Text":"and in both cases where we have non-sequential sampling."},{"Start":"00:35.780 ","End":"00:39.810","Text":"Now, how many numbers are selected or sampled?"},{"Start":"00:39.810 ","End":"00:41.640","Text":"Well, 4 numbers were sampled,"},{"Start":"00:41.640 ","End":"00:43.410","Text":"so that means k here,"},{"Start":"00:43.410 ","End":"00:45.720","Text":"that has to equal to 4,"},{"Start":"00:45.720 ","End":"00:52.860","Text":"and we have a total of 20 numbers to choose from so n here that equals to 20."},{"Start":"00:52.860 ","End":"00:55.425","Text":"Again, we have"},{"Start":"00:55.425 ","End":"00:58.080","Text":"non-sequential sampling right in"},{"Start":"00:58.080 ","End":"01:01.995","Text":"both cases and in the first case we have sampling without returns."},{"Start":"01:01.995 ","End":"01:04.940","Text":"In the second case we have sampling with returns."},{"Start":"01:04.940 ","End":"01:10.860","Text":"Let\u0027s go to the summary table and see what the number of combinations are."},{"Start":"01:11.470 ","End":"01:14.240","Text":"Here\u0027s our summary table,"},{"Start":"01:14.240 ","End":"01:18.170","Text":"and here\u0027s our non-sequential sampling column."},{"Start":"01:18.170 ","End":"01:24.950","Text":"Now we have here with returns and without returns so in section a we\u0027ll be using"},{"Start":"01:24.950 ","End":"01:33.450","Text":"this formula right here and this will be the formula for section b."},{"Start":"01:33.450 ","End":"01:36.910","Text":"Let\u0027s get back to our question now."},{"Start":"01:36.950 ","End":"01:43.910","Text":"In a we have non-sequential sampling without returns,"},{"Start":"01:43.910 ","End":"01:48.800","Text":"so that was n over k. Now in our case,"},{"Start":"01:48.800 ","End":"01:57.380","Text":"that equals to 20 over 4 and that equals to 4,850."},{"Start":"01:57.380 ","End":"01:59.870","Text":"But we weren\u0027t asked about the number of combinations,"},{"Start":"01:59.870 ","End":"02:02.435","Text":"we\u0027re asked about the chances of guessing."},{"Start":"02:02.435 ","End":"02:08.780","Text":"That means that we\u0027re looking for the probability of guessing correctly or winning."},{"Start":"02:08.780 ","End":"02:15.950","Text":"All that is we\u0027re guessing 1 out of the total number of combinations here,"},{"Start":"02:15.950 ","End":"02:22.555","Text":"which is 4,850 so the probability is 1 over 4,850."},{"Start":"02:22.555 ","End":"02:26.370","Text":"Now in b, we have sampling with return,"},{"Start":"02:26.370 ","End":"02:30.710","Text":"and again this is non-sequential so the number of combinations would"},{"Start":"02:30.710 ","End":"02:36.260","Text":"be k plus n minus 1 over k. In our case,"},{"Start":"02:36.260 ","End":"02:43.605","Text":"k is 4 n is 20 minus 1 over 4."},{"Start":"02:43.605 ","End":"02:48.630","Text":"Now, that comes out to 23 over 4."},{"Start":"02:48.630 ","End":"02:58.985","Text":"That equals to 8,855 so the probability of a win here,"},{"Start":"02:58.985 ","End":"03:04.500","Text":"again, we\u0027re guessing 1 out of 8,855"},{"Start":"03:04.500 ","End":"03:10.110","Text":"possibilities here so we have 1 divided by 8,855."},{"Start":"03:10.110 ","End":"03:14.625","Text":"This would be the probability of guessing"},{"Start":"03:14.625 ","End":"03:21.000","Text":"all 4 numbers where we can select the same number over and over again."}],"ID":12839},{"Watched":false,"Name":"Exercise 9","Duration":"3m 20s","ChapterTopicVideoID":24819,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/24819.jpeg","UploadDate":"2021-02-16T10:26:37.6230000","DurationForVideoObject":"PT3M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this question, we have a lottery and"},{"Start":"00:02.220 ","End":"00:05.805","Text":"4 numbers must be selected from the numbers 1 to 20."},{"Start":"00:05.805 ","End":"00:09.120","Text":"Now the order in which the numbers are selected is not important."},{"Start":"00:09.120 ","End":"00:13.995","Text":"We\u0027re asked, what are the chances of guessing all 4 numbers if in a,"},{"Start":"00:13.995 ","End":"00:17.120","Text":"the same number cannot be selected more than once,"},{"Start":"00:17.120 ","End":"00:21.420","Text":"and in b, the same number can be selected more than once."},{"Start":"00:21.420 ","End":"00:25.350","Text":"In a, we have sampling without returns,"},{"Start":"00:25.350 ","End":"00:28.500","Text":"and in B we have sampling with returns,"},{"Start":"00:28.500 ","End":"00:30.420","Text":"and in both cases, well,"},{"Start":"00:30.420 ","End":"00:34.780","Text":"we have non-sequential sampling."},{"Start":"00:34.970 ","End":"00:39.630","Text":"Now, how many numbers are selected or sampled?"},{"Start":"00:39.630 ","End":"00:41.630","Text":"Well, 4 numbers are sampled,"},{"Start":"00:41.630 ","End":"00:43.415","Text":"so that means k here,"},{"Start":"00:43.415 ","End":"00:45.995","Text":"that has to equal to 4."},{"Start":"00:45.995 ","End":"00:49.580","Text":"We have a total of 20 numbers to choose from,"},{"Start":"00:49.580 ","End":"00:53.370","Text":"so n here, that equals to 20."},{"Start":"00:53.370 ","End":"00:59.090","Text":"Again, we have non-sequential sampling in both cases,"},{"Start":"00:59.090 ","End":"01:00.320","Text":"and in the first case,"},{"Start":"01:00.320 ","End":"01:01.800","Text":"we have sampling would that returns,"},{"Start":"01:01.800 ","End":"01:03.170","Text":"and in the second case,"},{"Start":"01:03.170 ","End":"01:04.670","Text":"we have sampling with returns."},{"Start":"01:04.670 ","End":"01:10.870","Text":"Let\u0027s go to the summary table and see what the number of combinations are."},{"Start":"01:11.480 ","End":"01:14.240","Text":"Here\u0027s our summary table,"},{"Start":"01:14.240 ","End":"01:18.170","Text":"and here\u0027s our non-sequential sampling column."},{"Start":"01:18.170 ","End":"01:22.170","Text":"Now we have here with returns and without returns."},{"Start":"01:22.170 ","End":"01:24.955","Text":"In section a, we\u0027ll be using"},{"Start":"01:24.955 ","End":"01:33.450","Text":"this formula right here and this will be the formula for section b."},{"Start":"01:33.450 ","End":"01:36.910","Text":"Let\u0027s get back to our question now."},{"Start":"01:36.950 ","End":"01:43.910","Text":"In a, we have non-sequential sampling without returns,"},{"Start":"01:43.910 ","End":"01:48.800","Text":"so that was n over k. Now in our case,"},{"Start":"01:48.800 ","End":"01:57.380","Text":"that equals to 20/ 4 and that equals to 4,850."},{"Start":"01:57.380 ","End":"01:59.870","Text":"But we weren\u0027t asked about the number of combinations,"},{"Start":"01:59.870 ","End":"02:02.180","Text":"we\u0027re asked about the chances of guessing."},{"Start":"02:02.180 ","End":"02:08.780","Text":"That means that we\u0027re looking for the probability of guessing correctly or winning."},{"Start":"02:08.780 ","End":"02:13.220","Text":"All that is is we\u0027re guessing 1 out of"},{"Start":"02:13.220 ","End":"02:18.260","Text":"the total number of combinations here, which is 4,850."},{"Start":"02:18.260 ","End":"02:22.555","Text":"The probability is 1/4,850."},{"Start":"02:22.555 ","End":"02:24.195","Text":"Now in b,"},{"Start":"02:24.195 ","End":"02:26.370","Text":"we have sampling with return."},{"Start":"02:26.370 ","End":"02:28.725","Text":"Again, this is non-sequential,"},{"Start":"02:28.725 ","End":"02:34.370","Text":"so the number of combinations would be k plus n minus 1/k."},{"Start":"02:34.370 ","End":"02:37.830","Text":"In our case, k is 4,"},{"Start":"02:37.830 ","End":"02:42.490","Text":"n is 20 minus 1/4."},{"Start":"02:42.620 ","End":"02:48.240","Text":"Now, that comes out to 23/4."},{"Start":"02:48.240 ","End":"02:55.875","Text":"Now that equals to 8,855."},{"Start":"02:55.875 ","End":"02:59.870","Text":"The probability of a win here, again,"},{"Start":"02:59.870 ","End":"03:05.760","Text":"we\u0027re guessing 1 out of 8,855 possibilities here,"},{"Start":"03:05.760 ","End":"03:10.110","Text":"so we have 1 divided by 8,855."},{"Start":"03:10.110 ","End":"03:14.625","Text":"This would be the probability of guessing"},{"Start":"03:14.625 ","End":"03:21.000","Text":"all 4 numbers where we can select the same number over and over again."}],"ID":25732},{"Watched":false,"Name":"Exercise 9 - Parts a-d","Duration":"5m 25s","ChapterTopicVideoID":15223,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/15223.jpeg","UploadDate":"2019-01-20T11:51:36.4070000","DurationForVideoObject":"PT5M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this question, we have 5 balls that are put into to 6 boxes and we\u0027re asked to"},{"Start":"00:03.900 ","End":"00:08.009","Text":"calculate the number of possible ways of putting the balls into the boxes."},{"Start":"00:08.009 ","End":"00:10.380","Text":"Now, before we answer any of the questions,"},{"Start":"00:10.380 ","End":"00:17.235","Text":"let\u0027s just summarize all the possible combinations that we have here. Here we go."},{"Start":"00:17.235 ","End":"00:19.170","Text":"If we\u0027re told that we have different balls,"},{"Start":"00:19.170 ","End":"00:21.330","Text":"then we\u0027re dealing with sequential sampling."},{"Start":"00:21.330 ","End":"00:23.085","Text":"If the balls are identical,"},{"Start":"00:23.085 ","End":"00:26.010","Text":"then we\u0027re dealing with non-sequential sampling."},{"Start":"00:26.010 ","End":"00:30.870","Text":"If the box can contain more than 1 ball then we have sampling with returns"},{"Start":"00:30.870 ","End":"00:35.465","Text":"and if the the box cannot contain more than 1 ball,"},{"Start":"00:35.465 ","End":"00:39.180","Text":"then we have sampling without returns."},{"Start":"00:40.320 ","End":"00:43.025","Text":"What do I have here?"},{"Start":"00:43.025 ","End":"00:45.435","Text":"The balls are different."},{"Start":"00:45.435 ","End":"00:50.750","Text":"We\u0027re dealing with sequential sampling and the box can contain more than 1 ball."},{"Start":"00:50.750 ","End":"00:55.160","Text":"So we have sampling with returns."},{"Start":"00:55.160 ","End":"01:01.435","Text":"Let\u0027s just take a look at our summary table to see where we\u0027re at."},{"Start":"01:01.435 ","End":"01:03.920","Text":"Here\u0027s our summary table."},{"Start":"01:03.920 ","End":"01:08.300","Text":"Here\u0027s a column of sequential sampling and this is with returns."},{"Start":"01:08.300 ","End":"01:09.800","Text":"That\u0027s this row right here."},{"Start":"01:09.800 ","End":"01:16.350","Text":"The number of possible combinations is n^k."},{"Start":"01:16.730 ","End":"01:21.435","Text":"Now, what\u0027s n and what\u0027s k?"},{"Start":"01:21.435 ","End":"01:25.460","Text":"Well, we know that we have 5 balls that we have to choose,"},{"Start":"01:25.460 ","End":"01:26.870","Text":"that we have to sample."},{"Start":"01:26.870 ","End":"01:31.300","Text":"Well, that means that k equals to 5."},{"Start":"01:31.300 ","End":"01:34.185","Text":"We have 6 boxes."},{"Start":"01:34.185 ","End":"01:37.135","Text":"N here that equals to 6."},{"Start":"01:37.135 ","End":"01:40.655","Text":"We have to remember n is the number of"},{"Start":"01:40.655 ","End":"01:45.335","Text":"boxes that we have in case the number of objects that are sampled."},{"Start":"01:45.335 ","End":"01:48.095","Text":"In this case right here,"},{"Start":"01:48.095 ","End":"01:53.610","Text":"the number of possible combinations of putting the balls into the boxes when"},{"Start":"01:53.610 ","End":"01:59.685","Text":"the balls are different and the box can contain more than 1 ball, would be n^k."},{"Start":"01:59.685 ","End":"02:08.750","Text":"In our case that\u0027ll be 6^5 and that equals to 7,776."},{"Start":"02:08.750 ","End":"02:12.560","Text":"In this section, we want to calculate the number of combinations where the balls"},{"Start":"02:12.560 ","End":"02:16.460","Text":"are identical and the box can contain more than 1 ball."},{"Start":"02:16.460 ","End":"02:18.320","Text":"Well, if the balls are identical,"},{"Start":"02:18.320 ","End":"02:21.281","Text":"then we have non-sequential sampling,"},{"Start":"02:21.281 ","End":"02:24.365","Text":"and if the box can contain more than 1 ball,"},{"Start":"02:24.365 ","End":"02:26.960","Text":"then we have with returns."},{"Start":"02:26.960 ","End":"02:30.980","Text":"Again, non-sequential sampling with returns."},{"Start":"02:30.980 ","End":"02:34.370","Text":"Let\u0027s go back to our summary table."},{"Start":"02:34.370 ","End":"02:37.010","Text":"Here\u0027s our summary table and we\u0027re looking for"},{"Start":"02:37.010 ","End":"02:42.495","Text":"non-sequential sampling that\u0027s right here with returns, and that\u0027s right here."},{"Start":"02:42.495 ","End":"02:48.175","Text":"This then would be the way to calculate the number of combinations."},{"Start":"02:48.175 ","End":"02:53.700","Text":"That\u0027ll be k plus n minus 1/k."},{"Start":"02:53.700 ","End":"03:01.605","Text":"In our case, k is 5 plus 6 minus 1/5."},{"Start":"03:01.605 ","End":"03:11.650","Text":"Now, that equals to 10/5 and that equals to 252."},{"Start":"03:11.650 ","End":"03:15.590","Text":"In this section, we have that the balls are different from"},{"Start":"03:15.590 ","End":"03:19.310","Text":"each other and the box cannot contain more than 1 ball."},{"Start":"03:19.310 ","End":"03:20.990","Text":"The balls are different."},{"Start":"03:20.990 ","End":"03:27.598","Text":"That means that we have sequential sampling and the box cannot contain more than 1 ball,"},{"Start":"03:27.598 ","End":"03:30.635","Text":"so that means that we have without returns."},{"Start":"03:30.635 ","End":"03:33.530","Text":"Again, sequential sampling without returns."},{"Start":"03:33.530 ","End":"03:36.145","Text":"Let\u0027s go to our summary table."},{"Start":"03:36.145 ","End":"03:39.228","Text":"Sequential sampling, that\u0027s this column,"},{"Start":"03:39.228 ","End":"03:42.000","Text":"without returns, that\u0027s this right here."},{"Start":"03:42.000 ","End":"03:47.820","Text":"This is the way to calculate the number of combinations."},{"Start":"03:48.190 ","End":"03:53.790","Text":"We have here n_k and that equals to n"},{"Start":"03:53.790 ","End":"04:00.060","Text":"factorial divided by n minus k factorial."},{"Start":"04:00.060 ","End":"04:02.190","Text":"In our case n here,"},{"Start":"04:02.190 ","End":"04:03.360","Text":"that equals to 6."},{"Start":"04:03.360 ","End":"04:11.580","Text":"It\u0027ll be 6 factorial divided by 6 minus 5 factorial."},{"Start":"04:11.580 ","End":"04:18.810","Text":"That means it\u0027s 6 factorial divided by 1 factorial and that equals to 720."},{"Start":"04:18.810 ","End":"04:22.820","Text":"In this section, we\u0027re given that the balls are"},{"Start":"04:22.820 ","End":"04:27.095","Text":"identical and the box cannot contain more than 1 ball."},{"Start":"04:27.095 ","End":"04:29.360","Text":"The balls are identical that means we have"},{"Start":"04:29.360 ","End":"04:34.565","Text":"non-sequential sampling and the box cannot contain more than 1 ball."},{"Start":"04:34.565 ","End":"04:38.810","Text":"Well then we\u0027re dealing with sampling without returns."},{"Start":"04:38.810 ","End":"04:44.480","Text":"Let\u0027s go to our summary table. Here we go."},{"Start":"04:44.480 ","End":"04:46.490","Text":"We have non-sequential sampling,"},{"Start":"04:46.490 ","End":"04:51.065","Text":"that\u0027s this column, and without returns, that\u0027s this row."},{"Start":"04:51.065 ","End":"04:55.550","Text":"This is the way to calculate the number of combinations."},{"Start":"04:55.550 ","End":"04:59.740","Text":"We have here n/k."},{"Start":"04:59.740 ","End":"05:08.690","Text":"Now n/k, in our case that\u0027ll be 6/5 and that equals to 6. There we go."},{"Start":"05:08.690 ","End":"05:15.930","Text":"In this question, we\u0027ve covered basically all the 4 combinations of sampling."},{"Start":"05:15.930 ","End":"05:20.659","Text":"We have sampling with returns and without returns,"},{"Start":"05:20.659 ","End":"05:25.410","Text":"and we have sequential sampling and non-sequential sampling."}],"ID":16010},{"Watched":false,"Name":"Exercise 10","Duration":"2m 45s","ChapterTopicVideoID":15224,"CourseChapterTopicPlaylistID":264991,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/15224.jpeg","UploadDate":"2019-01-20T11:51:58.1770000","DurationForVideoObject":"PT2M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this question, we\u0027re given k balls to be put"},{"Start":"00:03.180 ","End":"00:06.300","Text":"into n boxes where n is greater than k. We\u0027re asked"},{"Start":"00:06.300 ","End":"00:10.030","Text":"to calculate the number of possible ways of putting the balls in"},{"Start":"00:10.030 ","End":"00:15.540","Text":"when we have all these combinations here."},{"Start":"00:15.540 ","End":"00:17.519","Text":"Now, if you recall,"},{"Start":"00:17.519 ","End":"00:20.850","Text":"when the balls are different from each other,"},{"Start":"00:20.850 ","End":"00:24.555","Text":"then we have sequential sampling,"},{"Start":"00:24.555 ","End":"00:27.600","Text":"and if they\u0027re the same,"},{"Start":"00:27.600 ","End":"00:30.300","Text":"then we have non-sequential sampling."},{"Start":"00:30.300 ","End":"00:33.360","Text":"If the box can contain more than 1 ball,"},{"Start":"00:33.360 ","End":"00:35.520","Text":"then we have sampling with returns,"},{"Start":"00:35.520 ","End":"00:39.165","Text":"and if the box can only contain 1 ball,"},{"Start":"00:39.165 ","End":"00:41.565","Text":"then we have sampling without returns."},{"Start":"00:41.565 ","End":"00:43.560","Text":"In a, we\u0027re asked,"},{"Start":"00:43.560 ","End":"00:46.370","Text":"what\u0027s the number of possible ways of putting the balls"},{"Start":"00:46.370 ","End":"00:49.610","Text":"in when the balls are different from each other and when we"},{"Start":"00:49.610 ","End":"00:57.395","Text":"have sequential sampling and the box can contain more than 1 ball with returns."},{"Start":"00:57.395 ","End":"01:02.430","Text":"Again, sequential sampling with returns."},{"Start":"01:02.840 ","End":"01:06.795","Text":"Sequential sampling, that\u0027s right here,"},{"Start":"01:06.795 ","End":"01:08.970","Text":"and with returns, that\u0027s right here,"},{"Start":"01:08.970 ","End":"01:14.560","Text":"so this would be the answer for section a."},{"Start":"01:14.560 ","End":"01:16.250","Text":"Now, here in section b,"},{"Start":"01:16.250 ","End":"01:22.170","Text":"we have the balls are identical and the box can contain more than 1 ball."},{"Start":"01:22.170 ","End":"01:24.530","Text":"So the balls are identical means that we have"},{"Start":"01:24.530 ","End":"01:29.675","Text":"non-sequential sampling and the box can contain more than 1 ball."},{"Start":"01:29.675 ","End":"01:32.569","Text":"Then that means that we have with returns."},{"Start":"01:32.569 ","End":"01:35.960","Text":"Again, non-sequential sampling with returns."},{"Start":"01:35.960 ","End":"01:39.440","Text":"Where here\u0027s non-sequential and here\u0027s with returns,"},{"Start":"01:39.440 ","End":"01:43.205","Text":"so this would be the answer for section b."},{"Start":"01:43.205 ","End":"01:50.270","Text":"Now, the balls are different from each other and a box cannot contain more than 1 ball."},{"Start":"01:50.270 ","End":"01:51.950","Text":"Well, the balls are different,"},{"Start":"01:51.950 ","End":"01:55.820","Text":"that means that we have sequential sampling and the box"},{"Start":"01:55.820 ","End":"02:00.410","Text":"cannot contain more than 1 ball means that we have without returns,"},{"Start":"02:00.410 ","End":"02:03.950","Text":"so we have sequential sampling without returns."},{"Start":"02:03.950 ","End":"02:05.990","Text":"Here\u0027s sequential sampling,"},{"Start":"02:05.990 ","End":"02:07.790","Text":"here\u0027s without returns,"},{"Start":"02:07.790 ","End":"02:10.885","Text":"this would be the answer to c,"},{"Start":"02:10.885 ","End":"02:17.056","Text":"and d means that the balls are identical and the box cannot contain more than 1 ball."},{"Start":"02:17.056 ","End":"02:18.360","Text":"Well, the balls are identical,"},{"Start":"02:18.360 ","End":"02:23.090","Text":"we have non-sequential sampling and the box cannot contain more than 1 ball,"},{"Start":"02:23.090 ","End":"02:25.825","Text":"then we have without returns."},{"Start":"02:25.825 ","End":"02:30.120","Text":"Here, non-sequential sampling without returns,"},{"Start":"02:30.120 ","End":"02:35.085","Text":"so this would be the answer for d. Here we\u0027ve summarized,"},{"Start":"02:35.085 ","End":"02:39.290","Text":"again, all the possible combinations of sampling,"},{"Start":"02:39.290 ","End":"02:42.020","Text":"sequential sampling or non-sequential,"},{"Start":"02:42.020 ","End":"02:45.540","Text":"sampling with or without returns."}],"ID":16011}],"Thumbnail":null,"ID":264991},{"Name":"Summary Questions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"44s","ChapterTopicVideoID":12364,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12364.jpeg","UploadDate":"2019-01-31T23:26:11.1130000","DurationForVideoObject":"PT44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"This chapter summarizes all the stuff that we\u0027ve been learning about combinatorics."},{"Start":"00:04.710 ","End":"00:08.723","Text":"Now, there\u0027s no theoretical background to the chapter,"},{"Start":"00:08.723 ","End":"00:10.695","Text":"so as an introduction, I\u0027ll say this."},{"Start":"00:10.695 ","End":"00:13.845","Text":"First of all, the exercises go from easy to hard."},{"Start":"00:13.845 ","End":"00:17.400","Text":"Start by trying to solve the questions by"},{"Start":"00:17.400 ","End":"00:21.360","Text":"yourself and look at the solutions only when you get stuck."},{"Start":"00:21.360 ","End":"00:26.010","Text":"Now, 1 of the factors that contribute to"},{"Start":"00:26.010 ","End":"00:31.125","Text":"the difficulty in this chapter is the fact that it is a summary chapter, right?"},{"Start":"00:31.125 ","End":"00:35.430","Text":"All the subjects are mixed in together and you\u0027re going to have to decide"},{"Start":"00:35.430 ","End":"00:40.879","Text":"what techniques should be used to solve each and every 1 of the exercises."},{"Start":"00:40.879 ","End":"00:42.970","Text":"Good luck."}],"ID":12843},{"Watched":false,"Name":"Exercise 1 - Parts a-c","Duration":"4m 40s","ChapterTopicVideoID":12365,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12365.jpeg","UploadDate":"2018-07-22T17:06:47.3970000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this question, we have 40 students in a class."},{"Start":"00:02.460 ","End":"00:05.880","Text":"Now, a class committee with 5 positions is selected."},{"Start":"00:05.880 ","End":"00:09.060","Text":"Then we\u0027re asked, in how many ways can the committee be"},{"Start":"00:09.060 ","End":"00:12.794","Text":"selected if the committee has 5 different positions,"},{"Start":"00:12.794 ","End":"00:16.844","Text":"and a student can serve in more than 1 position."},{"Start":"00:16.844 ","End":"00:22.200","Text":"Well, here we have sequential sampling with returns."},{"Start":"00:22.200 ","End":"00:24.015","Text":"Now, why is that?"},{"Start":"00:24.015 ","End":"00:30.825","Text":"Well, it\u0027s sequential sampling because we have 5 different positions."},{"Start":"00:30.825 ","End":"00:32.400","Text":"Why is it with returns?"},{"Start":"00:32.400 ","End":"00:36.075","Text":"Because a student can serve in more than 1 position."},{"Start":"00:36.075 ","End":"00:39.120","Text":"Whenever we have sequential sampling with returns,"},{"Start":"00:39.120 ","End":"00:44.340","Text":"the number of combinations that we have is n^k."},{"Start":"00:44.340 ","End":"00:46.650","Text":"What\u0027s our n, and what\u0027s our k?"},{"Start":"00:46.650 ","End":"00:49.665","Text":"Well, n is 40."},{"Start":"00:49.665 ","End":"00:53.285","Text":"That\u0027s the total number of students in the class."},{"Start":"00:53.285 ","End":"00:56.435","Text":"K would be the number of students that we sample."},{"Start":"00:56.435 ","End":"00:58.640","Text":"Well, we sample 5 students,"},{"Start":"00:58.640 ","End":"01:00.680","Text":"out of the 5 positions."},{"Start":"01:00.680 ","End":"01:03.590","Text":"Now, in our case,"},{"Start":"01:03.590 ","End":"01:07.820","Text":"n^k would be 40^5."},{"Start":"01:07.820 ","End":"01:09.080","Text":"Now this is a very large number,"},{"Start":"01:09.080 ","End":"01:10.580","Text":"so I can leave it like this."},{"Start":"01:10.580 ","End":"01:14.780","Text":"Now, I could have also thought of the solution differently."},{"Start":"01:14.780 ","End":"01:18.020","Text":"I could say that for the first position,"},{"Start":"01:18.020 ","End":"01:20.930","Text":"how many students do I have to choose from?"},{"Start":"01:20.930 ","End":"01:23.255","Text":"I have 40 students to choose from."},{"Start":"01:23.255 ","End":"01:25.220","Text":"In the second position, well,"},{"Start":"01:25.220 ","End":"01:28.610","Text":"since a student can serve in more than 1 position,"},{"Start":"01:28.610 ","End":"01:31.990","Text":"then I still have 40 students to choose from,"},{"Start":"01:31.990 ","End":"01:34.490","Text":"and the same for the third position,"},{"Start":"01:34.490 ","End":"01:36.403","Text":"and the fourth position,"},{"Start":"01:36.403 ","End":"01:38.275","Text":"and the fifth position."},{"Start":"01:38.275 ","End":"01:43.105","Text":"That equals to 40^5 as well."},{"Start":"01:43.105 ","End":"01:47.510","Text":"In this section, we\u0027re given that the committee has 5 different positions,"},{"Start":"01:47.510 ","End":"01:51.110","Text":"but a student cannot serve in more than 1 position."},{"Start":"01:51.110 ","End":"01:57.290","Text":"Well, again, here we have sequential sampling because I have 5 different positions."},{"Start":"01:57.290 ","End":"02:01.490","Text":"But now, since a student cannot serve in more than 1 position,"},{"Start":"02:01.490 ","End":"02:06.340","Text":"then we have sampling without returns."},{"Start":"02:06.740 ","End":"02:13.880","Text":"How many combinations do I have when I have sequential sampling without returns?"},{"Start":"02:13.880 ","End":"02:17.035","Text":"Well, again, n here that equals to 40,"},{"Start":"02:17.035 ","End":"02:19.170","Text":"k that equals to 5,"},{"Start":"02:19.170 ","End":"02:23.780","Text":"so the number of combinations would be n_k."},{"Start":"02:23.780 ","End":"02:32.810","Text":"Now, that equals to n factorial divided by n minus k factorial."},{"Start":"02:32.810 ","End":"02:35.105","Text":"Now, in our case,"},{"Start":"02:35.105 ","End":"02:42.290","Text":"that\u0027ll be 40 factorial divided by 40 minus 5 factorial,"},{"Start":"02:42.290 ","End":"02:47.440","Text":"that equals to 40 factorial divided by 35 factorial."},{"Start":"02:47.440 ","End":"02:50.750","Text":"That again, is a very large number."},{"Start":"02:50.750 ","End":"02:53.495","Text":"Now again, I could have thought of it like this."},{"Start":"02:53.495 ","End":"02:55.490","Text":"For the first position,"},{"Start":"02:55.490 ","End":"02:57.320","Text":"how many students do I have to choose from?"},{"Start":"02:57.320 ","End":"03:00.290","Text":"Well, I have 40 students to choose from."},{"Start":"03:00.290 ","End":"03:02.630","Text":"Now for the second position,"},{"Start":"03:02.630 ","End":"03:05.870","Text":"I have 39 position a student to choose from,"},{"Start":"03:05.870 ","End":"03:10.100","Text":"because a student cannot serve in more than 1 position."},{"Start":"03:10.100 ","End":"03:11.640","Text":"For the third position,"},{"Start":"03:11.640 ","End":"03:13.365","Text":"I have 38 students,"},{"Start":"03:13.365 ","End":"03:14.780","Text":"for the fourth position,"},{"Start":"03:14.780 ","End":"03:16.520","Text":"I have 37 students,"},{"Start":"03:16.520 ","End":"03:18.140","Text":"and for the fifth position,"},{"Start":"03:18.140 ","End":"03:23.960","Text":"I have 36 students to choose from."},{"Start":"03:23.960 ","End":"03:27.850","Text":"This would be the solution for section B."},{"Start":"03:27.850 ","End":"03:32.345","Text":"In this section, we\u0027re given that there are no different positions on the committee."},{"Start":"03:32.345 ","End":"03:37.720","Text":"Here we have non-sequential sampling without returns."},{"Start":"03:37.720 ","End":"03:39.655","Text":"Well, why is that?"},{"Start":"03:39.655 ","End":"03:42.115","Text":"Well, it\u0027s not sequential sampling,"},{"Start":"03:42.115 ","End":"03:46.930","Text":"because we don\u0027t differentiate between the different positions,"},{"Start":"03:46.930 ","End":"03:50.470","Text":"all we want are 5 students to be on a committee."},{"Start":"03:50.470 ","End":"03:56.020","Text":"Now, it\u0027s without returns because we want 5 different people on the committee,"},{"Start":"03:56.020 ","End":"04:00.910","Text":"we can\u0027t have the same person on a 5 person committee."},{"Start":"04:00.910 ","End":"04:05.320","Text":"This is non-sequential sampling without returns as I said."},{"Start":"04:05.320 ","End":"04:06.820","Text":"Now n here, if we recall,"},{"Start":"04:06.820 ","End":"04:09.115","Text":"that\u0027s 40 and k here,"},{"Start":"04:09.115 ","End":"04:11.260","Text":"we\u0027re sampling 5 students."},{"Start":"04:11.260 ","End":"04:12.710","Text":"That equals to 5. Now,"},{"Start":"04:12.710 ","End":"04:15.260","Text":"whenever we have non-sequential sampling without returns,"},{"Start":"04:15.260 ","End":"04:22.260","Text":"the number of combinations would be n over k. In our case that\u0027ll be 40 over 5,"},{"Start":"04:22.260 ","End":"04:32.030","Text":"and that equals to 40 factorial divided by 5 factorial times 35 factorial."},{"Start":"04:32.030 ","End":"04:40.710","Text":"Now, this comes out to 658,008."}],"ID":12844},{"Watched":false,"Name":"Exercise 2","Duration":"3m 21s","ChapterTopicVideoID":12366,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12366.jpeg","UploadDate":"2018-07-22T17:06:55.5700000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this question, we have 30 employees in an office."},{"Start":"00:02.640 ","End":"00:06.915","Text":"Now, an overseas delegation with 4 positions is selected."},{"Start":"00:06.915 ","End":"00:12.210","Text":"We\u0027re asked, in how many ways can the delegation be selected if the delegation has"},{"Start":"00:12.210 ","End":"00:18.900","Text":"4 different tasks to carry out and each employee can do more than 1 task?"},{"Start":"00:18.900 ","End":"00:26.315","Text":"Well, here we have sequential sampling with returns. Again, why is that?"},{"Start":"00:26.315 ","End":"00:32.580","Text":"Well, we have sequential sampling because we have 4 different tests to carry out,"},{"Start":"00:32.580 ","End":"00:37.250","Text":"and it\u0027s with returns because an employee can do more than 1 task."},{"Start":"00:37.250 ","End":"00:40.190","Text":"Whenever we have sequential sampling with returns,"},{"Start":"00:40.190 ","End":"00:45.335","Text":"then the number of possible combinations would be n to the power of k. Now n here,"},{"Start":"00:45.335 ","End":"00:47.185","Text":"well, that equals to 30."},{"Start":"00:47.185 ","End":"00:49.760","Text":"We have 30 employees to choose from,"},{"Start":"00:49.760 ","End":"00:55.160","Text":"and we choose or we sample 4 employees out of the 30."},{"Start":"00:55.160 ","End":"01:02.765","Text":"Here, the number of possible combinations would be 30 to the power of 4."},{"Start":"01:02.765 ","End":"01:10.135","Text":"Now, that comes out to 810,000 possible combinations."},{"Start":"01:10.135 ","End":"01:14.120","Text":"In this section, the delegation has 4 different tasks to do."},{"Start":"01:14.120 ","End":"01:18.335","Text":"But now, each employee cannot do more than 1 task."},{"Start":"01:18.335 ","End":"01:22.925","Text":"Here, we have sequential sampling without returns."},{"Start":"01:22.925 ","End":"01:24.380","Text":"Now, why is that?"},{"Start":"01:24.380 ","End":"01:26.405","Text":"Well, sequential sampling, again,"},{"Start":"01:26.405 ","End":"01:29.495","Text":"that\u0027s because we have 4 different tasks to do."},{"Start":"01:29.495 ","End":"01:35.095","Text":"Now, it\u0027s without returns because each employee cannot do more than 1 test."},{"Start":"01:35.095 ","End":"01:38.060","Text":"Whenever we have sequential sampling without returns,"},{"Start":"01:38.060 ","End":"01:42.240","Text":"then the number of possible combinations would be n_k."},{"Start":"01:42.610 ","End":"01:49.910","Text":"That equals to n factorial divided by n minus k factorial."},{"Start":"01:49.910 ","End":"01:52.555","Text":"Now, n here, that equals to 30,"},{"Start":"01:52.555 ","End":"01:55.425","Text":"and k, that equals to 4."},{"Start":"01:55.425 ","End":"02:00.590","Text":"In our case, that\u0027ll be 30_4 and"},{"Start":"02:00.590 ","End":"02:06.035","Text":"that equals to 30 factorial divided by 26 factorial."},{"Start":"02:06.035 ","End":"02:14.855","Text":"Now, that comes out to 657,720 different combinations."},{"Start":"02:14.855 ","End":"02:18.375","Text":"In this section, we have 4 different people."},{"Start":"02:18.375 ","End":"02:22.955","Text":"They will be chosen for a delegation in which everyone has the same position."},{"Start":"02:22.955 ","End":"02:29.885","Text":"Well, here we have non-sequential sampling without returns. Why is that?"},{"Start":"02:29.885 ","End":"02:34.310","Text":"Well, it\u0027s non-sequential because the positions are the same,"},{"Start":"02:34.310 ","End":"02:36.770","Text":"there\u0027s no differentiation in the positions,"},{"Start":"02:36.770 ","End":"02:41.380","Text":"and it\u0027s without returns because we\u0027re looking at 4 different employees."},{"Start":"02:41.380 ","End":"02:46.325","Text":"We can\u0027t have 1 employee in a 4 employee delegation."},{"Start":"02:46.325 ","End":"02:49.790","Text":"So whenever we have non-sequential sampling without returns,"},{"Start":"02:49.790 ","End":"02:54.260","Text":"then the number of combinations would be n over k. Now n here,"},{"Start":"02:54.260 ","End":"02:59.565","Text":"that equals to 30 k. We\u0027re choosing 4 people out of the 30,"},{"Start":"02:59.565 ","End":"03:01.220","Text":"so in our case,"},{"Start":"03:01.220 ","End":"03:03.875","Text":"that\u0027ll be 30 over 4."},{"Start":"03:03.875 ","End":"03:05.420","Text":"Now, 30 over 4,"},{"Start":"03:05.420 ","End":"03:12.890","Text":"that equals to 30 factorial divided by 4 factorial times 26 factorial,"},{"Start":"03:12.890 ","End":"03:21.240","Text":"and that comes out to 27,405 different combinations."}],"ID":12845},{"Watched":false,"Name":"Exercise 3","Duration":"6m 20s","ChapterTopicVideoID":12367,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12367.jpeg","UploadDate":"2018-07-22T17:07:09.5000000","DurationForVideoObject":"PT6M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this question, a secret code is being created."},{"Start":"00:02.460 ","End":"00:05.220","Text":"Now the code is composed of 2 different digits and"},{"Start":"00:05.220 ","End":"00:10.020","Text":"3 different English letters and we\u0027re asked how many different codes can be created?"},{"Start":"00:10.020 ","End":"00:12.480","Text":"Now, there are many ways to calculate this,"},{"Start":"00:12.480 ","End":"00:17.535","Text":"but let me show you the way that I choose because it\u0027s the easiest for me."},{"Start":"00:17.535 ","End":"00:21.390","Text":"I had to choose 2 different digits out of 10,"},{"Start":"00:21.390 ","End":"00:23.655","Text":"so that\u0027ll be 10 over 2."},{"Start":"00:23.655 ","End":"00:28.355","Text":"Now, I have to choose 3 different letters out of 26,"},{"Start":"00:28.355 ","End":"00:31.250","Text":"so that\u0027ll be 26 over 3."},{"Start":"00:31.250 ","End":"00:34.475","Text":"Here, this product shows me"},{"Start":"00:34.475 ","End":"00:41.345","Text":"the possible number of combinations of 2 digits and 3 letters."},{"Start":"00:41.345 ","End":"00:47.525","Text":"But I still have to order the 5 characters in the code."},{"Start":"00:47.525 ","End":"00:52.940","Text":"That means I have to multiply all this by 5 factorial."},{"Start":"00:52.940 ","End":"01:01.805","Text":"Now, again, this would be the choice of my 5 characters,2 digits and 3 letters."},{"Start":"01:01.805 ","End":"01:09.860","Text":"Here, I order the 5 characters in all the possible combinations."},{"Start":"01:09.860 ","End":"01:19.420","Text":"Now, that comes out to 14 million and 40,000 combinations."},{"Start":"01:19.420 ","End":"01:22.916","Text":"In this section, we\u0027re asked how many different codes can be created if"},{"Start":"01:22.916 ","End":"01:26.645","Text":"the code begins with a digit and ends with a digit?"},{"Start":"01:26.645 ","End":"01:28.730","Text":"In the first place,"},{"Start":"01:28.730 ","End":"01:31.920","Text":"I have to choose 1 out of 10 digits."},{"Start":"01:31.920 ","End":"01:33.830","Text":"In the last place well,"},{"Start":"01:33.830 ","End":"01:37.340","Text":"I have to choose 1 out of the 9 remaining digits."},{"Start":"01:37.340 ","End":"01:39.650","Text":"Now, in the middle 3 positions,"},{"Start":"01:39.650 ","End":"01:43.250","Text":"I have to choose out of 26 letters so I have to choose 3 letters."},{"Start":"01:43.250 ","End":"01:44.870","Text":"Now, these 3 letters,"},{"Start":"01:44.870 ","End":"01:46.490","Text":"I have to order them."},{"Start":"01:46.490 ","End":"01:50.585","Text":"Now, another way of thinking of this is this way;"},{"Start":"01:50.585 ","End":"01:55.010","Text":"how many digits do I have to choose from in the first place of the code?"},{"Start":"01:55.010 ","End":"01:58.370","Text":"Well, I have 10 digits to choose from and in the last place,"},{"Start":"01:58.370 ","End":"02:01.154","Text":"I have 9 digits to choose,"},{"Start":"02:01.154 ","End":"02:03.170","Text":"the remaining 9 digits."},{"Start":"02:03.170 ","End":"02:07.365","Text":"Now, for the middle 3 letters here,"},{"Start":"02:07.365 ","End":"02:10.690","Text":"well, that\u0027ll be 26_3."},{"Start":"02:10.690 ","End":"02:14.000","Text":"Now, why are these equal?"},{"Start":"02:14.000 ","End":"02:18.590","Text":"Well, 10 over 1 that equals to 10 and 9 over 1, that equals 9."},{"Start":"02:18.590 ","End":"02:26.705","Text":"Now, 26 over 3 times 3 factorial is equal to 26_3. Now why is that?"},{"Start":"02:26.705 ","End":"02:31.310","Text":"Well, 26 over 3 times 3 factorial,"},{"Start":"02:31.310 ","End":"02:40.815","Text":"that equals to 26 factorial divided by 3 factorial times 23 factorial times 3 factorial."},{"Start":"02:40.815 ","End":"02:43.250","Text":"The 3 factorials cancel each other out,"},{"Start":"02:43.250 ","End":"02:47.615","Text":"so we\u0027re left with 26 factorial divided by 23 factorial."},{"Start":"02:47.615 ","End":"02:53.740","Text":"Now, this comes out to 1,404,000."},{"Start":"02:53.740 ","End":"02:58.055","Text":"Now, again, why does this equal to that?"},{"Start":"02:58.055 ","End":"03:01.370","Text":"Well, again, what\u0027s 26_3?"},{"Start":"03:01.370 ","End":"03:05.150","Text":"Well, that\u0027s 26 factorial divided by"},{"Start":"03:05.150 ","End":"03:13.250","Text":"26 minus 3 factorial and that equals to 26 factorial divided by 23 factorial,"},{"Start":"03:13.250 ","End":"03:15.290","Text":"which is exactly what we have here."},{"Start":"03:15.290 ","End":"03:22.940","Text":"Again, we have 1,404,000 different codes that can be created."},{"Start":"03:22.940 ","End":"03:25.940","Text":"In section C, we\u0027re asked how many codes can be"},{"Start":"03:25.940 ","End":"03:29.860","Text":"created if the digits must be next to each other?"},{"Start":"03:29.860 ","End":"03:34.350","Text":"First of all, we have to choose the 2 digits and 3 letters."},{"Start":"03:34.350 ","End":"03:40.640","Text":"That\u0027ll be 10 over 2 times 26 over 3."},{"Start":"03:40.640 ","End":"03:45.140","Text":"Now, we\u0027re given that the digits must be next to each"},{"Start":"03:45.140 ","End":"03:50.045","Text":"other so let\u0027s say we have the 5 characters here."},{"Start":"03:50.045 ","End":"03:54.479","Text":"We have 1,"},{"Start":"03:54.479 ","End":"03:57.935","Text":"2, 3, 4, and 5 characters."},{"Start":"03:57.935 ","End":"04:01.205","Text":"Now we\u0027re told that the digits must be next to each other."},{"Start":"04:01.205 ","End":"04:08.485","Text":"Let\u0027s say X_1 and X_2 are the digits and they have to be together all the time."},{"Start":"04:08.485 ","End":"04:10.940","Text":"How many objects do I have here?"},{"Start":"04:10.940 ","End":"04:15.120","Text":"Well, I have 1, 2, 3, and 4 objects."},{"Start":"04:15.120 ","End":"04:18.905","Text":"The 1st object is the 2 digits together,"},{"Start":"04:18.905 ","End":"04:20.780","Text":"the 2nd object is X_3,"},{"Start":"04:20.780 ","End":"04:22.220","Text":"the 3rd object is X_4,"},{"Start":"04:22.220 ","End":"04:26.215","Text":"and the 4th object is X_5."},{"Start":"04:26.215 ","End":"04:30.665","Text":"So how many ways can I order 4 objects in a row?"},{"Start":"04:30.665 ","End":"04:33.995","Text":"Well, 4 factorial."},{"Start":"04:33.995 ","End":"04:39.545","Text":"Once I\u0027ve chosen the 5 characters in my code,"},{"Start":"04:39.545 ","End":"04:44.415","Text":"I have to order them by 4 factorial."},{"Start":"04:44.415 ","End":"04:46.790","Text":"I have to multiply this by 4 factorial,"},{"Start":"04:46.790 ","End":"04:50.770","Text":"but I\u0027m not done yet because now I have to order these."},{"Start":"04:50.770 ","End":"04:53.810","Text":"I have the internal order of the digits,"},{"Start":"04:53.810 ","End":"04:57.545","Text":"so I have to multiply this by 2 factorial."},{"Start":"04:57.545 ","End":"05:07.229","Text":"Now, this comes out to 5,616,000 possible ways."},{"Start":"05:07.280 ","End":"05:13.625","Text":"In this section we are asked how many codes to the digits not appear consecutively?"},{"Start":"05:13.625 ","End":"05:17.750","Text":"We know the total number of codes that I can generate."},{"Start":"05:17.750 ","End":"05:20.075","Text":"We\u0027ve done that in section a."},{"Start":"05:20.075 ","End":"05:29.282","Text":"That\u0027ll be 10 over 2 times 26 over 3 times 5 factorial."},{"Start":"05:29.282 ","End":"05:36.110","Text":"Now that came out to 14,040,000."},{"Start":"05:36.110 ","End":"05:39.425","Text":"Now, I need to take away from that,"},{"Start":"05:39.425 ","End":"05:44.690","Text":"that total number of codes that I can create when the digits are together."},{"Start":"05:44.690 ","End":"05:54.455","Text":"Now we did this in section c. That was 10 over 2 times 26 over 3 times"},{"Start":"05:54.455 ","End":"06:04.865","Text":"4 factorial times 2 factorial and that came out to 5,616,000."},{"Start":"06:04.865 ","End":"06:08.495","Text":"Now, this comes out to"},{"Start":"06:08.495 ","End":"06:20.040","Text":"8,424,000 possible ways of creating a code when the 2 digits are not together."}],"ID":12846},{"Watched":false,"Name":"Exercise 4","Duration":"5m 46s","ChapterTopicVideoID":12368,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12368.jpeg","UploadDate":"2018-07-22T17:07:26.7130000","DurationForVideoObject":"PT5M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this question, we have 4 drawers in a closet."},{"Start":"00:03.000 ","End":"00:07.290","Text":"Now a mother asks her child to arrange 6 different games in the closet."},{"Start":"00:07.290 ","End":"00:10.200","Text":"The child puts the games randomly into"},{"Start":"00:10.200 ","End":"00:14.130","Text":"the drawers and each drawer can contain all of the games together."},{"Start":"00:14.130 ","End":"00:17.430","Text":"Now, we\u0027re asked, what\u0027s the probability that the child would"},{"Start":"00:17.430 ","End":"00:20.660","Text":"put all the games into the top drawer."},{"Start":"00:20.660 ","End":"00:27.188","Text":"So we have here a closet here 4 drawers and just draw number 1,"},{"Start":"00:27.188 ","End":"00:29.140","Text":"2, 3 and 4."},{"Start":"00:29.140 ","End":"00:36.305","Text":"We\u0027re looking for the probability that all the games are in the top drawer."},{"Start":"00:36.305 ","End":"00:43.655","Text":"Now, before we can answer that let\u0027s just see what\u0027s the size of the sample space."},{"Start":"00:43.655 ","End":"00:46.880","Text":"So we have here the size of the sample space."},{"Start":"00:46.880 ","End":"00:52.880","Text":"Now we have here 6 games and we have 4 drawers."},{"Start":"00:52.880 ","End":"00:57.960","Text":"Now, if we take 1 game,"},{"Start":"00:57.960 ","End":"01:03.870","Text":"right, the first game well how many options do I have putting the game in a drawer?"},{"Start":"01:03.870 ","End":"01:05.760","Text":"Well, I have 4 options."},{"Start":"01:05.760 ","End":"01:07.205","Text":"Because I have 4 drawers."},{"Start":"01:07.205 ","End":"01:09.275","Text":"Now, what about the second game?"},{"Start":"01:09.275 ","End":"01:14.420","Text":"Well, the second game also has 4 drawers or 4 options."},{"Start":"01:14.420 ","End":"01:17.780","Text":"What about the third game 4 again,"},{"Start":"01:17.780 ","End":"01:21.170","Text":"and the fourth and the fifth and the sixth."},{"Start":"01:21.170 ","End":"01:24.590","Text":"So we have here 4^6."},{"Start":"01:24.590 ","End":"01:30.095","Text":"So that will be the size of the sample space."},{"Start":"01:30.095 ","End":"01:34.670","Text":"Now, we\u0027re looking for the probability that all the game,"},{"Start":"01:34.670 ","End":"01:35.720","Text":"so in the top drawer,"},{"Start":"01:35.720 ","End":"01:37.700","Text":"in this drawer right here."},{"Start":"01:37.700 ","End":"01:42.305","Text":"Well, we have here right then the first game,"},{"Start":"01:42.305 ","End":"01:44.815","Text":"well, we have 1 out of 4."},{"Start":"01:44.815 ","End":"01:47.895","Text":"So we have 1 choice."},{"Start":"01:47.895 ","End":"01:51.260","Text":"The second game well that also has to be in this drawer,"},{"Start":"01:51.260 ","End":"01:53.900","Text":"and the third and the fourth and the fifth and the sixth."},{"Start":"01:53.900 ","End":"02:02.920","Text":"We have 1 the power of 6 divided by the size of the sample space, which is 4^6."},{"Start":"02:02.920 ","End":"02:08.550","Text":"Here that\u0027ll be equal to 1 divided by 4^6,"},{"Start":"02:08.550 ","End":"02:14.145","Text":"and that equals to 1 divided by 4,096."},{"Start":"02:14.145 ","End":"02:19.715","Text":"This is the probability that all the games will be in the top drawer."},{"Start":"02:19.715 ","End":"02:21.290","Text":"This section we\u0027re asked,"},{"Start":"02:21.290 ","End":"02:25.360","Text":"what\u0027s the probability that the child would put all the games into the same drawer?"},{"Start":"02:25.360 ","End":"02:27.980","Text":"Now in the previous section we\u0027re asked about"},{"Start":"02:27.980 ","End":"02:31.805","Text":"the probability that all the games would be in the top drawer."},{"Start":"02:31.805 ","End":"02:36.110","Text":"Now, it really doesn\u0027t matter which drawer is chosen but we"},{"Start":"02:36.110 ","End":"02:41.185","Text":"want that all the games would be put into that 1 drawer."},{"Start":"02:41.185 ","End":"02:43.620","Text":"Let\u0027s get started."},{"Start":"02:43.620 ","End":"02:49.670","Text":"In the denominator, we have 4^6."},{"Start":"02:49.670 ","End":"02:53.008","Text":"That\u0027s the size of the sample space. Now in the numerator."},{"Start":"02:53.008 ","End":"02:54.110","Text":"Now, as I said,"},{"Start":"02:54.110 ","End":"02:56.930","Text":"it really doesn\u0027t matter which drawers chosen,"},{"Start":"02:56.930 ","End":"03:00.960","Text":"so we have to choose 1 drawer out of 4,"},{"Start":"03:00.960 ","End":"03:03.445","Text":"so that\u0027s 4 over 1."},{"Start":"03:03.445 ","End":"03:05.854","Text":"Now having chosen a drawer,"},{"Start":"03:05.854 ","End":"03:10.115","Text":"all the games have to be put into that 1 drawer."},{"Start":"03:10.115 ","End":"03:13.263","Text":"The choice here is 1^6."},{"Start":"03:13.263 ","End":"03:16.100","Text":"Because having chosen a drawer,"},{"Start":"03:16.100 ","End":"03:18.890","Text":"well then for the first game,"},{"Start":"03:18.890 ","End":"03:23.090","Text":"while I have only that 1 drawer to choose,"},{"Start":"03:23.090 ","End":"03:24.780","Text":"and for the second game and third,"},{"Start":"03:24.780 ","End":"03:26.525","Text":"and fourth, and fifth and sixth."},{"Start":"03:26.525 ","End":"03:30.890","Text":"I have here 1^6 times 4 over 1,"},{"Start":"03:30.890 ","End":"03:33.290","Text":"which is the choice of the drawer."},{"Start":"03:33.290 ","End":"03:40.755","Text":"This then is equal to 4 divided by 4,096,"},{"Start":"03:40.755 ","End":"03:46.150","Text":"and in decimal that would be 0.00098."},{"Start":"03:50.960 ","End":"03:53.300","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"03:53.300 ","End":"03:56.180","Text":"the probability that the game of chess will be put into"},{"Start":"03:56.180 ","End":"04:00.845","Text":"the uppermost drawer and the other games will be put into the other drawers."},{"Start":"04:00.845 ","End":"04:03.875","Text":"Let\u0027s get to work."},{"Start":"04:03.875 ","End":"04:06.200","Text":"This is a probability."},{"Start":"04:06.200 ","End":"04:08.900","Text":"We\u0027re looking for the probability that chess is in"},{"Start":"04:08.900 ","End":"04:13.040","Text":"the top drawer and all the other games in the other drawers right here."},{"Start":"04:13.040 ","End":"04:14.930","Text":"Well, in the denominator,"},{"Start":"04:14.930 ","End":"04:19.070","Text":"we have the size of the sample space, which is 4^6."},{"Start":"04:19.070 ","End":"04:20.599","Text":"Now in the numerator."},{"Start":"04:20.599 ","End":"04:23.840","Text":"Well, Chess,1 of the games is in the top drawer."},{"Start":"04:23.840 ","End":"04:26.375","Text":"So we only have 1 choice for chess."},{"Start":"04:26.375 ","End":"04:28.370","Text":"That\u0027ll be 1 here."},{"Start":"04:28.370 ","End":"04:30.740","Text":"Now, how many other games do we have?"},{"Start":"04:30.740 ","End":"04:32.555","Text":"We have 5 other games,"},{"Start":"04:32.555 ","End":"04:35.810","Text":"and we have 3 choices for each of the 5 games."},{"Start":"04:35.810 ","End":"04:38.285","Text":"That\u0027ll be 3^5."},{"Start":"04:38.285 ","End":"04:49.550","Text":"Now, that\u0027s the probability and that equals to 0.50593."},{"Start":"04:49.550 ","End":"04:53.405","Text":"In this section, we\u0027re asked what\u0027s the probability that the game of"},{"Start":"04:53.405 ","End":"04:57.410","Text":"chess will not be put into the uppermost drawer."},{"Start":"04:57.410 ","End":"05:00.380","Text":"Let\u0027s get started."},{"Start":"05:00.770 ","End":"05:03.330","Text":"That\u0027ll be equal to the following."},{"Start":"05:03.330 ","End":"05:04.820","Text":"Well, in the denominator,"},{"Start":"05:04.820 ","End":"05:08.905","Text":"we have the size of the sample space which is 4^6."},{"Start":"05:08.905 ","End":"05:11.300","Text":"Now, chess 1 of the games is not in"},{"Start":"05:11.300 ","End":"05:14.390","Text":"the top drawer so how many choices do I have for the game of chess?"},{"Start":"05:14.390 ","End":"05:16.090","Text":"Well, have 3 choices."},{"Start":"05:16.090 ","End":"05:17.800","Text":"That\u0027ll be 3 here."},{"Start":"05:17.800 ","End":"05:20.090","Text":"Now, how many games are left?"},{"Start":"05:20.090 ","End":"05:21.785","Text":"Well 5 games are left,"},{"Start":"05:21.785 ","End":"05:25.700","Text":"and each game has 4 choices,"},{"Start":"05:25.700 ","End":"05:29.970","Text":"so that\u0027ll be 4^5."},{"Start":"05:29.970 ","End":"05:33.090","Text":"Now, 4^5 and 4^6,"},{"Start":"05:33.090 ","End":"05:35.060","Text":"well the 4\u0027s cancel each other out,"},{"Start":"05:35.060 ","End":"05:38.525","Text":"so 4 to the power 5 cancels out this,"},{"Start":"05:38.525 ","End":"05:45.420","Text":"so we\u0027re left with 3/4 and that equals to 0.75."}],"ID":12847},{"Watched":false,"Name":"Exercise 5","Duration":"5m 10s","ChapterTopicVideoID":12369,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12369.jpeg","UploadDate":"2019-01-31T23:20:44.3670000","DurationForVideoObject":"PT5M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.305","Text":"In this question, we\u0027re given a city and we have 4 political parties,"},{"Start":"00:04.305 ","End":"00:07.680","Text":"A, B, C, and D, that are running for the city council."},{"Start":"00:07.680 ","End":"00:10.890","Text":"Now 6 people don\u0027t know from whom to vote,"},{"Start":"00:10.890 ","End":"00:12.690","Text":"so they choose a party randomly."},{"Start":"00:12.690 ","End":"00:18.270","Text":"We\u0027re asked what\u0027s the probability that all 6 people will vote for the same party?"},{"Start":"00:18.270 ","End":"00:23.775","Text":"Before answering this, let\u0027s just calculate the size of the sample space."},{"Start":"00:23.775 ","End":"00:27.449","Text":"Now that\u0027ll be equal to the following."},{"Start":"00:27.449 ","End":"00:32.700","Text":"Well, we have 4 parties and 6 people."},{"Start":"00:32.700 ","End":"00:38.078","Text":"Just think of this as a trial with 6 stages,"},{"Start":"00:38.078 ","End":"00:44.100","Text":"for the 6 people where in the first stage I have 4 parties to choose from."},{"Start":"00:44.100 ","End":"00:49.745","Text":"Then the second stage I also have 4 parties to choose from and so on and so forth."},{"Start":"00:49.745 ","End":"00:55.060","Text":"In essence, I have 4^6,"},{"Start":"00:55.060 ","End":"00:59.160","Text":"which is equal to 4,096."},{"Start":"00:59.160 ","End":"01:07.180","Text":"I have a total of 4,096 possible combinations of 6 people choosing 4 parties."},{"Start":"01:07.180 ","End":"01:10.640","Text":"In section a, I\u0027m asked what\u0027s the probability"},{"Start":"01:10.640 ","End":"01:14.840","Text":"that all 6 people will vote for the same party?"},{"Start":"01:15.420 ","End":"01:18.994","Text":"This is the probability that I have to calculate."},{"Start":"01:18.994 ","End":"01:20.855","Text":"Now in the denominator,"},{"Start":"01:20.855 ","End":"01:23.240","Text":"I have the size of the sample space,"},{"Start":"01:23.240 ","End":"01:28.160","Text":"which is 4^6, or 4,096."},{"Start":"01:28.160 ","End":"01:30.500","Text":"Now in the numerator, well,"},{"Start":"01:30.500 ","End":"01:33.530","Text":"all the people choose the same party."},{"Start":"01:33.530 ","End":"01:37.580","Text":"Now, we didn\u0027t say which party,"},{"Start":"01:37.580 ","End":"01:41.630","Text":"so we have to choose 1 out of the 4 parties."},{"Start":"01:41.630 ","End":"01:45.365","Text":"That\u0027ll be 4/1 times now,"},{"Start":"01:45.365 ","End":"01:48.425","Text":"each person or they don\u0027t really have a choice,"},{"Start":"01:48.425 ","End":"01:51.350","Text":"so they choose 1 out of the 4 parties."},{"Start":"01:51.350 ","End":"01:59.630","Text":"That\u0027ll be 1^6. Now that equals to 4 divided by 4,096,"},{"Start":"01:59.630 ","End":"02:07.160","Text":"and that equals to 0.00098 approximately."},{"Start":"02:07.160 ","End":"02:12.545","Text":"In this section, we\u0027re asked what\u0027s the probability that party A will get no votes?"},{"Start":"02:12.545 ","End":"02:20.780","Text":"We\u0027re looking for the probability that A gets no votes."},{"Start":"02:20.780 ","End":"02:24.125","Text":"Now, if that\u0027s the case on the denominator,"},{"Start":"02:24.125 ","End":"02:28.250","Text":"where we have the size of the sample space which is 4,096."},{"Start":"02:28.250 ","End":"02:29.885","Text":"Now in the numerator well,"},{"Start":"02:29.885 ","End":"02:36.960","Text":"all 6 people vote for 1 out of the 3 remaining parties,"},{"Start":"02:36.960 ","End":"02:40.520","Text":"party B, C, and D. How many parties do we have?"},{"Start":"02:40.520 ","End":"02:41.810","Text":"We have 3 parties,"},{"Start":"02:41.810 ","End":"02:45.215","Text":"and each person gets to choose 1 out of 3."},{"Start":"02:45.215 ","End":"02:48.005","Text":"We have 3^6 in the numerator."},{"Start":"02:48.005 ","End":"02:54.355","Text":"Now that equals to 0.17798."},{"Start":"02:54.355 ","End":"02:56.240","Text":"In this section,"},{"Start":"02:56.240 ","End":"02:59.450","Text":"we\u0027re asked what\u0027s the probability that party A will receive"},{"Start":"02:59.450 ","End":"03:05.520","Text":"exactly 3 votes and the other parties will get only 1 vote each."},{"Start":"03:05.810 ","End":"03:11.570","Text":"Again, we\u0027re looking at the probability that A gets 3 votes and B,"},{"Start":"03:11.570 ","End":"03:14.585","Text":"C, D get 1 vote each."},{"Start":"03:14.585 ","End":"03:16.730","Text":"In the denominator again,"},{"Start":"03:16.730 ","End":"03:21.170","Text":"we have the size of the sample space, which is 4,096."},{"Start":"03:21.170 ","End":"03:23.570","Text":"Now if party A gets 3 votes,"},{"Start":"03:23.570 ","End":"03:27.040","Text":"then out of the 6 people we have to choose 3."},{"Start":"03:27.040 ","End":"03:31.200","Text":"Now, there are 3 other people that remains,"},{"Start":"03:31.200 ","End":"03:38.955","Text":"so out of the 3 people we need 1 to vote for a party B."},{"Start":"03:38.955 ","End":"03:44.940","Text":"Out of the 2 remaining people we need 1 to vote for party C,"},{"Start":"03:44.940 ","End":"03:51.295","Text":"and out of the remaining person we need to vote for party"},{"Start":"03:51.295 ","End":"03:57.820","Text":"D. That equals to 20 times 3 times 2 times 1,"},{"Start":"03:57.820 ","End":"04:09.710","Text":"divided by 4,096, and that equals to 0.02929."},{"Start":"04:09.710 ","End":"04:14.090","Text":"In this section, we\u0027re asked what\u0027s the probability that party A will get 2 votes,"},{"Start":"04:14.090 ","End":"04:15.620","Text":"party B will get 2 votes,"},{"Start":"04:15.620 ","End":"04:18.125","Text":"and party C will get 2 votes."},{"Start":"04:18.125 ","End":"04:21.060","Text":"Let\u0027s calculate that."},{"Start":"04:21.290 ","End":"04:24.620","Text":"Here we go in the denominator again,"},{"Start":"04:24.620 ","End":"04:29.270","Text":"we have the size of the sample space, which is 4,096."},{"Start":"04:29.270 ","End":"04:32.960","Text":"Now, if party A gets 2 votes,"},{"Start":"04:32.960 ","End":"04:35.080","Text":"then out of the 6 people,"},{"Start":"04:35.080 ","End":"04:38.730","Text":"2 people vote for party A."},{"Start":"04:38.730 ","End":"04:42.405","Text":"We have to choose 2 out of the 6."},{"Start":"04:42.405 ","End":"04:45.315","Text":"Now we have 4 people that remain."},{"Start":"04:45.315 ","End":"04:47.670","Text":"Now out of the 4 remaining people,"},{"Start":"04:47.670 ","End":"04:51.555","Text":"we need 2 to vote for party B."},{"Start":"04:51.555 ","End":"04:55.545","Text":"Out of the 2 remaining people,"},{"Start":"04:55.545 ","End":"05:00.250","Text":"we need those 2 people to vote for party C,"},{"Start":"05:00.250 ","End":"05:04.380","Text":"so that comes out to 0.02197."}],"ID":12848},{"Watched":false,"Name":"Exercise 6 - Parts a-e","Duration":"6m 32s","ChapterTopicVideoID":12370,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12370.jpeg","UploadDate":"2019-01-31T23:21:07.5400000","DurationForVideoObject":"PT6M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this question, we have 5 friends that meet at the theater to see a movie."},{"Start":"00:04.080 ","End":"00:10.185","Text":"There are 8 different movies showing and each person selects a movie randomly."},{"Start":"00:10.185 ","End":"00:15.510","Text":"We\u0027re asked, what\u0027s the probability that everyone will select the same movie?"},{"Start":"00:15.510 ","End":"00:19.110","Text":"The first thing that we need to do before we calculate"},{"Start":"00:19.110 ","End":"00:24.870","Text":"any probability is to calculate the size of the sample space."},{"Start":"00:24.870 ","End":"00:27.780","Text":"We have here, as we said,"},{"Start":"00:27.780 ","End":"00:31.545","Text":"5 friends and we have 8 different movies."},{"Start":"00:31.545 ","End":"00:37.845","Text":"The first person has 1 out of 8 or 8 different movies to choose from,"},{"Start":"00:37.845 ","End":"00:41.885","Text":"and the second person has 8 movies to choose from,"},{"Start":"00:41.885 ","End":"00:43.355","Text":"and so on and so forth."},{"Start":"00:43.355 ","End":"00:47.820","Text":"In essence, we have 8^5 different"},{"Start":"00:47.820 ","End":"00:53.700","Text":"combinations of 5 friends choosing 8 different movies,"},{"Start":"00:53.700 ","End":"00:55.905","Text":"1 out of 8 different movies."},{"Start":"00:55.905 ","End":"00:59.810","Text":"We\u0027re asked now, what\u0027s the probability that everyone will"},{"Start":"00:59.810 ","End":"01:04.325","Text":"select the same movie? Here we go."},{"Start":"01:04.325 ","End":"01:09.350","Text":"In the denominator, we have the size of the sample space,"},{"Start":"01:09.350 ","End":"01:11.545","Text":"which is 8^5,"},{"Start":"01:11.545 ","End":"01:13.234","Text":"and in the numerator,"},{"Start":"01:13.234 ","End":"01:16.080","Text":"well, if everyone chooses the same movie,"},{"Start":"01:16.080 ","End":"01:21.245","Text":"they have 1 choice out of 8 to make of choosing a movie."},{"Start":"01:21.245 ","End":"01:23.445","Text":"Having chosen a movie,"},{"Start":"01:23.445 ","End":"01:25.580","Text":"everyone chooses that movie,"},{"Start":"01:25.580 ","End":"01:27.755","Text":"so there\u0027s really no choice."},{"Start":"01:27.755 ","End":"01:31.505","Text":"All the 5 people choose this same movie right here."},{"Start":"01:31.505 ","End":"01:41.185","Text":"Now, that equals to 8 divided by 8^5 or 1 divided by 8^4."},{"Start":"01:41.185 ","End":"01:48.960","Text":"That comes out to 1 divided by 4,096."},{"Start":"01:48.960 ","End":"01:54.700","Text":"That\u0027s the probability of everyone choosing the same movie."},{"Start":"01:54.700 ","End":"01:57.620","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"01:57.620 ","End":"02:00.140","Text":"that everyone will choose the movie Alien?"},{"Start":"02:00.140 ","End":"02:02.300","Text":"That\u0027s 1 of the 8 movies."},{"Start":"02:02.300 ","End":"02:04.745","Text":"Let\u0027s get to work."},{"Start":"02:04.745 ","End":"02:08.030","Text":"Well, in the denominator,"},{"Start":"02:08.030 ","End":"02:14.300","Text":"we have the size of the sample space, which is 8^5."},{"Start":"02:14.300 ","End":"02:18.525","Text":"Now, everyone chooses the movie Alien."},{"Start":"02:18.525 ","End":"02:22.650","Text":"So basically, they have 1 out of the 8,"},{"Start":"02:22.650 ","End":"02:24.450","Text":"that\u0027s the movement that they chose,"},{"Start":"02:24.450 ","End":"02:27.140","Text":"and 5 of them chose the same movie."},{"Start":"02:27.140 ","End":"02:31.975","Text":"That\u0027s 1 divided by 8^5,"},{"Start":"02:31.975 ","End":"02:38.835","Text":"and that equals to 1 divided by 32,768."},{"Start":"02:38.835 ","End":"02:40.680","Text":"In this section,"},{"Start":"02:40.680 ","End":"02:45.425","Text":"we\u0027re asked what\u0027s the probability that each friend will select a different movie?"},{"Start":"02:45.425 ","End":"02:47.870","Text":"Let\u0027s calculate that."},{"Start":"02:47.870 ","End":"02:51.755","Text":"In the denominator, what do we have here?"},{"Start":"02:51.755 ","End":"02:56.360","Text":"We have the size of the sample space which is 8^5."},{"Start":"02:56.360 ","End":"02:59.645","Text":"If each person selects a different movie, well,"},{"Start":"02:59.645 ","End":"03:03.515","Text":"the first person has 8 movies to choose from,"},{"Start":"03:03.515 ","End":"03:07.025","Text":"the second person has 7 movies to choose from,"},{"Start":"03:07.025 ","End":"03:11.085","Text":"the third person has 6 movies to choose from,"},{"Start":"03:11.085 ","End":"03:15.230","Text":"the fourth person has 5 movies to choose from,"},{"Start":"03:15.230 ","End":"03:19.795","Text":"and the fifth person has 4 movies to choose from."},{"Start":"03:19.795 ","End":"03:23.700","Text":"All in all, that equals to the following."},{"Start":"03:23.700 ","End":"03:25.425","Text":"Well, this thing right here,"},{"Start":"03:25.425 ","End":"03:31.280","Text":"that\u0027s 8 by 5 divided by 8^5."},{"Start":"03:31.280 ","End":"03:32.870","Text":"Why is that so?"},{"Start":"03:32.870 ","End":"03:37.565","Text":"This is just like some sequential sampling without returns."},{"Start":"03:37.565 ","End":"03:42.359","Text":"It\u0027s without returns because we\u0027re selecting different movies and"},{"Start":"03:42.359 ","End":"03:44.600","Text":"it\u0027s sequential sampling because there\u0027s"},{"Start":"03:44.600 ","End":"03:48.114","Text":"meaning to the order of the movies being sampled."},{"Start":"03:48.114 ","End":"03:52.720","Text":"That equals to 0.205."},{"Start":"03:54.610 ","End":"03:57.050","Text":"In this question, we\u0027re asked,"},{"Start":"03:57.050 ","End":"04:00.080","Text":"what\u0027s the probability that at least 2 friends will select"},{"Start":"04:00.080 ","End":"04:03.985","Text":"the same movie? Let\u0027s get to work."},{"Start":"04:03.985 ","End":"04:06.725","Text":"When we say at least 2 friends,"},{"Start":"04:06.725 ","End":"04:08.930","Text":"we\u0027re talking about friend Number 1 and 2,"},{"Start":"04:08.930 ","End":"04:10.730","Text":"friend Number 1 and 3,"},{"Start":"04:10.730 ","End":"04:12.665","Text":"2 and 4, and so on and so forth."},{"Start":"04:12.665 ","End":"04:16.385","Text":"There are many combination of 2 friends and"},{"Start":"04:16.385 ","End":"04:21.360","Text":"we want this combination to select a movie from 1-8 movies."},{"Start":"04:21.880 ","End":"04:24.950","Text":"That\u0027s a lot of combinations and it\u0027s very"},{"Start":"04:24.950 ","End":"04:28.910","Text":"complicated to calculate the probability of this event."},{"Start":"04:28.910 ","End":"04:36.635","Text":"So what we need to do is we need to look at the complimentary set."},{"Start":"04:36.635 ","End":"04:41.060","Text":"Now, the complimentary set of at least 2 friends selecting"},{"Start":"04:41.060 ","End":"04:47.580","Text":"the same movie would be that each friend chooses a different movie,"},{"Start":"04:47.580 ","End":"04:51.570","Text":"so the probability of this set is equal to"},{"Start":"04:51.570 ","End":"04:56.530","Text":"1 minus the probability of the complementary set."},{"Start":"04:56.530 ","End":"05:01.291","Text":"Now, the probability of each friend selecting a different movie,"},{"Start":"05:01.291 ","End":"05:04.685","Text":"we\u0027ve calculated that in the last section."},{"Start":"05:04.685 ","End":"05:11.020","Text":"That equals to 0.205."},{"Start":"05:11.020 ","End":"05:16.880","Text":"The probability of at least 2 friends selecting the same movie is equal to"},{"Start":"05:16.880 ","End":"05:24.120","Text":"1 minus 0.205 which equals to 0.795."},{"Start":"05:25.360 ","End":"05:30.130","Text":"In this section, we\u0027re asked what\u0027s the probability that Danny and Eliot,"},{"Start":"05:30.130 ","End":"05:33.440","Text":"2 of the 5 friends will select the movie Alien,"},{"Start":"05:33.440 ","End":"05:34.985","Text":"1 of the 8 movies,"},{"Start":"05:34.985 ","End":"05:39.470","Text":"and all the other friends will select other movies?"},{"Start":"05:39.470 ","End":"05:42.490","Text":"What\u0027s that probability?"},{"Start":"05:42.490 ","End":"05:44.570","Text":"This is the probability."},{"Start":"05:44.570 ","End":"05:53.575","Text":"In the denominator, we have the size of the sample space which is equal to 8^5."},{"Start":"05:53.575 ","End":"05:56.520","Text":"Now, Danny and Eliot,"},{"Start":"05:56.520 ","End":"05:58.079","Text":"that\u0027s 2 friends,"},{"Start":"05:58.079 ","End":"06:03.550","Text":"they choose 1 specific movie which is Aliens, so that\u0027s 1^2."},{"Start":"06:04.490 ","End":"06:08.190","Text":"Now, the others."},{"Start":"06:08.190 ","End":"06:10.230","Text":"We have 3 other friends,"},{"Start":"06:10.230 ","End":"06:14.235","Text":"they have 7 other movies to choose from,"},{"Start":"06:14.235 ","End":"06:18.360","Text":"so that\u0027ll be 7^3."},{"Start":"06:18.360 ","End":"06:24.345","Text":"Now, that equals to 0.0105."},{"Start":"06:24.345 ","End":"06:27.780","Text":"That\u0027s the probability of Danny and Eliot choosing"},{"Start":"06:27.780 ","End":"06:32.730","Text":"Aliens and the other friends choosing other movies."}],"ID":12849},{"Watched":false,"Name":"Exercise 6 - Parts f-g","Duration":"3m 3s","ChapterTopicVideoID":12371,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12371.jpeg","UploadDate":"2019-01-31T23:21:15.6630000","DurationForVideoObject":"PT3M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this section we\u0027re asked what\u0027s the probability"},{"Start":"00:02.580 ","End":"00:07.000","Text":"that none of the friends will select the movie Alien?"},{"Start":"00:07.460 ","End":"00:12.090","Text":"Again, in the denominator we have the size of our sample space,"},{"Start":"00:12.090 ","End":"00:14.355","Text":"which is 8 to the power of 5."},{"Start":"00:14.355 ","End":"00:17.205","Text":"Now, if no 1 chooses Alien,"},{"Start":"00:17.205 ","End":"00:22.335","Text":"then all the 5 friends choose 1 out of the 7 movie."},{"Start":"00:22.335 ","End":"00:24.855","Text":"We have 7 options,"},{"Start":"00:24.855 ","End":"00:30.825","Text":"or 7 movies to choose from for all 5 friends."},{"Start":"00:30.825 ","End":"00:38.600","Text":"Now, that equals to 0.51-9."},{"Start":"00:38.600 ","End":"00:43.430","Text":"In this section, we\u0027re told that all 8 movies are presented to the friends as a list,"},{"Start":"00:43.430 ","End":"00:47.050","Text":"that means that a list of the 8 movies is created."},{"Start":"00:47.050 ","End":"00:50.190","Text":"Now, 3 of the 8 movies are horror movies,"},{"Start":"00:50.190 ","End":"00:52.700","Text":"and we\u0027re asked what\u0027s the probability that"},{"Start":"00:52.700 ","End":"00:57.160","Text":"the 3 horror movies will appear consecutively on the list?"},{"Start":"00:57.160 ","End":"01:01.490","Text":"The first thing that we need to understand in this section right here,"},{"Start":"01:01.490 ","End":"01:04.925","Text":"is that the size of the sample space has changed."},{"Start":"01:04.925 ","End":"01:11.825","Text":"We\u0027re not looking at the 5 friends choosing 1 out of the 8 movies."},{"Start":"01:11.825 ","End":"01:18.895","Text":"What we\u0027re doing here is we\u0027re looking at ordering 8 movies in a row,"},{"Start":"01:18.895 ","End":"01:24.605","Text":"so the size of the sample space would be 8 factorial."},{"Start":"01:24.605 ","End":"01:30.020","Text":"Now, the next thing that we have to understand is that we"},{"Start":"01:30.020 ","End":"01:36.245","Text":"have 3 horror movies that need to be together."},{"Start":"01:36.245 ","End":"01:38.195","Text":"If we have the 8 movies,"},{"Start":"01:38.195 ","End":"01:39.390","Text":"A, B,"},{"Start":"01:39.390 ","End":"01:41.025","Text":"C, D,"},{"Start":"01:41.025 ","End":"01:43.065","Text":"E, F, G,"},{"Start":"01:43.065 ","End":"01:47.010","Text":"and H, let\u0027s assume that A, B,"},{"Start":"01:47.010 ","End":"01:48.930","Text":"and C are the horror movie,"},{"Start":"01:48.930 ","End":"01:51.705","Text":"and these 3 movies need to be together."},{"Start":"01:51.705 ","End":"01:54.485","Text":"How many objects do we have here?"},{"Start":"01:54.485 ","End":"01:56.225","Text":"Well, we have 6 objects,"},{"Start":"01:56.225 ","End":"01:58.220","Text":"we have this object right here,"},{"Start":"01:58.220 ","End":"02:00.740","Text":"which contains the 3 movies,"},{"Start":"02:00.740 ","End":"02:06.200","Text":"and we have each other object which is a different movie right here,"},{"Start":"02:06.200 ","End":"02:08.322","Text":"so this would be 2, 3,"},{"Start":"02:08.322 ","End":"02:09.845","Text":"4, 5 and 6,"},{"Start":"02:09.845 ","End":"02:11.720","Text":"we have 6 objects."},{"Start":"02:11.720 ","End":"02:16.235","Text":"Now, we\u0027re looking at the probability then of A,"},{"Start":"02:16.235 ","End":"02:20.270","Text":"B, and C being together."},{"Start":"02:20.270 ","End":"02:23.870","Text":"Now, in this probability,"},{"Start":"02:23.870 ","End":"02:27.060","Text":"in the denominator, we have the size of the sample space,"},{"Start":"02:27.060 ","End":"02:29.360","Text":"now what about the numerator?"},{"Start":"02:29.360 ","End":"02:36.710","Text":"Well, how many ways can I order the 6 movies now or the 6 objects in a row?"},{"Start":"02:36.710 ","End":"02:39.920","Text":"Well, that\u0027s 6 factorial, but I\u0027m not done,"},{"Start":"02:39.920 ","End":"02:43.655","Text":"because I have to take into consideration"},{"Start":"02:43.655 ","End":"02:48.755","Text":"the internal ordering of this object right here of the 3 movies,"},{"Start":"02:48.755 ","End":"02:51.430","Text":"so that\u0027ll be 3 factorial."},{"Start":"02:51.430 ","End":"02:55.100","Text":"This end would be the probability of A, B,"},{"Start":"02:55.100 ","End":"03:01.290","Text":"C being together, and that equals to 0.1071."}],"ID":12850},{"Watched":false,"Name":"Exercise 7","Duration":"4m 30s","ChapterTopicVideoID":12372,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12372.jpeg","UploadDate":"2019-01-31T23:21:26.6730000","DurationForVideoObject":"PT4M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"In this question, there are 10 people in a group."},{"Start":"00:02.805 ","End":"00:05.925","Text":"Now, 2 different committees must be assembled from the group."},{"Start":"00:05.925 ","End":"00:09.000","Text":"1 with 4 people and the other was 3 people."},{"Start":"00:09.000 ","End":"00:12.975","Text":"Each person can be selected to only 1 committee."},{"Start":"00:12.975 ","End":"00:16.920","Text":"Now, we\u0027re asked to calculate the number of different ways of assembling"},{"Start":"00:16.920 ","End":"00:22.305","Text":"these committees when there are no different positions on the committees."},{"Start":"00:22.305 ","End":"00:26.250","Text":"First of all, we\u0027re dealing with pure combinatorics here,"},{"Start":"00:26.250 ","End":"00:28.350","Text":"we\u0027re not dealing with probabilities."},{"Start":"00:28.350 ","End":"00:31.620","Text":"What we need to do is we want to count the number of"},{"Start":"00:31.620 ","End":"00:36.480","Text":"different combinations of assembling these committees."},{"Start":"00:36.480 ","End":"00:38.250","Text":"Let\u0027s get started."},{"Start":"00:38.250 ","End":"00:40.225","Text":"Well, when thinking about this,"},{"Start":"00:40.225 ","End":"00:46.235","Text":"we have basically non-sequential sampling without returns. Now why is that?"},{"Start":"00:46.235 ","End":"00:49.520","Text":"Well, it\u0027s not sequential sampling because there aren\u0027t"},{"Start":"00:49.520 ","End":"00:53.665","Text":"any different positions on the committees, everybody\u0027s the same."},{"Start":"00:53.665 ","End":"00:58.670","Text":"It\u0027s without returns because each person can be"},{"Start":"00:58.670 ","End":"01:03.260","Text":"selected to only 1 committee. Let\u0027s get started."},{"Start":"01:03.260 ","End":"01:05.180","Text":"Let\u0027s create the first committee."},{"Start":"01:05.180 ","End":"01:09.140","Text":"We need to choose 4 people out of 10."},{"Start":"01:09.140 ","End":"01:12.020","Text":"We have here 10 over 4,"},{"Start":"01:12.020 ","End":"01:17.240","Text":"but now we have to choose the 3 people for the second committee."},{"Start":"01:17.240 ","End":"01:20.320","Text":"Now, we have to choose the 3 people from how many?"},{"Start":"01:20.320 ","End":"01:22.535","Text":"Well, from 6."},{"Start":"01:22.535 ","End":"01:26.265","Text":"We have to choose 3 people out of 6 people. Why is that?"},{"Start":"01:26.265 ","End":"01:31.235","Text":"Because 4 have already been taken to the first committee."},{"Start":"01:31.235 ","End":"01:36.080","Text":"This then would be the number of combinations and that equals to"},{"Start":"01:36.080 ","End":"01:42.490","Text":"4,200 ways of assembling the 2 committees."},{"Start":"01:42.490 ","End":"01:47.270","Text":"In this section, we\u0027re asked to count the number of ways we"},{"Start":"01:47.270 ","End":"01:51.440","Text":"can assemble the committees when each committee has a leader."},{"Start":"01:51.440 ","End":"01:55.835","Text":"Okay, so again, under the 10 original people,"},{"Start":"01:55.835 ","End":"01:59.555","Text":"we choose 4 for the first committee."},{"Start":"01:59.555 ","End":"02:04.580","Text":"Now, out of the 4 people in the first committee,"},{"Start":"02:04.580 ","End":"02:07.940","Text":"we need to choose 1 which would be a leader."},{"Start":"02:07.940 ","End":"02:16.280","Text":"Now, we have to choose 3 out of the 6 remaining people to be in the second committee."},{"Start":"02:16.280 ","End":"02:18.215","Text":"Out of the 3 people,"},{"Start":"02:18.215 ","End":"02:20.945","Text":"we need to choose a leader."},{"Start":"02:20.945 ","End":"02:26.420","Text":"This then is the number of combinations that we can do that."},{"Start":"02:26.420 ","End":"02:29.730","Text":"Now here, turnover 4 and 6 over 3,"},{"Start":"02:29.730 ","End":"02:34.140","Text":"we\u0027ve already calculated that in the first section."},{"Start":"02:34.140 ","End":"02:36.465","Text":"That equals to 4,200,"},{"Start":"02:36.465 ","End":"02:38.895","Text":"and 4 over 1,"},{"Start":"02:38.895 ","End":"02:41.985","Text":"well, that equals to 4 and 3 over 1,"},{"Start":"02:41.985 ","End":"02:44.805","Text":"well, that equals to 3."},{"Start":"02:44.805 ","End":"02:50.650","Text":"That equals to 50,400."},{"Start":"02:50.650 ","End":"02:55.490","Text":"In this section, we\u0027re asked to count the number of combinations of"},{"Start":"02:55.490 ","End":"03:00.290","Text":"assembling the committees when each committee has different positions."},{"Start":"03:00.290 ","End":"03:04.070","Text":"Well, when each committee has a different possession,"},{"Start":"03:04.070 ","End":"03:07.490","Text":"then we\u0027re dealing with sequential sampling without returns."},{"Start":"03:07.490 ","End":"03:12.140","Text":"Again, without returns because a person can be chosen to only 1 committee."},{"Start":"03:12.140 ","End":"03:17.840","Text":"Now, it\u0027s sequential sampling because each person has a different role,"},{"Start":"03:17.840 ","End":"03:20.525","Text":"a different position on that committee."},{"Start":"03:20.525 ","End":"03:24.710","Text":"Now whenever we have sequential sampling without returns, well,"},{"Start":"03:24.710 ","End":"03:31.410","Text":"in the first committee we have 10 people and we choose 4 times,"},{"Start":"03:31.410 ","End":"03:37.665","Text":"so that\u0027ll be 10 by 4 times out of the 6 remaining people, we choose 3."},{"Start":"03:37.665 ","End":"03:39.635","Text":"We have 6 by 3."},{"Start":"03:39.635 ","End":"03:48.170","Text":"Now that equals to 6,004 or 604,800."},{"Start":"03:48.170 ","End":"03:51.715","Text":"Now, we could also calculate it like this."},{"Start":"03:51.715 ","End":"03:58.875","Text":"We can say that we chose 4 out of the 10 people for the first committee,"},{"Start":"03:58.875 ","End":"04:02.390","Text":"but because each committee has a different position,"},{"Start":"04:02.390 ","End":"04:04.730","Text":"then we have 4 factorial,"},{"Start":"04:04.730 ","End":"04:10.665","Text":"we have to multiply that by 4 factorial times 6 over 3,"},{"Start":"04:10.665 ","End":"04:14.780","Text":"choosing 3 people for the second committee out of 6."},{"Start":"04:14.780 ","End":"04:19.250","Text":"But then for each person in the second committee,"},{"Start":"04:19.250 ","End":"04:21.350","Text":"we have a different role there."},{"Start":"04:21.350 ","End":"04:24.170","Text":"We have to multiply that by 3 factorial."},{"Start":"04:24.170 ","End":"04:28.860","Text":"That also equals to 604,800."}],"ID":12851},{"Watched":false,"Name":"Exercise 8","Duration":"5m 19s","ChapterTopicVideoID":12373,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12373.jpeg","UploadDate":"2019-01-31T23:21:39.9830000","DurationForVideoObject":"PT5M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this question, we have 4 men and 3 women that sit in 1 row containing"},{"Start":"00:05.130 ","End":"00:08.340","Text":"10 chairs and we\u0027re asked how many ways can they be"},{"Start":"00:08.340 ","End":"00:12.210","Text":"seated if anyone can sit anywhere they want."},{"Start":"00:12.210 ","End":"00:17.475","Text":"First of all, we have 4 men and we have 3 women."},{"Start":"00:17.475 ","End":"00:21.915","Text":"Now, that means that they sit in 7 chairs."},{"Start":"00:21.915 ","End":"00:27.120","Text":"So we have 3 chairs that remain empty."},{"Start":"00:27.120 ","End":"00:29.085","Text":"Now, these 3 chairs are identical."},{"Start":"00:29.085 ","End":"00:31.620","Text":"We can\u0027t differentiate between the 3 chairs here."},{"Start":"00:31.620 ","End":"00:34.770","Text":"We can differentiate between the men and the women."},{"Start":"00:34.770 ","End":"00:38.780","Text":"Each man is a different person."},{"Start":"00:38.780 ","End":"00:40.970","Text":"Each woman that\u0027s a different person as well,"},{"Start":"00:40.970 ","End":"00:43.565","Text":"but the 3 chairs are identical."},{"Start":"00:43.565 ","End":"00:47.480","Text":"So how many objects in total do I have here?"},{"Start":"00:47.480 ","End":"00:51.518","Text":"Well, I have 10 objects,"},{"Start":"00:51.518 ","End":"00:56.510","Text":"and how many ways can I put 10 objects in a row?"},{"Start":"00:56.510 ","End":"01:00.080","Text":"Well, that\u0027s 10 factorial."},{"Start":"01:00.080 ","End":"01:04.790","Text":"The thing is that the 3 chairs that are left empty they\u0027re identical."},{"Start":"01:04.790 ","End":"01:06.845","Text":"I don\u0027t know how to differentiate between them."},{"Start":"01:06.845 ","End":"01:11.735","Text":"I had to divide 10 factorial by 3 factorial."},{"Start":"01:11.735 ","End":"01:16.910","Text":"That\u0027s the internal ordering of the identical chairs."},{"Start":"01:16.910 ","End":"01:22.630","Text":"Now, that equals to 604,800."},{"Start":"01:22.630 ","End":"01:27.665","Text":"That\u0027s the total number of combinations of sitting"},{"Start":"01:27.665 ","End":"01:33.770","Text":"4 men and 3 women in a row that contains 10 chairs."},{"Start":"01:33.930 ","End":"01:40.030","Text":"In this section we\u0027re asked how many ways can I sit the 4 men and 3 women in"},{"Start":"01:40.030 ","End":"01:42.550","Text":"a row of 10 seats where all the men need to be"},{"Start":"01:42.550 ","End":"01:46.045","Text":"seated together and all the women need to be seated together."},{"Start":"01:46.045 ","End":"01:49.285","Text":"I have here again, 4 men,"},{"Start":"01:49.285 ","End":"01:50.920","Text":"I have 3 women,"},{"Start":"01:50.920 ","End":"01:54.055","Text":"and I have 3 remaining chairs."},{"Start":"01:54.055 ","End":"01:58.795","Text":"Now, just a reminder the chairs are identical I can\u0027t differentiate between them."},{"Start":"01:58.795 ","End":"02:02.810","Text":"I want the 4 men to sit together,"},{"Start":"02:02.810 ","End":"02:04.070","Text":"so I form a group."},{"Start":"02:04.070 ","End":"02:08.580","Text":"The 4 men become 1 group and the 3 women,"},{"Start":"02:08.580 ","End":"02:12.365","Text":"they become 1 group as well because they need to be seated together."},{"Start":"02:12.365 ","End":"02:16.460","Text":"Now the 3 identical chairs remain 3 different groups."},{"Start":"02:16.460 ","End":"02:19.790","Text":"So how many groups do I have in total?"},{"Start":"02:19.790 ","End":"02:21.035","Text":"How many objects here."},{"Start":"02:21.035 ","End":"02:25.070","Text":"I have here 5 objects."},{"Start":"02:25.070 ","End":"02:29.245","Text":"Now in how many ways can I put 5 objects in a row?"},{"Start":"02:29.245 ","End":"02:34.850","Text":"Well, that\u0027s 5 factorial now because 3 of the objects are identical,"},{"Start":"02:34.850 ","End":"02:39.302","Text":"we have to divide 5 factorial by 3 factorial."},{"Start":"02:39.302 ","End":"02:41.840","Text":"But I\u0027m not done yet because I have to take into"},{"Start":"02:41.840 ","End":"02:46.850","Text":"consideration the internal ordering of the 4 men."},{"Start":"02:46.850 ","End":"02:53.180","Text":"I have to multiply this by 4 factorial and I have to do the same thing with the women."},{"Start":"02:53.180 ","End":"02:57.110","Text":"I have to take into consideration the internal ordering of the women."},{"Start":"02:57.110 ","End":"03:00.275","Text":"So I have to multiply this by 3 factorial."},{"Start":"03:00.275 ","End":"03:04.640","Text":"Now this comes out to 2,880,"},{"Start":"03:04.640 ","End":"03:11.405","Text":"so that\u0027s the number of different ways I can seat all the men and all the women"},{"Start":"03:11.405 ","End":"03:14.810","Text":"in a row of 10 seats where all the men"},{"Start":"03:14.810 ","End":"03:19.295","Text":"need to be seated together and all the women need to be seated together."},{"Start":"03:19.295 ","End":"03:21.470","Text":"In this section, we\u0027re asked"},{"Start":"03:21.470 ","End":"03:25.260","Text":"how many sitting combinations do I have when 2 men are seated"},{"Start":"03:25.260 ","End":"03:30.617","Text":"on 1 end of the row and the other 2 men are seated on the other end of the row?"},{"Start":"03:30.617 ","End":"03:33.840","Text":"Again, we have 4 men,"},{"Start":"03:33.840 ","End":"03:35.520","Text":"we have 3 women,"},{"Start":"03:35.520 ","End":"03:42.585","Text":"and we have the 3 free chairs that are identical."},{"Start":"03:42.585 ","End":"03:46.695","Text":"Now, we said the 2 men,"},{"Start":"03:46.695 ","End":"03:51.615","Text":"2 of the 4 men are seated on 1 end of the row"},{"Start":"03:51.615 ","End":"03:57.455","Text":"and the other 2 men are seated on the other end of the row."},{"Start":"03:57.455 ","End":"03:59.785","Text":"So we have 2 here,"},{"Start":"03:59.785 ","End":"04:02.480","Text":"we have 2 here and in the middle,"},{"Start":"04:02.480 ","End":"04:06.995","Text":"we have the 3 women plus the 3 chairs."},{"Start":"04:06.995 ","End":"04:09.650","Text":"So what do we have here?"},{"Start":"04:09.650 ","End":"04:15.965","Text":"How can we calculate all the possible combinations of this thing right here?"},{"Start":"04:15.965 ","End":"04:18.485","Text":"Well, let\u0027s start with the men."},{"Start":"04:18.485 ","End":"04:24.085","Text":"First of all, we have 4 men sitting in the 4 seats,"},{"Start":"04:24.085 ","End":"04:26.045","Text":"there are 4 specific seats."},{"Start":"04:26.045 ","End":"04:29.510","Text":"So in how many ways can I seat the 4 men?"},{"Start":"04:29.510 ","End":"04:31.745","Text":"Well, that\u0027s 4 factorial."},{"Start":"04:31.745 ","End":"04:34.400","Text":"Now, what about the rest?"},{"Start":"04:34.400 ","End":"04:37.925","Text":"Well, I have 3 women and the 3 chairs here."},{"Start":"04:37.925 ","End":"04:40.280","Text":"So how many objects do I have?"},{"Start":"04:40.280 ","End":"04:42.305","Text":"I have 6 objects."},{"Start":"04:42.305 ","End":"04:46.640","Text":"I need to order the 6 objects in a row,"},{"Start":"04:46.640 ","End":"04:51.950","Text":"so that\u0027ll be 6 factorial and now I have to divide"},{"Start":"04:51.950 ","End":"04:58.340","Text":"the 6 factorial by 3 factorial because the 3 chairs are identical here."},{"Start":"04:58.340 ","End":"05:00.320","Text":"So all in all,"},{"Start":"05:00.320 ","End":"05:04.680","Text":"that equals to 2,880"},{"Start":"05:04.680 ","End":"05:11.333","Text":"possible combinations of sitting the 4 men;"},{"Start":"05:11.333 ","End":"05:14.730","Text":"2 in the beginning of the row, 2 in the end of the row and the 3 women and"},{"Start":"05:14.730 ","End":"05:19.210","Text":"the 3 free chairs in the middle."}],"ID":12852},{"Watched":false,"Name":"Exercise 9","Duration":"2m 1s","ChapterTopicVideoID":12374,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12374.jpeg","UploadDate":"2019-01-31T23:21:45.6800000","DurationForVideoObject":"PT2M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"In this question, there are 10 numbers from 1-10 in a lottery."},{"Start":"00:03.675 ","End":"00:06.360","Text":"Now, 5 numbers are chosen randomly."},{"Start":"00:06.360 ","End":"00:09.360","Text":"We\u0027re asked, what\u0027s the probability that the number"},{"Start":"00:09.360 ","End":"00:13.140","Text":"7 is the second largest of the numbers selected."},{"Start":"00:13.140 ","End":"00:17.745","Text":"The first thing that we want to do when we\u0027re asked about probabilities,"},{"Start":"00:17.745 ","End":"00:21.434","Text":"is to calculate the size of the sample space."},{"Start":"00:21.434 ","End":"00:24.420","Text":"Now, we have 10 numbers,"},{"Start":"00:24.420 ","End":"00:27.690","Text":"and we have to choose 5 of the numbers."},{"Start":"00:27.690 ","End":"00:30.765","Text":"The size is 10 over 5."},{"Start":"00:30.765 ","End":"00:42.120","Text":"Now we\u0027re asked for the probability that the number 7 is second largest."},{"Start":"00:44.300 ","End":"00:46.785","Text":"If that\u0027s the case, well,"},{"Start":"00:46.785 ","End":"00:48.915","Text":"here are the numbers that I have."},{"Start":"00:48.915 ","End":"00:52.320","Text":"From 1-10, I have to choose 5 of the numbers."},{"Start":"00:52.320 ","End":"00:55.115","Text":"Now if we\u0027re looking for this probability, well,"},{"Start":"00:55.115 ","End":"00:58.920","Text":"in the denominator, we have 10 over 5."},{"Start":"00:58.920 ","End":"01:00.770","Text":"That\u0027s the size of the sample space."},{"Start":"01:00.770 ","End":"01:02.375","Text":"Now what about the numerator?"},{"Start":"01:02.375 ","End":"01:06.980","Text":"We\u0027re told that the number 7 is the second largest number,"},{"Start":"01:06.980 ","End":"01:08.690","Text":"so out of the 5 numbers,"},{"Start":"01:08.690 ","End":"01:11.620","Text":"we have the number 7."},{"Start":"01:11.620 ","End":"01:16.830","Text":"We choose 1 number out of the 7."},{"Start":"01:16.830 ","End":"01:18.060","Text":"Out of this number right here,"},{"Start":"01:18.060 ","End":"01:19.695","Text":"so it\u0027s 1 over 1."},{"Start":"01:19.695 ","End":"01:24.930","Text":"Now, this number 7 has to be the second number,"},{"Start":"01:24.930 ","End":"01:27.330","Text":"so out of these 3 numbers,"},{"Start":"01:27.330 ","End":"01:31.065","Text":"we have to choose 1 number."},{"Start":"01:31.065 ","End":"01:35.900","Text":"Out of these numbers which are less than 7, well,"},{"Start":"01:35.900 ","End":"01:40.175","Text":"we have to choose 3 numbers. What do we have here?"},{"Start":"01:40.175 ","End":"01:41.975","Text":"Out of these 3 numbers,"},{"Start":"01:41.975 ","End":"01:44.125","Text":"to 8, 9, 10,"},{"Start":"01:44.125 ","End":"01:48.585","Text":"we have to choose 1 number, times,"},{"Start":"01:48.585 ","End":"01:50.970","Text":"now out of these 6 numbers,"},{"Start":"01:50.970 ","End":"01:53.685","Text":"we have to choose 3 numbers."},{"Start":"01:53.685 ","End":"01:55.470","Text":"That\u0027s the probability,"},{"Start":"01:55.470 ","End":"01:59.200","Text":"and that comes out to 0.238."}],"ID":12853},{"Watched":false,"Name":"Exercise 10 - Parts a-d","Duration":"5m 18s","ChapterTopicVideoID":12375,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12375.jpeg","UploadDate":"2019-01-31T23:21:59.7530000","DurationForVideoObject":"PT5M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this question, we have 6 people that get on a bus that has 10 stops."},{"Start":"00:03.870 ","End":"00:09.215","Text":"Now, each person independently and randomly selects at which stop to get off."},{"Start":"00:09.215 ","End":"00:14.730","Text":"We\u0027re asked, what\u0027s the probability that each person gets off at a different station?"},{"Start":"00:14.730 ","End":"00:19.725","Text":"The first thing that we need to do is calculate the size of the sample space,"},{"Start":"00:19.725 ","End":"00:24.855","Text":"and that\u0027ll be equal to 10 to the power of 6. Now why is that?"},{"Start":"00:24.855 ","End":"00:27.885","Text":"Well, we have 10 stops and we have 6 people."},{"Start":"00:27.885 ","End":"00:32.220","Text":"The first person, has 10 options where to get off,"},{"Start":"00:32.220 ","End":"00:35.995","Text":"and same for the second person and the third person and so on and so forth."},{"Start":"00:35.995 ","End":"00:38.350","Text":"That\u0027ll be 10 times 10 times 10,"},{"Start":"00:38.350 ","End":"00:41.320","Text":"6 times, so that\u0027s 10 to the power of 6."},{"Start":"00:41.320 ","End":"00:43.865","Text":"We\u0027re asked now, in a,"},{"Start":"00:43.865 ","End":"00:49.505","Text":"what\u0027s the probability that each person gets off at a different station?"},{"Start":"00:49.505 ","End":"00:52.235","Text":"Now, what does that equal to?"},{"Start":"00:52.235 ","End":"00:57.380","Text":"In the denominator, we have the size of the sample space,"},{"Start":"00:57.380 ","End":"00:58.970","Text":"that\u0027ll be 10 to the power 6."},{"Start":"00:58.970 ","End":"01:00.515","Text":"What about the numerator?"},{"Start":"01:00.515 ","End":"01:03.095","Text":"If each person gets off at a different station,"},{"Start":"01:03.095 ","End":"01:06.925","Text":"the first-person get has 10 options in which to get off."},{"Start":"01:06.925 ","End":"01:10.395","Text":"The second person has 9 options,"},{"Start":"01:10.395 ","End":"01:12.450","Text":"the third has 8 options,"},{"Start":"01:12.450 ","End":"01:15.509","Text":"the fourth has 7 options."},{"Start":"01:15.509 ","End":"01:18.230","Text":"The fifth has 6 options,"},{"Start":"01:18.230 ","End":"01:21.080","Text":"and the sixth has 5 options."},{"Start":"01:21.080 ","End":"01:28.895","Text":"Now, that equals to 10 by 6 divided by 10 to the power of 6,"},{"Start":"01:28.895 ","End":"01:34.775","Text":"and that comes out to 0.1512."},{"Start":"01:34.775 ","End":"01:36.155","Text":"In this section, we\u0027re asked,"},{"Start":"01:36.155 ","End":"01:43.610","Text":"what\u0027s the probability that exactly 3 people get off at the fifth stop?"},{"Start":"01:43.610 ","End":"01:47.660","Text":"In the denominator, we have the size of the sample space,"},{"Start":"01:47.660 ","End":"01:50.195","Text":"which is 10 to the power of 6."},{"Start":"01:50.195 ","End":"01:54.300","Text":"Now, we have to have 3 people,"},{"Start":"01:54.300 ","End":"01:58.455","Text":"and only 3 people that get off at the fifth stop."},{"Start":"01:58.455 ","End":"02:02.500","Text":"We have to choose 3 people out of the 6."},{"Start":"02:02.500 ","End":"02:06.784","Text":"This is like non-sequential sampling without returns."},{"Start":"02:06.784 ","End":"02:09.830","Text":"They have to get up at the fifth stop,"},{"Start":"02:09.830 ","End":"02:11.510","Text":"they don\u0027t have any options."},{"Start":"02:11.510 ","End":"02:15.525","Text":"They get off only at this specific stop."},{"Start":"02:15.525 ","End":"02:17.880","Text":"Now, what about the other 3 people?"},{"Start":"02:17.880 ","End":"02:21.165","Text":"They do not get off at the fifth stop."},{"Start":"02:21.165 ","End":"02:23.985","Text":"How many stops do they have to choose from?"},{"Start":"02:23.985 ","End":"02:29.145","Text":"9 stops and each person, has 9 options."},{"Start":"02:29.145 ","End":"02:32.140","Text":"That\u0027ll be 9 to the power of 3."},{"Start":"02:33.250 ","End":"02:37.880","Text":"That\u0027ll be the probability of exactly 3 people getting"},{"Start":"02:37.880 ","End":"02:45.160","Text":"off at the fifth stop and that equals to 0.0146."},{"Start":"02:45.160 ","End":"02:48.080","Text":"In this section we\u0027re asked what\u0027s the probability that"},{"Start":"02:48.080 ","End":"02:52.980","Text":"Sharon is the only passenger to get off at the second stop."},{"Start":"02:53.180 ","End":"02:58.695","Text":"Again, we have here in the denominator,"},{"Start":"02:58.695 ","End":"03:00.080","Text":"10 to the power 6,"},{"Start":"03:00.080 ","End":"03:01.460","Text":"the size of the sample space."},{"Start":"03:01.460 ","End":"03:02.960","Text":"Now if only 1 person,"},{"Start":"03:02.960 ","End":"03:05.135","Text":"Sharon gets off at the second stop,"},{"Start":"03:05.135 ","End":"03:07.570","Text":"then we have 1,"},{"Start":"03:07.570 ","End":"03:09.540","Text":"that\u0027s her, times,"},{"Start":"03:09.540 ","End":"03:18.480","Text":"now we have 5 other people that have 9 options or 9 stops in which to get off."},{"Start":"03:18.480 ","End":"03:28.210","Text":"That means that we have 9 to the power of 5 and that equals to 0.059."},{"Start":"03:28.220 ","End":"03:35.660","Text":"That\u0027s the probability that Sharon is the only person to get off at the second stop."},{"Start":"03:35.660 ","End":"03:38.945","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"03:38.945 ","End":"03:44.075","Text":"all the passengers get off at either the fifth or the sixth station,"},{"Start":"03:44.075 ","End":"03:49.440","Text":"with at least 1 person getting off at each of these stops?"},{"Start":"03:49.940 ","End":"03:56.540","Text":"Here, again in the denominator we have 10 to the power of 6,"},{"Start":"03:56.540 ","End":"03:58.325","Text":"the size of the sample space."},{"Start":"03:58.325 ","End":"04:03.650","Text":"If all the people get off either at the fifth or sixth station,"},{"Start":"04:03.650 ","End":"04:07.640","Text":"there are only 2 stops that they have to choose from."},{"Start":"04:07.640 ","End":"04:11.105","Text":"We have 2 to the power of 6."},{"Start":"04:11.105 ","End":"04:17.515","Text":"We have 2 stops and 6 people and the 6 people choose either 1 or the other stop."},{"Start":"04:17.515 ","End":"04:21.425","Text":"Now, what about the second condition?"},{"Start":"04:21.425 ","End":"04:25.890","Text":"With at least 1 person getting off at each of these stops."},{"Start":"04:25.890 ","End":"04:29.790","Text":"Well, out of the 2 to the power of 6 options,"},{"Start":"04:29.790 ","End":"04:35.270","Text":"we have only 2 options where all the people get off on"},{"Start":"04:35.270 ","End":"04:40.665","Text":"the fifth station and all the people get off on the sixth stations,"},{"Start":"04:40.665 ","End":"04:45.590","Text":"so that we want to take away from this number of combinations."},{"Start":"04:45.590 ","End":"04:49.550","Text":"We want to take away 2 possible combinations."},{"Start":"04:49.550 ","End":"04:53.480","Text":"The first 1 is where all the people get off on the fifth station,"},{"Start":"04:53.480 ","End":"04:58.560","Text":"and the second 1 where all people get off in the sixth station."},{"Start":"04:58.720 ","End":"05:01.885","Text":"That\u0027s the probability that"},{"Start":"05:01.885 ","End":"05:05.575","Text":"all the people get off either in the fifth or the sixth stop,"},{"Start":"05:05.575 ","End":"05:09.995","Text":"and there\u0027s at least 1 person that gets off at each stop."},{"Start":"05:09.995 ","End":"05:17.650","Text":"Now, that equals to 62 divided by 10 to the power of 6."}],"ID":12854},{"Watched":false,"Name":"Exercise 11 - Parts a-c","Duration":"6m 40s","ChapterTopicVideoID":12376,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12376.jpeg","UploadDate":"2019-01-31T23:22:17.4970000","DurationForVideoObject":"PT6M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.725","Text":"In this question, we have a train."},{"Start":"00:01.725 ","End":"00:05.570","Text":"Now. Here we see a cabin within the train where"},{"Start":"00:05.570 ","End":"00:09.959","Text":"4 seats are facing forward and seats are facing backwards."},{"Start":"00:09.959 ","End":"00:15.030","Text":"Now we have 4 couples that sit randomly in the seats and we\u0027re asked,"},{"Start":"00:15.030 ","End":"00:20.535","Text":"in how many different ways can the 4 couples or 8 people sit in seats?"},{"Start":"00:20.535 ","End":"00:24.750","Text":"This is basically the size of the sample space."},{"Start":"00:24.750 ","End":"00:30.345","Text":"How many different combinations can 8 people sit in 8 seats?"},{"Start":"00:30.345 ","End":"00:31.710","Text":"We have the first seat,"},{"Start":"00:31.710 ","End":"00:33.540","Text":"the second, the third, and the fourth,"},{"Start":"00:33.540 ","End":"00:38.040","Text":"seats 5,6,7, and 8 seats."},{"Start":"00:38.040 ","End":"00:42.285","Text":"We know that this is 8 factorial."},{"Start":"00:42.285 ","End":"00:46.580","Text":"The first person has 8 seats to choose from."},{"Start":"00:46.580 ","End":"00:49.505","Text":"The second person has 7 seats to choose from,"},{"Start":"00:49.505 ","End":"00:51.155","Text":"and so on and so forth."},{"Start":"00:51.155 ","End":"00:59.465","Text":"We have 8 factorial ways or combinations of seating 8 people into 8 seats."},{"Start":"00:59.465 ","End":"01:05.045","Text":"Now that equals to 4,320."},{"Start":"01:05.045 ","End":"01:08.300","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"01:08.300 ","End":"01:13.280","Text":"the Smiths would sit beside each other facing forward."},{"Start":"01:13.280 ","End":"01:16.265","Text":"This would be the cabin with the 8 seats."},{"Start":"01:16.265 ","End":"01:20.915","Text":"We have 4 seats facing forward and 4 seats facing backwards."},{"Start":"01:20.915 ","End":"01:22.355","Text":"Just for argument\u0027s sake,"},{"Start":"01:22.355 ","End":"01:26.330","Text":"let\u0027s say that this direction right here is forward,"},{"Start":"01:26.330 ","End":"01:32.585","Text":"all the seats face forward and all these seats face backwards."},{"Start":"01:32.585 ","End":"01:41.955","Text":"I know now that the Smith they sit together in 2 of the 4 seats here."},{"Start":"01:41.955 ","End":"01:45.420","Text":"What\u0027s the probability that the Smiths seat"},{"Start":"01:45.420 ","End":"01:49.440","Text":"beside each other facing forward. Let\u0027s get started."},{"Start":"01:49.440 ","End":"01:51.825","Text":"Well, first of all,"},{"Start":"01:51.825 ","End":"01:56.945","Text":"we know in the denominator that we have the size of the sample space,"},{"Start":"01:56.945 ","End":"01:58.925","Text":"which is 8 factorial."},{"Start":"01:58.925 ","End":"02:02.625","Text":"Now, let\u0027s start calculating the numerator."},{"Start":"02:02.625 ","End":"02:06.950","Text":"We have 2 people that are the Smith and we want to choose both of them,"},{"Start":"02:06.950 ","End":"02:08.495","Text":"so out of the 2 Smiths,"},{"Start":"02:08.495 ","End":"02:10.295","Text":"I want to choose both of them."},{"Start":"02:10.295 ","End":"02:16.235","Text":"Now I have 2 other people that I have to choose from,"},{"Start":"02:16.235 ","End":"02:18.965","Text":"from the remaining 6 people."},{"Start":"02:18.965 ","End":"02:20.970","Text":"Out of the remaining 6 people,"},{"Start":"02:20.970 ","End":"02:23.040","Text":"I need to choose 2 people."},{"Start":"02:23.040 ","End":"02:29.650","Text":"Now I have 4 people that are sitting facing forward."},{"Start":"02:29.900 ","End":"02:32.720","Text":"Now, what do I need to do?"},{"Start":"02:32.720 ","End":"02:37.190","Text":"Well, we\u0027re told that the Smith sit beside each other."},{"Start":"02:37.190 ","End":"02:40.900","Text":"That means I have the male Smith and the female"},{"Start":"02:40.900 ","End":"02:44.890","Text":"Smith and I have 2 other people that seat forward."},{"Start":"02:44.890 ","End":"02:49.680","Text":"Now the male Smith and the female Smith they need to be together."},{"Start":"02:49.680 ","End":"02:51.520","Text":"How many objects do I have here?"},{"Start":"02:51.520 ","End":"02:54.385","Text":"Well I have 1,2 and 3 objects."},{"Start":"02:54.385 ","End":"02:58.645","Text":"In how many ways can I order the 3 objects?"},{"Start":"02:58.645 ","End":"03:00.755","Text":"3 factorial."},{"Start":"03:00.755 ","End":"03:03.720","Text":"The 4 people right here,"},{"Start":"03:03.720 ","End":"03:08.185","Text":"facing forward that I\u0027ve already chosen where 2 of them are the Smith."},{"Start":"03:08.185 ","End":"03:09.430","Text":"I have to choose,"},{"Start":"03:09.430 ","End":"03:12.725","Text":"I have to multiply that by 3 factorial."},{"Start":"03:12.725 ","End":"03:15.115","Text":"Now, apart from that,"},{"Start":"03:15.115 ","End":"03:17.965","Text":"I have to multiply that by 2 factorial,"},{"Start":"03:17.965 ","End":"03:22.615","Text":"which would be the internal ordering of the Smith."},{"Start":"03:22.615 ","End":"03:24.660","Text":"Now, apart from that,"},{"Start":"03:24.660 ","End":"03:29.810","Text":"I still have 4 other people that are facing backwards."},{"Start":"03:29.810 ","End":"03:34.520","Text":"In how many ways can 4 people sit in 4 seats?"},{"Start":"03:34.520 ","End":"03:37.195","Text":"That\u0027s 4 factorial."},{"Start":"03:37.195 ","End":"03:41.330","Text":"This then would be the probability that the Smiths sit"},{"Start":"03:41.330 ","End":"03:46.770","Text":"beside each other facing forward and that equals to 0.1071."},{"Start":"03:47.140 ","End":"03:50.360","Text":"Let\u0027s just review what we\u0027ve done here."},{"Start":"03:50.360 ","End":"03:54.860","Text":"First of all, we\u0027ve chosen the 2 people that are the Smith."},{"Start":"03:54.860 ","End":"03:58.560","Text":"There\u0027ll be 2 out of the 2 and they sit here."},{"Start":"03:58.560 ","End":"04:01.760","Text":"Now, out of the 6 remaining people,"},{"Start":"04:01.760 ","End":"04:04.295","Text":"I still have to choose 2 of them."},{"Start":"04:04.295 ","End":"04:07.235","Text":"That\u0027s 6 over 2 and that\u0027s right here."},{"Start":"04:07.235 ","End":"04:11.120","Text":"I filled up the 4 seats that face forward."},{"Start":"04:11.120 ","End":"04:15.170","Text":"Now. I need for the Smiths to sit together."},{"Start":"04:15.170 ","End":"04:17.105","Text":"How many objects do I have?"},{"Start":"04:17.105 ","End":"04:18.740","Text":"I have 3 objects."},{"Start":"04:18.740 ","End":"04:20.825","Text":"The Smith, the male and female Smith,"},{"Start":"04:20.825 ","End":"04:26.000","Text":"that would be 1 object and I have 2 other people and so I have 3 objects altogether."},{"Start":"04:26.000 ","End":"04:29.769","Text":"Now, in how many ways can I order the 3 objects?"},{"Start":"04:29.769 ","End":"04:31.245","Text":"That\u0027s 3 factorial."},{"Start":"04:31.245 ","End":"04:33.890","Text":"I multiply this by 3 factorial."},{"Start":"04:33.890 ","End":"04:37.790","Text":"I also multiply this by 2 factorial,"},{"Start":"04:37.790 ","End":"04:41.440","Text":"which is the internal ordering of the Smiths."},{"Start":"04:41.440 ","End":"04:48.800","Text":"Now all I need to do is for the remaining 4 people to seat them in the 4 seats."},{"Start":"04:48.800 ","End":"04:50.390","Text":"Now, how many ways can I do that?"},{"Start":"04:50.390 ","End":"04:55.460","Text":"Well, 4 factorial, so this then would be the probability,"},{"Start":"04:55.460 ","End":"04:59.795","Text":"again, which would be equal to 0.1071."},{"Start":"04:59.795 ","End":"05:01.610","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"05:01.610 ","End":"05:05.270","Text":"the probability that the Smiths sit beside each other?"},{"Start":"05:05.270 ","End":"05:09.440","Text":"Well, this is very similar to the last section we were asked,"},{"Start":"05:09.440 ","End":"05:14.038","Text":"what\u0027s the probability that the Smiths sit beside each other,"},{"Start":"05:14.038 ","End":"05:16.850","Text":"but there were also facing forward?"},{"Start":"05:16.850 ","End":"05:19.310","Text":"We can use that. Why is that?"},{"Start":"05:19.310 ","End":"05:23.390","Text":"Because the probability, or let\u0027s say this,"},{"Start":"05:23.390 ","End":"05:26.405","Text":"the event that the Smiths sit beside each other."},{"Start":"05:26.405 ","End":"05:31.864","Text":"Well, we can split that up into 2 complimentary events."},{"Start":"05:31.864 ","End":"05:36.080","Text":"1 is that the Smiths sit beside each other facing forward"},{"Start":"05:36.080 ","End":"05:42.005","Text":"and the other event is at the Smiths sit beside each other facing backwards."},{"Start":"05:42.005 ","End":"05:44.290","Text":"Let\u0027s just write that down."},{"Start":"05:44.290 ","End":"05:46.685","Text":"Again, what did I do here?"},{"Start":"05:46.685 ","End":"05:51.875","Text":"I took this event and I split that up into 2 complimentary events."},{"Start":"05:51.875 ","End":"05:54.875","Text":"1 that they sit together facing forward and the"},{"Start":"05:54.875 ","End":"05:59.255","Text":"other that they sit together facing backwards."},{"Start":"05:59.255 ","End":"06:04.430","Text":"But we\u0027ve calculated this probability in the last section that equals"},{"Start":"06:04.430 ","End":"06:14.780","Text":"0.1071 and the probability that the Smiths sit beside each other facing backwards,"},{"Start":"06:14.780 ","End":"06:20.420","Text":"that\u0027s the same as the probability of them sitting beside each other facing forward."},{"Start":"06:20.420 ","End":"06:25.050","Text":"That also equals to 0.1071."},{"Start":"06:25.880 ","End":"06:33.425","Text":"The total probability, or the probability that the Smiths sit beside each other,"},{"Start":"06:33.425 ","End":"06:38.010","Text":"that equals to 0.2142."}],"ID":12855},{"Watched":false,"Name":"Exercise 11 - Parts d-h","Duration":"7m 48s","ChapterTopicVideoID":12377,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12377.jpeg","UploadDate":"2019-01-31T23:22:40.9700000","DurationForVideoObject":"PT7M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"00:03.000 ","End":"00:06.825","Text":"that each of the Smith has a window seat?"},{"Start":"00:06.825 ","End":"00:12.960","Text":"We can have the male Smith here in the female Smith here,"},{"Start":"00:12.960 ","End":"00:15.225","Text":"or the female Smith here,"},{"Start":"00:15.225 ","End":"00:17.460","Text":"and male Smith here."},{"Start":"00:17.460 ","End":"00:21.495","Text":"We really don\u0027t care what happens to the other guys."},{"Start":"00:21.495 ","End":"00:24.550","Text":"Let\u0027s just get started here."},{"Start":"00:26.060 ","End":"00:29.790","Text":"We have here in the denominator again,"},{"Start":"00:29.790 ","End":"00:33.585","Text":"8 factorial, which is the size of the sample space."},{"Start":"00:33.585 ","End":"00:41.425","Text":"Now, we have here 2 people that are the Smith and we have here 2 seats."},{"Start":"00:41.425 ","End":"00:47.610","Text":"In how many ways can I order 2 people in the 2 seats."},{"Start":"00:47.610 ","End":"00:51.055","Text":"That\u0027s 2 factorial, which is 2."},{"Start":"00:51.055 ","End":"00:55.070","Text":"Either the male Smith is here in the female Smith is here,"},{"Start":"00:55.070 ","End":"00:59.135","Text":"or the other way round where the female Smith is here and the male Smith is here."},{"Start":"00:59.135 ","End":"01:00.575","Text":"That\u0027s 2 factorial."},{"Start":"01:00.575 ","End":"01:07.605","Text":"Now we have 6 other people that sit in the 6 different seats."},{"Start":"01:07.605 ","End":"01:15.915","Text":"We have 6 factorial ways of seeding the 6 people in the 6 remaining seats."},{"Start":"01:15.915 ","End":"01:23.915","Text":"That\u0027s the probability and that equals to 0.0357."},{"Start":"01:23.915 ","End":"01:27.890","Text":"In this section, we\u0027re asked what\u0027s the probability that each of"},{"Start":"01:27.890 ","End":"01:32.910","Text":"the Smiths seat where 1 faces forward and the other face is backwards."},{"Start":"01:32.910 ","End":"01:35.810","Text":"For example, if the male sits here,"},{"Start":"01:35.810 ","End":"01:41.300","Text":"the female can sit in any 1 of the 4 seats here, for example here."},{"Start":"01:41.300 ","End":"01:44.450","Text":"Or if the female sits here, for example,"},{"Start":"01:44.450 ","End":"01:49.405","Text":"well the male can sit in any 1 of the 4 seats here, let\u0027s say here."},{"Start":"01:49.405 ","End":"01:54.010","Text":"Let\u0027s just build the probability here."},{"Start":"01:54.590 ","End":"01:59.055","Text":"We have that. Now, what is that equal to?"},{"Start":"01:59.055 ","End":"02:00.800","Text":"In the denominator again,"},{"Start":"02:00.800 ","End":"02:02.660","Text":"we have the size of the sample space,"},{"Start":"02:02.660 ","End":"02:05.310","Text":"which is 8 factorial."},{"Start":"02:05.470 ","End":"02:08.915","Text":"Let\u0027s calculate the numerator."},{"Start":"02:08.915 ","End":"02:10.430","Text":"Let\u0027s take 1 of the Smith,"},{"Start":"02:10.430 ","End":"02:12.985","Text":"so I\u0027m choosing 1 out of the 2."},{"Start":"02:12.985 ","End":"02:15.180","Text":"There\u0027ll be 2 over 1."},{"Start":"02:15.180 ","End":"02:23.610","Text":"Then let\u0027s say I put him in 1 of the seats facing forward."},{"Start":"02:23.610 ","End":"02:26.270","Text":"We said that this was facing forward,"},{"Start":"02:26.270 ","End":"02:28.480","Text":"this was facing backwards."},{"Start":"02:28.480 ","End":"02:32.010","Text":"I have to choose 1 out of the 4 here,"},{"Start":"02:32.010 ","End":"02:34.020","Text":"so that\u0027ll be 4 over 1."},{"Start":"02:34.020 ","End":"02:40.380","Text":"Now the other Smith will have to choose 1 out of the 4 facing the other way,"},{"Start":"02:40.380 ","End":"02:42.465","Text":"so that\u0027ll be 4 over 1."},{"Start":"02:42.465 ","End":"02:48.650","Text":"Now, I still have 6 remaining people to be seated in 6 remaining seats."},{"Start":"02:48.650 ","End":"02:51.875","Text":"That has to be times 6 factorial,"},{"Start":"02:51.875 ","End":"02:56.440","Text":"that equals to 0.5714."},{"Start":"02:56.440 ","End":"02:59.010","Text":"Let\u0027s just review again."},{"Start":"02:59.010 ","End":"03:02.415","Text":"I took 1 of the Smith, right."},{"Start":"03:02.415 ","End":"03:10.055","Text":"I chose 1 out of the 2 people and I placed them in 1 of the 4 seats."},{"Start":"03:10.055 ","End":"03:12.515","Text":"For example, facing forward."},{"Start":"03:12.515 ","End":"03:17.900","Text":"The other Smith while I chose 1 of the 4 seats facing the other direction,"},{"Start":"03:17.900 ","End":"03:24.950","Text":"and then I place the 6 remaining people in the 6 remaining seats,"},{"Start":"03:24.950 ","End":"03:27.680","Text":"and I divided that by 8 factorial."},{"Start":"03:27.680 ","End":"03:30.920","Text":"This then would be the probability."},{"Start":"03:30.920 ","End":"03:32.765","Text":"In this section, we\u0027re asked,"},{"Start":"03:32.765 ","End":"03:39.279","Text":"what\u0027s the probability that the Smiths sit facing each other or sit opposite 1 another."},{"Start":"03:39.279 ","End":"03:44.405","Text":"For example, if the male sits here then the female has to sit here,"},{"Start":"03:44.405 ","End":"03:46.490","Text":"or if the female sits here,"},{"Start":"03:46.490 ","End":"03:49.460","Text":"then the male has to sit here."},{"Start":"03:49.460 ","End":"03:54.750","Text":"Let\u0027s get started. Again,"},{"Start":"03:54.750 ","End":"03:55.980","Text":"in the denominator,"},{"Start":"03:55.980 ","End":"03:58.050","Text":"we have 8 factorial."},{"Start":"03:58.050 ","End":"04:01.545","Text":"Now, let\u0027s choose 1 of the Smith."},{"Start":"04:01.545 ","End":"04:04.845","Text":"I choose out of the 2 Smiths, I choose 1."},{"Start":"04:04.845 ","End":"04:12.110","Text":"Now, I have to choose 1 of the 4 seats that that person will sit in,"},{"Start":"04:12.110 ","End":"04:14.575","Text":"so that\u0027ll be 4 over 1."},{"Start":"04:14.575 ","End":"04:19.520","Text":"Now, the other Smith doesn\u0027t really have that much options,"},{"Start":"04:19.520 ","End":"04:23.675","Text":"so we have to multiply this by 1. Why is that?"},{"Start":"04:23.675 ","End":"04:27.245","Text":"Because again, if 1 Smith sits here,"},{"Start":"04:27.245 ","End":"04:28.970","Text":"then there\u0027s really no options."},{"Start":"04:28.970 ","End":"04:33.420","Text":"The other Smith has to sit opposite the partner."},{"Start":"04:33.420 ","End":"04:39.180","Text":"Again, we still have the 6 remaining people in the 6 remaining seats,"},{"Start":"04:39.180 ","End":"04:41.410","Text":"so that\u0027ll be 6 factorial."},{"Start":"04:41.410 ","End":"04:45.960","Text":"Now, that comes out to 0.1429."},{"Start":"04:49.240 ","End":"04:53.420","Text":"In this section, we\u0027re asked what\u0027s the probability that all the men"},{"Start":"04:53.420 ","End":"04:57.565","Text":"sit facing forward and all the woman\u0027s sit facing backwards."},{"Start":"04:57.565 ","End":"05:03.180","Text":"Again, this would be the direction of travel."},{"Start":"05:03.180 ","End":"05:09.335","Text":"These are the seats facing forward and these are the seats facing backwards."},{"Start":"05:09.335 ","End":"05:16.780","Text":"We want all the men to be here and all the woman to be here."},{"Start":"05:17.360 ","End":"05:21.790","Text":"Let\u0027s calculate this probability."},{"Start":"05:22.720 ","End":"05:25.930","Text":"In the denominator, again,"},{"Start":"05:25.930 ","End":"05:28.330","Text":"we have 8 factorial."},{"Start":"05:28.330 ","End":"05:31.600","Text":"Now, how many seats do I have here?"},{"Start":"05:31.600 ","End":"05:34.015","Text":"I have 4 seats facing forward."},{"Start":"05:34.015 ","End":"05:37.180","Text":"How many men do I have here? Well, 4 men."},{"Start":"05:37.180 ","End":"05:40.015","Text":"So that has to be 4 factorial."},{"Start":"05:40.015 ","End":"05:45.725","Text":"The number of combinations of seating for men in the 4 seats."},{"Start":"05:45.725 ","End":"05:49.469","Text":"I have 4 remaining seats facing backwards,"},{"Start":"05:49.469 ","End":"05:54.015","Text":"and I have 4 women that need to be seated in these 4 seats."},{"Start":"05:54.015 ","End":"05:56.425","Text":"How many combinations do I have here?"},{"Start":"05:56.425 ","End":"05:59.210","Text":"Well, 4 factorial."},{"Start":"05:59.210 ","End":"06:07.095","Text":"This comes out to 0.01429."},{"Start":"06:07.095 ","End":"06:09.560","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"06:09.560 ","End":"06:13.295","Text":"the probability that each couple sit facing each other?"},{"Start":"06:13.295 ","End":"06:17.030","Text":"That means that the first couple would sit here,"},{"Start":"06:17.030 ","End":"06:19.175","Text":"the second couple here,"},{"Start":"06:19.175 ","End":"06:21.425","Text":"the third couple would sit here,"},{"Start":"06:21.425 ","End":"06:24.500","Text":"and the fourth couple would sit here."},{"Start":"06:24.500 ","End":"06:29.720","Text":"What\u0027s this probability? First of all,"},{"Start":"06:29.720 ","End":"06:36.170","Text":"in the denominator we have the size of the sample space, That\u0027s 8 factorial."},{"Start":"06:36.170 ","End":"06:37.940","Text":"Now here I\u0027m looking at couples."},{"Start":"06:37.940 ","End":"06:40.310","Text":"I have basically 4 objects."},{"Start":"06:40.310 ","End":"06:42.110","Text":"I don\u0027t have 8 people,"},{"Start":"06:42.110 ","End":"06:44.345","Text":"am not looking at each individual person,"},{"Start":"06:44.345 ","End":"06:47.215","Text":"but I am looking at each couple."},{"Start":"06:47.215 ","End":"06:49.880","Text":"I have 4 couples or 4 objects."},{"Start":"06:49.880 ","End":"06:55.130","Text":"Now I have here then 4 places."},{"Start":"06:55.130 ","End":"06:58.400","Text":"I have this place right here."},{"Start":"06:58.400 ","End":"07:01.565","Text":"I have this place right here,"},{"Start":"07:01.565 ","End":"07:03.410","Text":"I have this place right here,"},{"Start":"07:03.410 ","End":"07:04.970","Text":"and I have this place right here."},{"Start":"07:04.970 ","End":"07:10.970","Text":"How many ways can I order the 4 couples in each 1 of the places?"},{"Start":"07:10.970 ","End":"07:13.085","Text":"Well, that\u0027s 4 factorial."},{"Start":"07:13.085 ","End":"07:17.240","Text":"Now, I need to deal with the internal ordering."},{"Start":"07:17.240 ","End":"07:21.305","Text":"It can have the male and the female here, or vice versa,"},{"Start":"07:21.305 ","End":"07:25.130","Text":"that the female would be here and the male would be here enough for 1 couple,"},{"Start":"07:25.130 ","End":"07:27.320","Text":"there\u0027ll be 2 factorial."},{"Start":"07:27.320 ","End":"07:29.690","Text":"Now, I need to do this 4 times,"},{"Start":"07:29.690 ","End":"07:34.175","Text":"so I need to take the 2 factorial and raise it to the power of 4."},{"Start":"07:34.175 ","End":"07:42.645","Text":"Now, all of that comes out to 0.0095."},{"Start":"07:42.645 ","End":"07:48.600","Text":"That\u0027s the probability that each couple would sit facing each other."}],"ID":12856},{"Watched":false,"Name":"Exercise 12 - Parts a-c","Duration":"5m 54s","ChapterTopicVideoID":12378,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12378.jpeg","UploadDate":"2019-01-31T23:22:56.3770000","DurationForVideoObject":"PT5M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this question, we\u0027re told that a password is composed of 5 characters,"},{"Start":"00:03.840 ","End":"00:08.535","Text":"where each character can be either a number from 0-9 or a letter in the English alphabet."},{"Start":"00:08.535 ","End":"00:12.390","Text":"Now, each character may repeat itself more than once and we\u0027re asked,"},{"Start":"00:12.390 ","End":"00:15.645","Text":"how many different passwords can be created?"},{"Start":"00:15.645 ","End":"00:19.455","Text":"A password is composed of 5 characters."},{"Start":"00:19.455 ","End":"00:23.310","Text":"That\u0027s 1, 2, 3, 4 and 5."},{"Start":"00:23.310 ","End":"00:24.690","Text":"That would be my password."},{"Start":"00:24.690 ","End":"00:34.260","Text":"Now, each 1 of the characters here can be either a number from 0-9 or a letter from A-Z."},{"Start":"00:34.260 ","End":"00:39.615","Text":"Now from 0-9 I have 10 characters and from A to Z I have 26 characters,"},{"Start":"00:39.615 ","End":"00:40.865","Text":"so all in all,"},{"Start":"00:40.865 ","End":"00:44.605","Text":"I have 36 characters to choose from."},{"Start":"00:44.605 ","End":"00:49.115","Text":"We can use the multiplication rule here that says well,"},{"Start":"00:49.115 ","End":"00:55.669","Text":"the first character needs to be 1 out of 36 possible characters."},{"Start":"00:55.669 ","End":"00:58.765","Text":"We have 36 combinations here,"},{"Start":"00:58.765 ","End":"01:01.885","Text":"times 36 for the second character,"},{"Start":"01:01.885 ","End":"01:03.710","Text":"and for the third character,"},{"Start":"01:03.710 ","End":"01:05.210","Text":"and for the fourth character,"},{"Start":"01:05.210 ","End":"01:06.845","Text":"and for the fifth character,"},{"Start":"01:06.845 ","End":"01:10.130","Text":"and that equals to 36 to the power of 5."},{"Start":"01:10.130 ","End":"01:16.090","Text":"Now, we can also think of this as sequential sampling with returns."},{"Start":"01:16.090 ","End":"01:19.855","Text":"Now why is that? Well, k here that equals to 5."},{"Start":"01:19.855 ","End":"01:23.100","Text":"I\u0027m sampling 5 characters."},{"Start":"01:23.100 ","End":"01:28.749","Text":"Now, it\u0027s sequential in that there\u0027s meaning to the order here."},{"Start":"01:28.749 ","End":"01:30.570","Text":"1, 2, 3, 4,"},{"Start":"01:30.570 ","End":"01:33.820","Text":"5 is totally different from 5, 4, 3,"},{"Start":"01:33.820 ","End":"01:38.030","Text":"2,1, so it is sequential sampling."},{"Start":"01:38.030 ","End":"01:40.615","Text":"Why is it now with returns?"},{"Start":"01:40.615 ","End":"01:44.125","Text":"Well, we\u0027re told a character may repeat itself."},{"Start":"01:44.125 ","End":"01:46.270","Text":"That\u0027s with returns right here."},{"Start":"01:46.270 ","End":"01:50.980","Text":"Now, n here, that equals to 36,"},{"Start":"01:50.980 ","End":"01:55.780","Text":"because I\u0027m sampling from 36 different characters here."},{"Start":"01:55.780 ","End":"01:59.750","Text":"When I\u0027m dealing with sequential sampling with returns,"},{"Start":"01:59.750 ","End":"02:08.750","Text":"then I have n to the power of k and that equals in our case to 36 to the power of 5."},{"Start":"02:08.750 ","End":"02:14.615","Text":"That\u0027s the total number of different passwords that I can create."},{"Start":"02:14.615 ","End":"02:18.585","Text":"In this section, we\u0027re asked how many passwords are there when all the characters,"},{"Start":"02:18.585 ","End":"02:21.770","Text":"the letters, and the numbers are different?"},{"Start":"02:21.770 ","End":"02:27.735","Text":"Again, we have our 5 characters for the password."},{"Start":"02:27.735 ","End":"02:35.510","Text":"We have to choose our characters out of 10 numbers and 26 letters,"},{"Start":"02:35.510 ","End":"02:39.200","Text":"a total of 36 different characters."},{"Start":"02:39.200 ","End":"02:42.950","Text":"Now, we\u0027re asked about the number of"},{"Start":"02:42.950 ","End":"02:46.940","Text":"combinations where all these characters are different."},{"Start":"02:46.940 ","End":"02:53.600","Text":"This is just like thinking about sequential sampling without returns"},{"Start":"02:53.600 ","End":"03:00.560","Text":"and the number of combinations there would be n by k. In our case,"},{"Start":"03:00.560 ","End":"03:04.160","Text":"that would be 36 by 5."},{"Start":"03:04.160 ","End":"03:07.760","Text":"Now if you use the multiplication rule, well,"},{"Start":"03:07.760 ","End":"03:12.950","Text":"the first character where we can choose 1 out of 36 characters,"},{"Start":"03:12.950 ","End":"03:15.170","Text":"but the second character,"},{"Start":"03:15.170 ","End":"03:17.330","Text":"because all the characters need to be different,"},{"Start":"03:17.330 ","End":"03:21.005","Text":"well then we only have 35 characters to choose from,"},{"Start":"03:21.005 ","End":"03:24.275","Text":"the third would be 34 characters to choose from"},{"Start":"03:24.275 ","End":"03:34.665","Text":"33 and 32 characters to choose from and these 2 expressions equal 1 another."},{"Start":"03:34.665 ","End":"03:37.880","Text":"That\u0027s the number of different combinations or"},{"Start":"03:37.880 ","End":"03:43.564","Text":"different passwords that I can create when all the characters are different."},{"Start":"03:43.564 ","End":"03:49.135","Text":"In this section, we\u0027re asked how many passwords have at least 1 number and 1 letter?"},{"Start":"03:49.135 ","End":"03:51.630","Text":"Again, this would be the password,"},{"Start":"03:51.630 ","End":"03:54.980","Text":"we have 10 characters which are numbers,"},{"Start":"03:54.980 ","End":"03:57.740","Text":"and 26 characters which are letters,"},{"Start":"03:57.740 ","End":"03:59.870","Text":"and we have a total of 36 letters."},{"Start":"03:59.870 ","End":"04:01.760","Text":"Now, in order to do that,"},{"Start":"04:01.760 ","End":"04:04.070","Text":"the first thing that I want to do is to calculate"},{"Start":"04:04.070 ","End":"04:08.030","Text":"the total number of passwords that I can create."},{"Start":"04:08.030 ","End":"04:09.200","Text":"Well, we\u0027ve done that,"},{"Start":"04:09.200 ","End":"04:12.065","Text":"that\u0027s 36 to the power of 5."},{"Start":"04:12.065 ","End":"04:16.340","Text":"Now, if I have this great big list of passwords,"},{"Start":"04:16.340 ","End":"04:20.030","Text":"what I want to do is I want to take away from that"},{"Start":"04:20.030 ","End":"04:26.570","Text":"all the passwords that are all letters and all the passwords that are all numbers."},{"Start":"04:26.570 ","End":"04:27.665","Text":"Now, by doing that,"},{"Start":"04:27.665 ","End":"04:34.040","Text":"what remains would be all the passwords that have at least 1 number and 1 letter."},{"Start":"04:34.040 ","End":"04:38.405","Text":"Let\u0027s do that. Now,"},{"Start":"04:38.405 ","End":"04:39.980","Text":"what would that be equal to?"},{"Start":"04:39.980 ","End":"04:43.760","Text":"Well, again, that\u0027s 36 to the power of 5 minus."},{"Start":"04:43.760 ","End":"04:47.960","Text":"Now, how many passwords do I have that are all numbers?"},{"Start":"04:47.960 ","End":"04:50.510","Text":"Well, how many numbers do I have in total?"},{"Start":"04:50.510 ","End":"04:56.495","Text":"I have 10 numbers in total to choose from for each 1 of the 5 characters."},{"Start":"04:56.495 ","End":"04:58.475","Text":"The first character, I\u0027d have"},{"Start":"04:58.475 ","End":"05:01.970","Text":"10 possible combinations or 10 possible numbers to choose from,"},{"Start":"05:01.970 ","End":"05:06.260","Text":"the second again, 10 numbers to choose from and so on and so forth."},{"Start":"05:06.260 ","End":"05:13.815","Text":"The number of passwords that are all numbers would be 10 to the power of 5 plus now,"},{"Start":"05:13.815 ","End":"05:17.890","Text":"how many passwords do I have that are all letters?"},{"Start":"05:17.890 ","End":"05:21.925","Text":"Well again, I have 26 letters to choose from"},{"Start":"05:21.925 ","End":"05:26.475","Text":"and so the first character I\u0027d have 26 letters to choose from,"},{"Start":"05:26.475 ","End":"05:29.900","Text":"the second character I\u0027d have 26 letters to choose from,"},{"Start":"05:29.900 ","End":"05:31.235","Text":"and so on and so forth,"},{"Start":"05:31.235 ","End":"05:32.825","Text":"for the 5 characters."},{"Start":"05:32.825 ","End":"05:37.040","Text":"That\u0027ll be 26 to the power of 5 as well."},{"Start":"05:37.040 ","End":"05:43.130","Text":"Again, if I take the total number of passwords and I subtract from this great list,"},{"Start":"05:43.130 ","End":"05:45.485","Text":"all the passwords that are all numbers and"},{"Start":"05:45.485 ","End":"05:48.470","Text":"all the passwords that are all letters then what"},{"Start":"05:48.470 ","End":"05:55.230","Text":"remains are passwords that have at least 1 number and 1 letter."}],"ID":12857},{"Watched":false,"Name":"Exercise 13 - Parts a-c","Duration":"2m 59s","ChapterTopicVideoID":12379,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12379.jpeg","UploadDate":"2019-01-31T23:23:04.2670000","DurationForVideoObject":"PT2M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this question, we have a committee of"},{"Start":"00:02.070 ","End":"00:05.010","Text":"3 people that\u0027s chosen out of n different people."},{"Start":"00:05.010 ","End":"00:08.985","Text":"We\u0027re asked how many different ways can the committee be chosen?"},{"Start":"00:08.985 ","End":"00:13.559","Text":"We\u0027re asked to give our answers in terms of n. What are the constraints?"},{"Start":"00:13.559 ","End":"00:15.780","Text":"In a, we\u0027re told that there are"},{"Start":"00:15.780 ","End":"00:20.955","Text":"no defined roles in the committee and 3 different people are chosen."},{"Start":"00:20.955 ","End":"00:28.475","Text":"This is exactly non-sequential sampling with L returns."},{"Start":"00:28.475 ","End":"00:31.010","Text":"Now, how do I know that?"},{"Start":"00:31.010 ","End":"00:36.940","Text":"Well, it\u0027s non-sequential because we\u0027re told that there are no defined roles,"},{"Start":"00:36.940 ","End":"00:43.055","Text":"and it\u0027s without returns because we\u0027re looking for 3 different people."},{"Start":"00:43.055 ","End":"00:50.085","Text":"Now, we\u0027re sampling 3 people right out of n people."},{"Start":"00:50.085 ","End":"00:54.049","Text":"Whenever we have non-sequential sampling without returns,"},{"Start":"00:54.049 ","End":"00:56.300","Text":"then the number of total combinations,"},{"Start":"00:56.300 ","End":"01:01.730","Text":"that\u0027ll be n over k. In our case where k equals 3,"},{"Start":"01:01.730 ","End":"01:04.685","Text":"that\u0027ll be n over 3."},{"Start":"01:04.685 ","End":"01:13.390","Text":"We know that n over k is n factorial divided by k factorial times n minus k factorial."},{"Start":"01:13.390 ","End":"01:18.565","Text":"In our case, that\u0027ll be n factorial divided by,"},{"Start":"01:18.565 ","End":"01:19.990","Text":"k is equal to 3,"},{"Start":"01:19.990 ","End":"01:26.155","Text":"that\u0027ll be 3 factorial times n minus 3 factorial."},{"Start":"01:26.155 ","End":"01:30.100","Text":"In this section, we\u0027re told that there are 3 defined roles in"},{"Start":"01:30.100 ","End":"01:34.060","Text":"the committee and each person can serve and only 1 role."},{"Start":"01:34.060 ","End":"01:40.135","Text":"This now is sequential sampling without returns."},{"Start":"01:40.135 ","End":"01:42.030","Text":"Again, why is that?"},{"Start":"01:42.030 ","End":"01:46.530","Text":"Sequential sampling, because now we have 3 defined roles"},{"Start":"01:46.530 ","End":"01:52.870","Text":"and it\u0027s without returns because a person can serve in only 1 role."},{"Start":"01:52.870 ","End":"01:56.930","Text":"Whenever we have sequential sampling without returns then the number of"},{"Start":"01:56.930 ","End":"02:02.315","Text":"possible combinations would be n by k. In our case,"},{"Start":"02:02.315 ","End":"02:08.105","Text":"that\u0027ll be equal to n by 3 and that equals to"},{"Start":"02:08.105 ","End":"02:15.335","Text":"n times n minus 1 times n minus 2."},{"Start":"02:15.335 ","End":"02:19.010","Text":"In this section, we\u0027re told that there are 3 defined roles in"},{"Start":"02:19.010 ","End":"02:23.665","Text":"the committee and each person can serve in more than 1 rule."},{"Start":"02:23.665 ","End":"02:28.985","Text":"Now we have sequential sampling because we have 3 defined roles."},{"Start":"02:28.985 ","End":"02:36.390","Text":"But now it\u0027s with returns because a person can serve in more than 1 role."},{"Start":"02:37.370 ","End":"02:44.690","Text":"What\u0027s the number of combinations when we sample k out of n,"},{"Start":"02:44.690 ","End":"02:47.300","Text":"when we have sequential sampling with returns,"},{"Start":"02:47.300 ","End":"02:50.195","Text":"that\u0027s n to the power of k,"},{"Start":"02:50.195 ","End":"02:52.310","Text":"and in our case,"},{"Start":"02:52.310 ","End":"02:54.230","Text":"k equals to 3."},{"Start":"02:54.230 ","End":"03:00.340","Text":"That\u0027ll be equal to n to the power of 3."}],"ID":12858},{"Watched":false,"Name":"Exercise 14","Duration":"4m 58s","ChapterTopicVideoID":12380,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12380.jpeg","UploadDate":"2019-01-31T23:23:18.7200000","DurationForVideoObject":"PT4M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"Question, we have 2 people tossing a coin each."},{"Start":"00:03.420 ","End":"00:08.610","Text":"We\u0027re asked what\u0027s the probability that after tossing the coin n times each,"},{"Start":"00:08.610 ","End":"00:11.565","Text":"that both people have the same number of heads,"},{"Start":"00:11.565 ","End":"00:15.135","Text":"we\u0027re asked to express your answer in terms of n."},{"Start":"00:15.135 ","End":"00:18.210","Text":"The first thing that I want to do when dealing with"},{"Start":"00:18.210 ","End":"00:21.630","Text":"probabilities is to calculate the size of my sample space."},{"Start":"00:21.630 ","End":"00:23.925","Text":"For 1 person,"},{"Start":"00:23.925 ","End":"00:28.050","Text":"the size of the sample space would be the number of"},{"Start":"00:28.050 ","End":"00:33.420","Text":"combinations of tossing a coin n times."},{"Start":"00:33.420 ","End":"00:37.215","Text":"That will be 2 to the power of n. Why is that?"},{"Start":"00:37.215 ","End":"00:40.335","Text":"I have 2 possible results for tossing a coin,"},{"Start":"00:40.335 ","End":"00:42.075","Text":"either a heads or tails."},{"Start":"00:42.075 ","End":"00:45.015","Text":"If I toss the coin n times,"},{"Start":"00:45.015 ","End":"00:48.200","Text":"then I have 2 to the power of n combinations."},{"Start":"00:48.200 ","End":"00:50.450","Text":"For 2 people,"},{"Start":"00:50.450 ","End":"00:56.510","Text":"well then the size of the sample space would be 2 to the power of"},{"Start":"00:56.510 ","End":"01:02.150","Text":"n for the first person times 2 to the power of n for the second person,"},{"Start":"01:02.150 ","End":"01:06.340","Text":"so that\u0027ll be the size of my sample space."},{"Start":"01:06.340 ","End":"01:11.670","Text":"Let\u0027s get to the probability. This is it."},{"Start":"01:11.670 ","End":"01:13.800","Text":"In the denominator,"},{"Start":"01:13.800 ","End":"01:16.280","Text":"we have the size of the sample space,"},{"Start":"01:16.280 ","End":"01:22.295","Text":"which is 2 to the power of n times 2 to the power of n. What about the numerator?"},{"Start":"01:22.295 ","End":"01:24.355","Text":"What\u0027s the idea here?"},{"Start":"01:24.355 ","End":"01:28.130","Text":"We want the same number of heads for both people,"},{"Start":"01:28.130 ","End":"01:33.130","Text":"for the first-person and for the second person after n tosses."},{"Start":"01:33.130 ","End":"01:41.045","Text":"What does that mean? Let\u0027s say that that after n tosses the first-person has only 1 head."},{"Start":"01:41.045 ","End":"01:46.045","Text":"We want the second person to have 1 head as well."},{"Start":"01:46.045 ","End":"01:52.055","Text":"Let\u0027s count the number of combinations that in n tosses we have 1 head."},{"Start":"01:52.055 ","End":"01:56.545","Text":"That\u0027ll be n over 1 for the first-person."},{"Start":"01:56.545 ","End":"02:01.760","Text":"We have to multiply that by the number of combinations of getting 1 head,"},{"Start":"02:01.760 ","End":"02:04.710","Text":"right for the second person."},{"Start":"02:05.480 ","End":"02:11.360","Text":"We don\u0027t necessarily need to have only 1 head in n tosses."},{"Start":"02:11.360 ","End":"02:13.145","Text":"We can have 2 heads."},{"Start":"02:13.145 ","End":"02:15.905","Text":"That\u0027ll be n over 2,"},{"Start":"02:15.905 ","End":"02:20.900","Text":"when counting the number of combinations of getting 2 heads in n"},{"Start":"02:20.900 ","End":"02:26.945","Text":"tosses for the first-person times n over 2 for the second person,"},{"Start":"02:26.945 ","End":"02:29.165","Text":"and so on and so forth."},{"Start":"02:29.165 ","End":"02:31.130","Text":"We started with 1 head,"},{"Start":"02:31.130 ","End":"02:33.110","Text":"but that\u0027s not necessarily so."},{"Start":"02:33.110 ","End":"02:36.930","Text":"There could be no heads in n tosses."},{"Start":"02:36.930 ","End":"02:46.370","Text":"We\u0027ll have n over 0 times n over 0 and we keep on going until we can have,"},{"Start":"02:46.370 ","End":"02:49.880","Text":"let\u0027s say, n heads out of n tosses for"},{"Start":"02:49.880 ","End":"02:56.990","Text":"the first-person times n heads out of n tosses for the second person."},{"Start":"02:56.990 ","End":"03:01.685","Text":"This is the probability."},{"Start":"03:01.685 ","End":"03:04.475","Text":"Let\u0027s simplify that a little bit."},{"Start":"03:04.475 ","End":"03:11.715","Text":"Let\u0027s say, that we see here that we have a counter,"},{"Start":"03:11.715 ","End":"03:14.970","Text":"0, 1, 2, 3, 4,"},{"Start":"03:14.970 ","End":"03:20.470","Text":"and so on and so forth until n. Let\u0027s call this counter i,"},{"Start":"03:20.470 ","End":"03:27.505","Text":"and that\u0027ll be the number of heads in n tosses."},{"Start":"03:27.505 ","End":"03:29.130","Text":"What do I have here."},{"Start":"03:29.130 ","End":"03:35.045","Text":"First of all, I have n over i. What do I need?"},{"Start":"03:35.045 ","End":"03:40.880","Text":"I need to multiply this by n over i. I have,"},{"Start":"03:40.880 ","End":"03:43.550","Text":"let\u0027s say n over 0 times n over 0,"},{"Start":"03:43.550 ","End":"03:46.775","Text":"n over 1 times n over 1, and so on and so forth."},{"Start":"03:46.775 ","End":"03:49.450","Text":"I have to sum,"},{"Start":"03:49.450 ","End":"03:55.100","Text":"I\u0027m summing all these products up where i goes from 0 right here,"},{"Start":"03:55.100 ","End":"04:01.430","Text":"all the way to n. I need to divide by this expression right here,"},{"Start":"04:01.430 ","End":"04:05.840","Text":"which is 2 to the power of n times 2 to the power"},{"Start":"04:05.840 ","End":"04:10.805","Text":"of n. Let\u0027s simplify this a little bit more."},{"Start":"04:10.805 ","End":"04:17.630","Text":"That\u0027ll be the sum where i goes from 0 to n of n over"},{"Start":"04:17.630 ","End":"04:26.055","Text":"i squared divided by 2 to the power of n times 2 to the power of n,"},{"Start":"04:26.055 ","End":"04:34.625","Text":"that\u0027s 2 to the power of 2 times n. We can even simplify this more by saying this is"},{"Start":"04:34.625 ","End":"04:44.385","Text":"the sum i goes from 0 to n of n over i squared divided by 2 to the power of 2n."},{"Start":"04:44.385 ","End":"04:49.065","Text":"That\u0027s 4 to the power of n. This would be"},{"Start":"04:49.065 ","End":"04:58.320","Text":"the probability that after n tosses both people will have the same number of heads."}],"ID":12859},{"Watched":false,"Name":"Exercise 15 - Parts a-c","Duration":"4m 46s","ChapterTopicVideoID":12381,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12381.jpeg","UploadDate":"2019-01-31T23:23:31.2800000","DurationForVideoObject":"PT4M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this question, we\u0027re told that a code has"},{"Start":"00:02.220 ","End":"00:05.430","Text":"a numbers and the numbers may be repeated in the code."},{"Start":"00:05.430 ","End":"00:09.285","Text":"We\u0027re asked to calculate the following probabilities."},{"Start":"00:09.285 ","End":"00:14.190","Text":"We\u0027re asked also to express the answers in terms of a."},{"Start":"00:14.190 ","End":"00:19.560","Text":"If the code has a numbers that we have this is the code x, x,"},{"Start":"00:19.560 ","End":"00:22.230","Text":"x, and so on and so forth,"},{"Start":"00:22.230 ","End":"00:25.454","Text":"and this is a times."},{"Start":"00:25.454 ","End":"00:30.540","Text":"Now the code is comprised of numbers."},{"Start":"00:30.540 ","End":"00:34.395","Text":"Now that means that the numbers are from 0-9."},{"Start":"00:34.395 ","End":"00:38.655","Text":"We have 10 numbers to choose from."},{"Start":"00:38.655 ","End":"00:42.875","Text":"We\u0027re looking now for the size of the sample space."},{"Start":"00:42.875 ","End":"00:48.500","Text":"What is the size of the sample space because we\u0027re dealing with probabilities."},{"Start":"00:48.500 ","End":"00:53.010","Text":"Well, for the first character in the code,"},{"Start":"00:53.010 ","End":"00:55.170","Text":"we have 10 numbers to choose from,"},{"Start":"00:55.170 ","End":"00:57.920","Text":"and for the second character in the code,"},{"Start":"00:57.920 ","End":"01:00.935","Text":"we have 10 numbers to choose from and so on and so forth."},{"Start":"01:00.935 ","End":"01:03.795","Text":"We have 10 times 10 times 10,"},{"Start":"01:03.795 ","End":"01:07.275","Text":"a times. That means 10^a."},{"Start":"01:07.275 ","End":"01:14.800","Text":"That\u0027s the total number of combinations for developing the code."},{"Start":"01:14.800 ","End":"01:25.050","Text":"Now in a, we\u0027re looking for the probability that the code has no number 5."},{"Start":"01:25.050 ","End":"01:29.015","Text":"Now, in the denominator,"},{"Start":"01:29.015 ","End":"01:33.665","Text":"we have the size of the sample space, which is 10^a."},{"Start":"01:33.665 ","End":"01:36.605","Text":"Now, if there\u0027s no number 5 in the code,"},{"Start":"01:36.605 ","End":"01:40.475","Text":"then that means that we have 9 numbers to choose from."},{"Start":"01:40.475 ","End":"01:42.130","Text":"That\u0027ll be 9,"},{"Start":"01:42.130 ","End":"01:44.480","Text":"and how many places do we have in the code?"},{"Start":"01:44.480 ","End":"01:46.400","Text":"Well, we have a places."},{"Start":"01:46.400 ","End":"01:51.905","Text":"That\u0027ll be 9^a and that equals to 9 over"},{"Start":"01:51.905 ","End":"01:57.620","Text":"10^a, or 0.9^a."},{"Start":"01:57.620 ","End":"02:02.585","Text":"That\u0027s the probability that the code does not contain the number 5."},{"Start":"02:02.585 ","End":"02:08.600","Text":"In this section, we\u0027re asked what\u0027s the probability that the code contains the number 3?"},{"Start":"02:08.600 ","End":"02:10.895","Text":"Well, by asking that,"},{"Start":"02:10.895 ","End":"02:14.630","Text":"I really don\u0027t know how many times the number 3 appears in the code."},{"Start":"02:14.630 ","End":"02:16.330","Text":"It could be once, twice, 3 times,"},{"Start":"02:16.330 ","End":"02:18.940","Text":"it could be more. I really don\u0027t know."},{"Start":"02:18.940 ","End":"02:22.700","Text":"The easiest way to calculate this is by taking"},{"Start":"02:22.700 ","End":"02:27.770","Text":"the complimentary set and calculating the probability the complimentary set,"},{"Start":"02:27.770 ","End":"02:29.030","Text":"and what are we talking about?"},{"Start":"02:29.030 ","End":"02:36.680","Text":"Well, that equals to 1 minus the probability that there\u0027s no number 3 in the code."},{"Start":"02:36.680 ","End":"02:40.145","Text":"Now, what\u0027s this probability right here?"},{"Start":"02:40.145 ","End":"02:43.705","Text":"The probability of the code not having the number 3,"},{"Start":"02:43.705 ","End":"02:49.855","Text":"that equals to the probability of the code not having the number 5 or 6."},{"Start":"02:49.855 ","End":"02:52.070","Text":"The event might be different,"},{"Start":"02:52.070 ","End":"02:55.295","Text":"but the probability of the event is the same."},{"Start":"02:55.295 ","End":"02:59.495","Text":"The event of not having a number 5 in the code"},{"Start":"02:59.495 ","End":"03:03.800","Text":"is different from the event of not having the number 3 in the code,"},{"Start":"03:03.800 ","End":"03:06.350","Text":"but the probabilities are the same."},{"Start":"03:06.350 ","End":"03:11.060","Text":"It\u0027s just like when you roll a die and you get a 4."},{"Start":"03:11.060 ","End":"03:12.635","Text":"Well, getting a 4,"},{"Start":"03:12.635 ","End":"03:16.095","Text":"you have a probability of 1 out of 6,"},{"Start":"03:16.095 ","End":"03:19.850","Text":"and the probability of getting a 5,"},{"Start":"03:19.850 ","End":"03:23.580","Text":"that\u0027s also 1 out of 6 or a 6."},{"Start":"03:23.580 ","End":"03:25.215","Text":"The events are different."},{"Start":"03:25.215 ","End":"03:27.145","Text":"We get a 4 or a 5,"},{"Start":"03:27.145 ","End":"03:30.455","Text":"but the probabilities are the same, a 6."},{"Start":"03:30.455 ","End":"03:34.600","Text":"Here, we\u0027ve calculated this probability."},{"Start":"03:34.600 ","End":"03:37.710","Text":"That equals to 1 minus,"},{"Start":"03:37.710 ","End":"03:41.645","Text":"now the probability of no number 3 in the code,"},{"Start":"03:41.645 ","End":"03:46.810","Text":"that equals to 0.9^a."},{"Start":"03:46.810 ","End":"03:48.320","Text":"In this section,"},{"Start":"03:48.320 ","End":"03:53.299","Text":"we\u0027re asked what\u0027s the probability that the code does not contain odd numbers?"},{"Start":"03:53.299 ","End":"03:56.090","Text":"Well, if it doesn\u0027t contain odd numbers,"},{"Start":"03:56.090 ","End":"04:00.260","Text":"then the numbers that the code has 0,"},{"Start":"04:00.260 ","End":"04:03.670","Text":"2, 4, 6, and 8."},{"Start":"04:03.670 ","End":"04:07.040","Text":"We\u0027re looking for this probability."},{"Start":"04:07.040 ","End":"04:09.920","Text":"Well, in the denominator,"},{"Start":"04:09.920 ","End":"04:13.085","Text":"we have 10^a,"},{"Start":"04:13.085 ","End":"04:15.620","Text":"and that\u0027s the size of the sample space."},{"Start":"04:15.620 ","End":"04:23.480","Text":"Now, if the code has only a pool of 5 numbers to choose from 0,"},{"Start":"04:23.480 ","End":"04:25.040","Text":"2, 4, 6, and 8,"},{"Start":"04:25.040 ","End":"04:31.885","Text":"then the number of possible combinations would be 5,"},{"Start":"04:31.885 ","End":"04:34.430","Text":"because we have 5 numbers, 0, 2, 4, 6,"},{"Start":"04:34.430 ","End":"04:36.290","Text":"and 8 to the power of a,"},{"Start":"04:36.290 ","End":"04:40.130","Text":"and that equals to 0.5^a."},{"Start":"04:40.130 ","End":"04:47.100","Text":"This would be the probability that the code does not contain odd numbers."}],"ID":12860},{"Watched":false,"Name":"Exercise 16","Duration":"5m 50s","ChapterTopicVideoID":12382,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12382.jpeg","UploadDate":"2019-01-31T23:23:48.1130000","DurationForVideoObject":"PT5M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"In this question, we\u0027re asked how many times should"},{"Start":"00:02.130 ","End":"00:04.695","Text":"a pair of dice be rolled so that we have"},{"Start":"00:04.695 ","End":"00:12.270","Text":"a probability of at least 1/2 of getting at least 1 roll with the sum of 12."},{"Start":"00:12.350 ","End":"00:16.500","Text":"The first thing that I want to do when I\u0027m dealing"},{"Start":"00:16.500 ","End":"00:20.355","Text":"with probabilities is to calculate the size of my sample space."},{"Start":"00:20.355 ","End":"00:25.620","Text":"What would be the size of my sample space when I\u0027m rolling 2 dice?"},{"Start":"00:25.620 ","End":"00:27.750","Text":"That\u0027ll be 6 times 6."},{"Start":"00:27.750 ","End":"00:31.466","Text":"I have 6 results in the first die and 6 results in the second die,"},{"Start":"00:31.466 ","End":"00:34.925","Text":"and that equals to 36 possible results."},{"Start":"00:34.925 ","End":"00:39.950","Text":"Now, the first thing that I want to do here is"},{"Start":"00:39.950 ","End":"00:46.225","Text":"to calculate the probability of getting a sum of 12."},{"Start":"00:46.225 ","End":"00:48.850","Text":"When getting a sum of 12,"},{"Start":"00:48.850 ","End":"00:52.595","Text":"I can get that only when I have a 6 and a 6."},{"Start":"00:52.595 ","End":"00:54.830","Text":"Getting a 6 and a 6, well,"},{"Start":"00:54.830 ","End":"00:58.730","Text":"I have that with a probability of 1 in 36."},{"Start":"00:58.730 ","End":"01:05.585","Text":"There\u0027s only 1 combination of results when rolling both dice of getting a 6 and a 6."},{"Start":"01:05.585 ","End":"01:13.020","Text":"Now, what\u0027s the probability of not getting a 12?"},{"Start":"01:13.730 ","End":"01:19.520","Text":"That equals to 1 minus this probability right here,"},{"Start":"01:19.520 ","End":"01:22.460","Text":"that\u0027s 1 minus 1 divided by 36,"},{"Start":"01:22.460 ","End":"01:27.100","Text":"and that equals to 35 divided by 36."},{"Start":"01:27.100 ","End":"01:33.350","Text":"Now that is the probability when I roll both dice,"},{"Start":"01:33.350 ","End":"01:35.060","Text":"when I roll them once."},{"Start":"01:35.060 ","End":"01:38.300","Text":"But now I\u0027m rolling them n times."},{"Start":"01:38.300 ","End":"01:47.675","Text":"Now let\u0027s define n. We\u0027ll define n as the number of rolls of the 2 dice right here."},{"Start":"01:47.675 ","End":"01:57.210","Text":"I want now to calculate the probability that at least 1 roll will have a sum of 12."},{"Start":"01:57.210 ","End":"02:00.166","Text":"This is what I\u0027m looking for right here."},{"Start":"02:00.166 ","End":"02:03.200","Text":"We want to calculate the probability of"},{"Start":"02:03.200 ","End":"02:07.115","Text":"at least 1/2 of getting at least 1 roll with the sum of 12."},{"Start":"02:07.115 ","End":"02:09.455","Text":"This is what we want right here."},{"Start":"02:09.455 ","End":"02:13.220","Text":"To calculate that, let\u0027s use the complimentary set."},{"Start":"02:13.220 ","End":"02:22.250","Text":"That\u0027ll be 1 minus the probability of getting no rolls with a sum of 12."},{"Start":"02:22.250 ","End":"02:26.450","Text":"Now that equals to 1 minus,"},{"Start":"02:26.450 ","End":"02:30.155","Text":"what\u0027s the probability of not getting a 12?"},{"Start":"02:30.155 ","End":"02:34.200","Text":"That\u0027ll be 35 divided by 36."},{"Start":"02:34.200 ","End":"02:36.810","Text":"But I have n rolls right here."},{"Start":"02:36.810 ","End":"02:44.750","Text":"This has to be to the power of n. This is what we have here."},{"Start":"02:44.750 ","End":"02:51.275","Text":"This is the probability that at least 1 roll has a sum of 12."},{"Start":"02:51.275 ","End":"02:56.540","Text":"But we\u0027re looking for the probability of"},{"Start":"02:56.540 ","End":"03:01.828","Text":"at least 1/2 of getting at least 1 roll with the sum of 12."},{"Start":"03:01.828 ","End":"03:04.025","Text":"So this expression right here,"},{"Start":"03:04.025 ","End":"03:10.955","Text":"1 minus 35 divided by 36^n,"},{"Start":"03:10.955 ","End":"03:15.500","Text":"well, then that has to be greater or equal to 1/2."},{"Start":"03:15.500 ","End":"03:18.240","Text":"That\u0027s what we\u0027re looking for."},{"Start":"03:19.400 ","End":"03:24.940","Text":"We have this expression right here and we have 1 variable, that\u0027s n,"},{"Start":"03:24.940 ","End":"03:27.880","Text":"and this is exactly what we want to calculate,"},{"Start":"03:27.880 ","End":"03:29.785","Text":"the value of n right here."},{"Start":"03:29.785 ","End":"03:33.730","Text":"Because we\u0027re asked how many times should a pair of"},{"Start":"03:33.730 ","End":"03:38.270","Text":"dice be rolled and n is defined as the number of rolls of the 2 dice."},{"Start":"03:38.270 ","End":"03:41.410","Text":"Once we have this expression right here,"},{"Start":"03:41.410 ","End":"03:47.470","Text":"let\u0027s just calculate for n. Let\u0027s just move things around here."},{"Start":"03:47.470 ","End":"03:52.450","Text":"That\u0027ll be 1/2 and that has to be greater or equal"},{"Start":"03:52.450 ","End":"03:59.185","Text":"to 35 divided by 36^n."},{"Start":"03:59.185 ","End":"04:05.340","Text":"Now, let\u0027s take a log of both sides here."},{"Start":"04:05.340 ","End":"04:13.785","Text":"This would be a log of 1/2 that has to be greater or equal to"},{"Start":"04:13.785 ","End":"04:22.325","Text":"a log of 35 divided by 36 to the power of n. Now,"},{"Start":"04:22.325 ","End":"04:24.223","Text":"what is that equal to?"},{"Start":"04:24.223 ","End":"04:30.320","Text":"That\u0027s log of 1/2 that has to be greater or equal"},{"Start":"04:30.320 ","End":"04:37.970","Text":"to n times a log of 35 divided by 36."},{"Start":"04:37.970 ","End":"04:40.790","Text":"I hope you remember your algebra."},{"Start":"04:40.790 ","End":"04:47.450","Text":"Now, since log 1/2 and log of 35 divided by 36,"},{"Start":"04:47.450 ","End":"04:50.285","Text":"both of them are negative numbers,"},{"Start":"04:50.285 ","End":"04:57.555","Text":"then we get the following: n has to be greater or equal to"},{"Start":"04:57.555 ","End":"05:07.625","Text":"log 1/2 divided by log of 35 divided by 36."},{"Start":"05:07.625 ","End":"05:17.410","Text":"That means that n has to be greater or equal to 24.6."},{"Start":"05:17.410 ","End":"05:20.484","Text":"Now we can answer the question."},{"Start":"05:20.484 ","End":"05:25.660","Text":"When we\u0027re asked how many times should a pair of dice be rolled so that we have"},{"Start":"05:25.660 ","End":"05:31.330","Text":"a probability of at least 1/2 of getting at least 1 roll with the sum of 12,"},{"Start":"05:31.330 ","End":"05:35.980","Text":"the answer is that n should be greater than this number,"},{"Start":"05:35.980 ","End":"05:37.780","Text":"but a roll is a whole number,"},{"Start":"05:37.780 ","End":"05:44.635","Text":"so n should be greater or equal to 25."},{"Start":"05:44.635 ","End":"05:50.150","Text":"We have to roll both dice at least 25 times."}],"ID":12861},{"Watched":false,"Name":"Exercise 17 - Parts a-b","Duration":"9m 48s","ChapterTopicVideoID":12383,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12383.jpeg","UploadDate":"2019-01-31T23:24:17.6970000","DurationForVideoObject":"PT9M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this question, we have a 6 digit number that is randomly chosen."},{"Start":"00:04.020 ","End":"00:09.760","Text":"We\u0027re asked what\u0027s the probability that the number 5 will appear only once?"},{"Start":"00:09.980 ","End":"00:12.915","Text":"This is our number,"},{"Start":"00:12.915 ","End":"00:15.885","Text":"x x x, x x x."},{"Start":"00:15.885 ","End":"00:18.360","Text":"This is our 6 digit number."},{"Start":"00:18.360 ","End":"00:24.260","Text":"Now, each place here can have a number from 0-9."},{"Start":"00:24.260 ","End":"00:30.050","Text":"That means, that it can have 10 numbers except for the first place right here,"},{"Start":"00:30.050 ","End":"00:33.800","Text":"this number right here, it cannot be a 0. Why is that?"},{"Start":"00:33.800 ","End":"00:37.430","Text":"Because if it\u0027s a 0, then this would become a 5 digit number,"},{"Start":"00:37.430 ","End":"00:39.310","Text":"not a 6 digit number."},{"Start":"00:39.310 ","End":"00:41.900","Text":"The first thing that I want to do,"},{"Start":"00:41.900 ","End":"00:46.325","Text":"is I want to calculate the size of my sample space here."},{"Start":"00:46.325 ","End":"00:47.945","Text":"Now, how do I do that?"},{"Start":"00:47.945 ","End":"00:52.430","Text":"Well, we have to think of this as a trial with 6 stages."},{"Start":"00:52.430 ","End":"00:56.420","Text":"The first stage is choosing the number in the first place."},{"Start":"00:56.420 ","End":"01:00.790","Text":"The second stage is choosing number for the second place and so on and so forth."},{"Start":"01:00.790 ","End":"01:04.940","Text":"Here, how many numbers do I have to choose from?"},{"Start":"01:04.940 ","End":"01:08.345","Text":"Well, I have 9 numbers to choose from."},{"Start":"01:08.345 ","End":"01:12.180","Text":"It can\u0027t be a 0. It could be from 1-9."},{"Start":"01:12.180 ","End":"01:15.470","Text":"Now, what about the other places right here?"},{"Start":"01:15.470 ","End":"01:19.085","Text":"Each one of the other places I can choose 0-9."},{"Start":"01:19.085 ","End":"01:21.875","Text":"That\u0027ll be 10^5."},{"Start":"01:21.875 ","End":"01:23.840","Text":"Here I can choose 10 numbers,"},{"Start":"01:23.840 ","End":"01:25.138","Text":"here I choose 10 numbers,"},{"Start":"01:25.138 ","End":"01:26.945","Text":"and so on and so forth, 5 times."},{"Start":"01:26.945 ","End":"01:29.780","Text":"That\u0027ll be 10 times 10 times 10 times 10 times 10."},{"Start":"01:29.780 ","End":"01:33.210","Text":"That\u0027s 10^5, times 9 numbers."},{"Start":"01:33.210 ","End":"01:34.560","Text":"That\u0027s this thing right here."},{"Start":"01:34.560 ","End":"01:38.105","Text":"This thing is the size of my sample space."},{"Start":"01:38.105 ","End":"01:44.780","Text":"Now, I can go on to calculate the probability that the number 5 will appear only once."},{"Start":"01:44.780 ","End":"01:47.510","Text":"Now, that\u0027ll be equal to the following."},{"Start":"01:47.510 ","End":"01:50.900","Text":"In the denominator, I have the size of the sample space,"},{"Start":"01:50.900 ","End":"01:53.225","Text":"which is 9 times 10^5."},{"Start":"01:53.225 ","End":"01:55.175","Text":"Now in the numerator, well,"},{"Start":"01:55.175 ","End":"02:02.535","Text":"let\u0027s split this event into 2 complementary events."},{"Start":"02:02.535 ","End":"02:07.115","Text":"1 event is that the number 5 appears in the first place,"},{"Start":"02:07.115 ","End":"02:13.800","Text":"and the other event is when the 5 appears in the other places."},{"Start":"02:13.800 ","End":"02:17.340","Text":"When 5 appears in the first place right here,"},{"Start":"02:17.340 ","End":"02:23.120","Text":"that\u0027ll be 1, because we have only 1 possible digit to choose from,"},{"Start":"02:23.120 ","End":"02:26.105","Text":"that\u0027s the 5 times,"},{"Start":"02:26.105 ","End":"02:27.950","Text":"now since 5 appears only once,"},{"Start":"02:27.950 ","End":"02:32.329","Text":"then in the other 5 places I have 9 numbers to choose from."},{"Start":"02:32.329 ","End":"02:35.015","Text":"So that\u0027ll be 9^5."},{"Start":"02:35.015 ","End":"02:39.755","Text":"Now, I have to add to that the other probability."},{"Start":"02:39.755 ","End":"02:46.415","Text":"The probability that 5 is somewhere here in these 5 places right here."},{"Start":"02:46.415 ","End":"02:48.830","Text":"Well, if that\u0027s the case, well,"},{"Start":"02:48.830 ","End":"02:52.760","Text":"how many numbers we\u0027ll have to choose from in the first place?"},{"Start":"02:52.760 ","End":"02:55.025","Text":"Well, I have 8 numbers to choose from."},{"Start":"02:55.025 ","End":"02:57.545","Text":"It can\u0027t be a 0 and it can\u0027t be a 5."},{"Start":"02:57.545 ","End":"03:01.415","Text":"I have 8 numbers to choose from here in this first place."},{"Start":"03:01.415 ","End":"03:08.550","Text":"Let\u0027s assume the 5 is in the second place, it\u0027s right here."},{"Start":"03:08.550 ","End":"03:13.850","Text":"Well, what\u0027s the number of combinations,"},{"Start":"03:13.850 ","End":"03:15.440","Text":"or how many numbers do I have to choose from?"},{"Start":"03:15.440 ","End":"03:18.890","Text":"I have to choose from 1 number because 5 is here."},{"Start":"03:18.890 ","End":"03:21.390","Text":"Now, what about the other 4 places?"},{"Start":"03:21.390 ","End":"03:23.100","Text":"Well in the other 4 places,"},{"Start":"03:23.100 ","End":"03:25.635","Text":"I have 9 numbers to choose from,"},{"Start":"03:25.635 ","End":"03:27.885","Text":"because 5 is already here."},{"Start":"03:27.885 ","End":"03:30.390","Text":"That would be 9^4."},{"Start":"03:30.390 ","End":"03:34.265","Text":"But, 5 can be also here,"},{"Start":"03:34.265 ","End":"03:36.335","Text":"or here, or here, or here."},{"Start":"03:36.335 ","End":"03:38.120","Text":"It could mean 5 places,"},{"Start":"03:38.120 ","End":"03:41.060","Text":"so I had to multiply this by 5."},{"Start":"03:41.060 ","End":"03:45.800","Text":"In this way, I calculate the probability that"},{"Start":"03:45.800 ","End":"03:50.880","Text":"the number 5 appears only once in this number right here."},{"Start":"03:50.880 ","End":"03:58.210","Text":"Now, this comes out to 0.35721."},{"Start":"03:58.210 ","End":"04:00.005","Text":"In this section we\u0027re asked,"},{"Start":"04:00.005 ","End":"04:03.620","Text":"what\u0027s the probability that the number 4 will appear at least once,"},{"Start":"04:03.620 ","End":"04:07.325","Text":"and the number 0 will also appear once?"},{"Start":"04:07.325 ","End":"04:11.413","Text":"This would be the number,"},{"Start":"04:11.413 ","End":"04:15.935","Text":"and we know that we\u0027re looking at numbers from 0-9."},{"Start":"04:15.935 ","End":"04:18.575","Text":"That means so we have 10 numbers to choose from."},{"Start":"04:18.575 ","End":"04:23.250","Text":"Now, let\u0027s first define 2 events."},{"Start":"04:23.250 ","End":"04:26.855","Text":"We\u0027ll define event A and B."},{"Start":"04:26.855 ","End":"04:33.655","Text":"We\u0027ll define A as having the number 4 at least once in this number,"},{"Start":"04:33.655 ","End":"04:41.200","Text":"and B as having the number 0 at least once in this number right here."},{"Start":"04:41.200 ","End":"04:43.960","Text":"We\u0027re looking for this probability."},{"Start":"04:43.960 ","End":"04:48.730","Text":"The probability of A intersect B is"},{"Start":"04:48.730 ","End":"04:51.790","Text":"the probability that the number 4 will appear at least"},{"Start":"04:51.790 ","End":"04:55.450","Text":"once and that the number 0 will also appear at least once."},{"Start":"04:55.450 ","End":"05:01.090","Text":"Now, we know according to the mortgage that the probability of A union B,"},{"Start":"05:01.090 ","End":"05:04.570","Text":"well that equals to the probability of A plus"},{"Start":"05:04.570 ","End":"05:09.990","Text":"the probability of B minus the probability of A intersect B."},{"Start":"05:09.990 ","End":"05:12.035","Text":"If we switch things around,"},{"Start":"05:12.035 ","End":"05:14.780","Text":"the probability of A intersect B,"},{"Start":"05:14.780 ","End":"05:19.190","Text":"well that equals to the probability of A plus the probability of"},{"Start":"05:19.190 ","End":"05:24.720","Text":"B minus the probability of A union B."},{"Start":"05:24.940 ","End":"05:27.140","Text":"Now that we have this,"},{"Start":"05:27.140 ","End":"05:30.030","Text":"let\u0027s just calculate these probabilities right here,"},{"Start":"05:30.030 ","End":"05:34.865","Text":"so then we can calculate the probability of A intersect B."},{"Start":"05:34.865 ","End":"05:41.716","Text":"Let\u0027s get started. The probability of A that we have 4 at least once,"},{"Start":"05:41.716 ","End":"05:46.655","Text":"well that equals to 1 minus the probability of A bar,"},{"Start":"05:46.655 ","End":"05:50.825","Text":"the probability that we don\u0027t have a 4 at all."},{"Start":"05:50.825 ","End":"05:52.865","Text":"Now, what does that equal to?"},{"Start":"05:52.865 ","End":"05:55.040","Text":"Well, that\u0027s 1 minus,"},{"Start":"05:55.040 ","End":"05:57.515","Text":"now in the denominator,"},{"Start":"05:57.515 ","End":"06:01.790","Text":"I have the size of the sample space which is 9 times 10^5."},{"Start":"06:01.790 ","End":"06:03.050","Text":"Now in the numerator,"},{"Start":"06:03.050 ","End":"06:05.690","Text":"well, if I don\u0027t have a 4,"},{"Start":"06:05.690 ","End":"06:12.710","Text":"that means that in the first place right here,"},{"Start":"06:12.710 ","End":"06:15.740","Text":"well then I have 8 numbers to choose from."},{"Start":"06:15.740 ","End":"06:18.523","Text":"I don\u0027t have a 0 and I don\u0027t have a 4,"},{"Start":"06:18.523 ","End":"06:20.480","Text":"and in the other 5 places,"},{"Start":"06:20.480 ","End":"06:22.985","Text":"I have 9 numbers to choose from."},{"Start":"06:22.985 ","End":"06:26.695","Text":"That would be 9^5."},{"Start":"06:26.695 ","End":"06:30.905","Text":"What about the probability of B?"},{"Start":"06:30.905 ","End":"06:35.360","Text":"Well, that equals to 1 minus the probability of B bar."},{"Start":"06:35.360 ","End":"06:37.455","Text":"Now that\u0027s 1 minus."},{"Start":"06:37.455 ","End":"06:39.470","Text":"Again, in the denominator,"},{"Start":"06:39.470 ","End":"06:42.200","Text":"I have the size of my sample space."},{"Start":"06:42.200 ","End":"06:43.640","Text":"Now what about the numerator?"},{"Start":"06:43.640 ","End":"06:45.589","Text":"Well, if I don\u0027t have a 0,"},{"Start":"06:45.589 ","End":"06:52.055","Text":"then that means that I have 9 numbers to choose from here in the first place,"},{"Start":"06:52.055 ","End":"06:56.810","Text":"and 9 numbers to choose from in each of the other 5 places."},{"Start":"06:56.810 ","End":"06:59.060","Text":"So in all 6 places,"},{"Start":"06:59.060 ","End":"07:01.175","Text":"I have 9 numbers to choose from."},{"Start":"07:01.175 ","End":"07:04.660","Text":"That\u0027ll be 9^6."},{"Start":"07:04.660 ","End":"07:12.000","Text":"Now I have to calculate the probability of A union B."},{"Start":"07:13.310 ","End":"07:19.625","Text":"The probability of A union B, well,"},{"Start":"07:19.625 ","End":"07:27.495","Text":"that equals to 1 minus the probability of A bar intersect B bar."},{"Start":"07:27.495 ","End":"07:31.010","Text":"The probability that I don\u0027t have a 4,"},{"Start":"07:31.010 ","End":"07:33.845","Text":"and I also don\u0027t have a 0."},{"Start":"07:33.845 ","End":"07:36.620","Text":"Now, that equals to 1 minus,"},{"Start":"07:36.620 ","End":"07:38.210","Text":"now in the denominator again,"},{"Start":"07:38.210 ","End":"07:42.620","Text":"we have the size of the sample space that\u0027s 9 times 10^5."},{"Start":"07:42.620 ","End":"07:46.476","Text":"Now if I don\u0027t have a 0 and I don\u0027t have a 4,"},{"Start":"07:46.476 ","End":"07:50.685","Text":"well then in all 6 places right here,"},{"Start":"07:50.685 ","End":"07:54.120","Text":"I have 8 numbers to choose from,"},{"Start":"07:54.120 ","End":"07:58.740","Text":"so that\u0027ll be 8^6."},{"Start":"07:58.740 ","End":"08:02.600","Text":"Excellent. Now that I have the probability of A,"},{"Start":"08:02.600 ","End":"08:06.245","Text":"probability of B and the probability of A union B,"},{"Start":"08:06.245 ","End":"08:11.650","Text":"then I can calculate the probability of A intersect B."},{"Start":"08:12.110 ","End":"08:19.010","Text":"Let\u0027s get started. The probability of A intersect B,"},{"Start":"08:19.010 ","End":"08:21.590","Text":"well that equals to the probability of A,"},{"Start":"08:21.590 ","End":"08:31.200","Text":"and that equals to 1 minus 8 times 9^5 divided by 9 times 10^5,"},{"Start":"08:31.200 ","End":"08:36.505","Text":"that\u0027s right here, plus the probability of B."},{"Start":"08:36.505 ","End":"08:46.520","Text":"What\u0027s that? That\u0027s 1 minus 9^6 divided by 9 times 10^5."},{"Start":"08:46.520 ","End":"08:54.435","Text":"That\u0027s right here. Minus the probability of A union B,"},{"Start":"08:54.435 ","End":"09:04.545","Text":"which is 1 minus A^6 divided by 9 times 10^5."},{"Start":"09:04.545 ","End":"09:07.065","Text":"That\u0027s the probability."},{"Start":"09:07.065 ","End":"09:08.915","Text":"This is what we\u0027re looking for."},{"Start":"09:08.915 ","End":"09:12.325","Text":"Now, let\u0027s just simplify things."},{"Start":"09:12.325 ","End":"09:16.005","Text":"That\u0027ll be equal to, 1 minus,"},{"Start":"09:16.005 ","End":"09:17.965","Text":"now in the denominator,"},{"Start":"09:17.965 ","End":"09:22.415","Text":"we have 9 times 10^5."},{"Start":"09:22.415 ","End":"09:27.950","Text":"Here we have 8 times 9^5."},{"Start":"09:27.950 ","End":"09:32.330","Text":"Here, we took the minus out,"},{"Start":"09:32.330 ","End":"09:40.700","Text":"so that\u0027ll be plus 9^6 minus, we have 8^6."},{"Start":"09:40.700 ","End":"09:46.930","Text":"Now, all this comes out to 0.1759."}],"ID":12862},{"Watched":false,"Name":"Exercise 18 - Parts a-d","Duration":"7m 40s","ChapterTopicVideoID":12384,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12384.jpeg","UploadDate":"2019-01-31T23:24:38.2330000","DurationForVideoObject":"PT7M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this question, we are told that Jane has"},{"Start":"00:02.460 ","End":"00:06.120","Text":"3 red binders and 2 blue binders on a shelf at her office."},{"Start":"00:06.120 ","End":"00:10.485","Text":"She wrote 2 notes and placed them randomly between the 5 binders."},{"Start":"00:10.485 ","End":"00:15.555","Text":"Each note may be placed in 1 of the 4 different places between the binders."},{"Start":"00:15.555 ","End":"00:17.580","Text":"Here we see the 5 binders,"},{"Start":"00:17.580 ","End":"00:19.065","Text":"3 red 2 blue."},{"Start":"00:19.065 ","End":"00:25.830","Text":"We see here the 4 different places where the notes may be placed and again,"},{"Start":"00:25.830 ","End":"00:26.940","Text":"we have 2 notes."},{"Start":"00:26.940 ","End":"00:31.395","Text":"The notes, both of them may be placed in the same place,"},{"Start":"00:31.395 ","End":"00:35.070","Text":"or they could be placed in different places."},{"Start":"00:35.070 ","End":"00:41.925","Text":"We\u0027re asked about the probability that the 2 notes will be placed in different places."},{"Start":"00:41.925 ","End":"00:46.490","Text":"In the denominator, we have the size of the sample space."},{"Start":"00:46.490 ","End":"00:51.235","Text":"Now, what is that? Let\u0027s calculate that."},{"Start":"00:51.235 ","End":"00:56.655","Text":"I have 5 different binders,"},{"Start":"00:56.655 ","End":"00:59.535","Text":"in how many ways can I order the 5 binders?"},{"Start":"00:59.535 ","End":"01:01.230","Text":"5 factorial."},{"Start":"01:01.230 ","End":"01:03.315","Text":"I have 2 notes."},{"Start":"01:03.315 ","End":"01:07.320","Text":"The first note, I can put this in 1 of 4 places,"},{"Start":"01:07.320 ","End":"01:11.430","Text":"so I have 4 options and the second note I also have 4 options,"},{"Start":"01:11.430 ","End":"01:15.570","Text":"so I have 4 squared options of putting the 2 notes into"},{"Start":"01:15.570 ","End":"01:23.810","Text":"4 different places where it\u0027s just like placing 2 balls into 4 baskets."},{"Start":"01:23.810 ","End":"01:26.230","Text":"This then would be the size of my sample space"},{"Start":"01:26.230 ","End":"01:29.000","Text":"and this is what I\u0027ll put in the denominator right here,"},{"Start":"01:29.000 ","End":"01:33.540","Text":"which is 5 factorial times 4 squared."},{"Start":"01:33.540 ","End":"01:42.305","Text":"In the numerator, the 2 notes that we want them to be placed in different places."},{"Start":"01:42.305 ","End":"01:46.100","Text":"Again, I can order the binders any which way I want."},{"Start":"01:46.100 ","End":"01:50.500","Text":"I have 5 factorial combinations."},{"Start":"01:50.500 ","End":"01:57.060","Text":"The first note, I can put this in 1 of 4 places and the second note,"},{"Start":"01:57.060 ","End":"01:59.370","Text":"I don\u0027t want the second note to be in"},{"Start":"01:59.370 ","End":"02:06.040","Text":"the same place that we have the first note so I have 3 different places to put it."},{"Start":"02:06.040 ","End":"02:11.775","Text":"That means that this then would be my probability and let\u0027s simplify that."},{"Start":"02:11.775 ","End":"02:15.225","Text":"5 factorial, 5 factorial cancel each other out."},{"Start":"02:15.225 ","End":"02:23.430","Text":"4 here and the square here cancel each other out so we\u0027re left with 3 quarters or 0.75,"},{"Start":"02:23.430 ","End":"02:29.210","Text":"so that\u0027s the probability that the 2 notes will be placed in different places."},{"Start":"02:29.210 ","End":"02:30.600","Text":"In this section we\u0027re asked,"},{"Start":"02:30.600 ","End":"02:33.465","Text":"what are the chances that between the 2 notes,"},{"Start":"02:33.465 ","End":"02:37.390","Text":"there are 2 red binders and no blue binders."},{"Start":"02:37.400 ","End":"02:40.635","Text":"Actually, there are"},{"Start":"02:40.635 ","End":"02:46.680","Text":"only 2 possible placement of notes where we have 2 binders between them."},{"Start":"02:46.680 ","End":"02:49.640","Text":"The nodes can be here and here,"},{"Start":"02:49.640 ","End":"02:54.305","Text":"or they can be here and here."},{"Start":"02:54.305 ","End":"02:58.860","Text":"Let\u0027s talk about the probability here."},{"Start":"02:59.600 ","End":"03:03.065","Text":"This is a probability that we\u0027re looking for."},{"Start":"03:03.065 ","End":"03:07.555","Text":"That there are 2 red binders and no blue binders between the notes."},{"Start":"03:07.555 ","End":"03:11.900","Text":"In the denominator, we have the size of the sample space,"},{"Start":"03:11.900 ","End":"03:17.765","Text":"which is 5 factorial times 4 squared."},{"Start":"03:17.765 ","End":"03:20.345","Text":"In the numerator, as I said,"},{"Start":"03:20.345 ","End":"03:24.810","Text":"we have 2 possible positions for the notes."},{"Start":"03:24.810 ","End":"03:29.175","Text":"The notes can be either here and here or here and here,"},{"Start":"03:29.175 ","End":"03:33.180","Text":"so we have 2 possible combinations."},{"Start":"03:33.180 ","End":"03:35.510","Text":"We have to take into consideration"},{"Start":"03:35.510 ","End":"03:38.630","Text":"the internal ordering of the notes and we have 2 notes,"},{"Start":"03:38.630 ","End":"03:41.320","Text":"so that\u0027ll be 2 factorial."},{"Start":"03:41.320 ","End":"03:46.605","Text":"We have to choose 2 red binders out of the 3,"},{"Start":"03:46.605 ","End":"03:48.960","Text":"so we have 3 over 2."},{"Start":"03:48.960 ","End":"03:52.665","Text":"These 2 red binders that we\u0027ve chosen,"},{"Start":"03:52.665 ","End":"03:55.200","Text":"we have to order them."},{"Start":"03:55.200 ","End":"04:01.125","Text":"We have 2 factorial ways of ordering the 2 binders that were chosen."},{"Start":"04:01.125 ","End":"04:05.385","Text":"We have the other 3 binders,"},{"Start":"04:05.385 ","End":"04:09.075","Text":"so we have to order them as well."},{"Start":"04:09.075 ","End":"04:16.490","Text":"There we go, this then is the probability of getting 2 notes,"},{"Start":"04:16.490 ","End":"04:20.760","Text":"or placing 2 notes whether 2 red binders between"},{"Start":"04:20.760 ","End":"04:28.205","Text":"them and no blue binders and that comes out to 0.075."},{"Start":"04:28.205 ","End":"04:29.510","Text":"In this section we\u0027re asked,"},{"Start":"04:29.510 ","End":"04:34.475","Text":"what are the chances that between the 2 notes there\u0027s exactly 1 binder."},{"Start":"04:34.475 ","End":"04:40.010","Text":"Here we have the 5 binders so I can put the 2 notes here."},{"Start":"04:40.010 ","End":"04:44.660","Text":"I can put the notes here,"},{"Start":"04:44.660 ","End":"04:46.770","Text":"or I can put the notes here."},{"Start":"04:46.770 ","End":"04:53.060","Text":"There are 3 possible configurations of placing the notes between the binders."},{"Start":"04:53.060 ","End":"04:56.860","Text":"Let\u0027s now calculate the probabilities."},{"Start":"04:56.860 ","End":"04:59.870","Text":"We have here in the denominator,"},{"Start":"04:59.870 ","End":"05:06.060","Text":"we have the size of the sample space which is 5 factorial times 4 squared."},{"Start":"05:06.060 ","End":"05:08.365","Text":"Let\u0027s take a look at the numerator."},{"Start":"05:08.365 ","End":"05:12.390","Text":"We have 3 configurations, 3 options."},{"Start":"05:12.390 ","End":"05:15.080","Text":"Within each 1 of the options,"},{"Start":"05:15.080 ","End":"05:18.320","Text":"we have the internal orderings of the notes,"},{"Start":"05:18.320 ","End":"05:22.380","Text":"so that has to be multiplied by 2 factorial."},{"Start":"05:22.380 ","End":"05:29.810","Text":"Since I didn\u0027t specify which binder or which color binder should be between the 2 notes,"},{"Start":"05:29.810 ","End":"05:36.970","Text":"then I have to order the 5 binders so I have 5 factorial ways of doing that."},{"Start":"05:36.970 ","End":"05:44.910","Text":"That\u0027s the probability right here and that comes out to 3 times 2,"},{"Start":"05:44.910 ","End":"05:47.945","Text":"the 5 factorial cancels each other out,"},{"Start":"05:47.945 ","End":"05:56.750","Text":"divided by 4 squared that equals to 6 divided by 16,"},{"Start":"05:56.750 ","End":"06:01.220","Text":"and that comes out to 0.375."},{"Start":"06:01.220 ","End":"06:02.750","Text":"In this section we\u0027re asked,"},{"Start":"06:02.750 ","End":"06:08.600","Text":"what are the chances that between the 2 notes there are 2 binders were 1 binder is blue?"},{"Start":"06:08.600 ","End":"06:14.610","Text":"Again, I can put the notes here and I have 2 binders"},{"Start":"06:14.610 ","End":"06:21.270","Text":"between them and I can put the notes here where I have 2 binders between them here,"},{"Start":"06:21.270 ","End":"06:27.785","Text":"so I have 2 possible ways of placing the notes where I have 2 binders between them."},{"Start":"06:27.785 ","End":"06:31.890","Text":"Let\u0027s now calculate the probability."},{"Start":"06:32.900 ","End":"06:36.530","Text":"In the denominator, we have the size of"},{"Start":"06:36.530 ","End":"06:40.865","Text":"the sample space which is 5 factorial times 4 squared."},{"Start":"06:40.865 ","End":"06:45.920","Text":"In the numerator, we have 2 possible ways of placing the notes,"},{"Start":"06:45.920 ","End":"06:51.000","Text":"so we\u0027ll have 2 binders between them so I need to put 2 here."},{"Start":"06:51.000 ","End":"06:53.475","Text":"Those are the 2 possible ways,"},{"Start":"06:53.475 ","End":"06:59.195","Text":"times 2 factorial, that\u0027s the internal orderings of the notes."},{"Start":"06:59.195 ","End":"07:04.815","Text":"If I have 2 binders between the 2 notes and 1 binder is blue,"},{"Start":"07:04.815 ","End":"07:07.620","Text":"that means that 1 binder is red so out of"},{"Start":"07:07.620 ","End":"07:13.460","Text":"the 3 red binders I have to choose 1 and out of the 2 blue binders,"},{"Start":"07:13.460 ","End":"07:17.880","Text":"I have to choose 1 and out of the 2 binders,"},{"Start":"07:17.880 ","End":"07:19.980","Text":"1 blue and 1 red,"},{"Start":"07:19.980 ","End":"07:24.775","Text":"I have the internal ordering of these binders."},{"Start":"07:24.775 ","End":"07:28.250","Text":"Now apart from that I have to multiply all this by"},{"Start":"07:28.250 ","End":"07:32.735","Text":"3 factorial which is the ordering of the rest of the binders."},{"Start":"07:32.735 ","End":"07:40.020","Text":"This then would be the probability and this comes out to 0.15."}],"ID":12863},{"Watched":false,"Name":"Exercise 19 - Parts a-c","Duration":"7m 3s","ChapterTopicVideoID":12385,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12385.jpeg","UploadDate":"2019-01-31T23:24:56.4500000","DurationForVideoObject":"PT7M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"In this question, I have 6 pens that are placed randomly in 3 different drawers."},{"Start":"00:04.545 ","End":"00:07.260","Text":"A drawer is chosen for each of the pens."},{"Start":"00:07.260 ","End":"00:12.015","Text":"We\u0027re asked, what are the chances that in exactly 2 drawers each,"},{"Start":"00:12.015 ","End":"00:14.685","Text":"there are exactly 2 pens."},{"Start":"00:14.685 ","End":"00:17.745","Text":"What is our trial here?"},{"Start":"00:17.745 ","End":"00:27.935","Text":"The trial is placing the 6 pens in 3 drawers."},{"Start":"00:27.935 ","End":"00:32.895","Text":"This is just like sequential sampling with returns."},{"Start":"00:32.895 ","End":"00:36.300","Text":"It\u0027s just like having 6 balls"},{"Start":"00:36.300 ","End":"00:40.045","Text":"and I have 3 cells and have to put the balls in the cells here."},{"Start":"00:40.045 ","End":"00:43.864","Text":"Now, whenever we have sequential sampling with returns,"},{"Start":"00:43.864 ","End":"00:46.430","Text":"what\u0027s the size of my sample space?"},{"Start":"00:46.430 ","End":"00:50.810","Text":"That equals to n to the power of k. Now,"},{"Start":"00:50.810 ","End":"00:54.890","Text":"n would be the placeholders,"},{"Start":"00:54.890 ","End":"00:56.450","Text":"the cells, the drawers,"},{"Start":"00:56.450 ","End":"00:58.735","Text":"and I have 3 of them."},{"Start":"00:58.735 ","End":"01:03.170","Text":"K would be the object that I allocate."},{"Start":"01:03.170 ","End":"01:08.290","Text":"K here would be the number of balls or the number of pens."},{"Start":"01:08.290 ","End":"01:09.630","Text":"I have 6 pens,"},{"Start":"01:09.630 ","End":"01:12.750","Text":"so that\u0027ll be 3 to the power of 6,"},{"Start":"01:12.750 ","End":"01:21.150","Text":"that\u0027ll be the size of my sample space and that equals to 729."},{"Start":"01:23.840 ","End":"01:26.370","Text":"Let\u0027s get to the question now."},{"Start":"01:26.370 ","End":"01:30.945","Text":"What are the chances that in exactly 2 drawers,"},{"Start":"01:30.945 ","End":"01:34.005","Text":"there are exactly 2 pens?"},{"Start":"01:34.005 ","End":"01:35.610","Text":"Let\u0027s take a look at this."},{"Start":"01:35.610 ","End":"01:43.889","Text":"I have 3 drawers and I want that in exactly 2 drawers,"},{"Start":"01:43.889 ","End":"01:48.045","Text":"I\u0027ll have exactly 2 pens."},{"Start":"01:48.045 ","End":"01:51.755","Text":"What happens to the remaining drawer here?"},{"Start":"01:51.755 ","End":"01:53.360","Text":"How many pens will that have?"},{"Start":"01:53.360 ","End":"01:58.355","Text":"That\u0027ll have 2 pens as well because I have a total of 6 pens."},{"Start":"01:58.355 ","End":"02:04.190","Text":"The answer to this question would be 0."},{"Start":"02:04.190 ","End":"02:06.020","Text":"Now, why is that?"},{"Start":"02:06.020 ","End":"02:11.690","Text":"We can\u0027t have exactly 2 drawers with 2 pens in"},{"Start":"02:11.690 ","End":"02:18.180","Text":"each because the remaining drawer will also have the 2 remaining pens."},{"Start":"02:18.180 ","End":"02:20.865","Text":"This is an impossibility."},{"Start":"02:20.865 ","End":"02:25.430","Text":"This section, we\u0027re asked what are the chances that in exactly 1 drawer,"},{"Start":"02:25.430 ","End":"02:28.085","Text":"there are exactly 2 pens."},{"Start":"02:28.085 ","End":"02:32.390","Text":"Again, let\u0027s say that I have my 3 drawers here."},{"Start":"02:32.390 ","End":"02:35.529","Text":"Now, we want that in exactly 1 drawer,"},{"Start":"02:35.529 ","End":"02:37.655","Text":"there are exactly 2 pens."},{"Start":"02:37.655 ","End":"02:41.415","Text":"Let\u0027s put the pens here for now in drawer number 1."},{"Start":"02:41.415 ","End":"02:47.565","Text":"Now, we have to allocate the 4 other pens into the 2 drawers."},{"Start":"02:47.565 ","End":"02:50.645","Text":"Now, there are 2 types of configurations here."},{"Start":"02:50.645 ","End":"02:55.820","Text":"1 is that the second drawer will have 3 pens,"},{"Start":"02:55.820 ","End":"02:59.005","Text":"and the third drawer will have 1 pen."},{"Start":"02:59.005 ","End":"03:05.555","Text":"The other configuration is that the second drawer will have 4 pens,"},{"Start":"03:05.555 ","End":"03:09.020","Text":"and the third drawer will have no pens whatsoever."},{"Start":"03:09.020 ","End":"03:10.770","Text":"We have 2,"},{"Start":"03:10.770 ","End":"03:12.465","Text":"3, and 1."},{"Start":"03:12.465 ","End":"03:13.980","Text":"2 pens, 3 pens,"},{"Start":"03:13.980 ","End":"03:16.560","Text":"and 1 pen, that\u0027s 1 configuration."},{"Start":"03:16.560 ","End":"03:21.505","Text":"The other configuration is that we\u0027ll have 2, 4, and 0."},{"Start":"03:21.505 ","End":"03:24.075","Text":"2 pens, 4 pens,"},{"Start":"03:24.075 ","End":"03:25.835","Text":"and no pens whatsoever."},{"Start":"03:25.835 ","End":"03:30.380","Text":"Now, the choice of drawers for the 2 pens,"},{"Start":"03:30.380 ","End":"03:32.990","Text":"3 or 1 pen, or the 2,"},{"Start":"03:32.990 ","End":"03:34.204","Text":"4, and 0 pens,"},{"Start":"03:34.204 ","End":"03:36.505","Text":"that doesn\u0027t have to be like this."},{"Start":"03:36.505 ","End":"03:40.940","Text":"The drawer with the 2 pens can be either here or here or here."},{"Start":"03:40.940 ","End":"03:43.580","Text":"The same for the drawers with"},{"Start":"03:43.580 ","End":"03:48.040","Text":"the 3 pens and 1 pen or the drawers with the 4 pens, 0 pens."},{"Start":"03:48.040 ","End":"03:52.070","Text":"How can we calculate this probability?"},{"Start":"03:52.070 ","End":"03:55.265","Text":"We know that in the denominator,"},{"Start":"03:55.265 ","End":"03:59.540","Text":"we have the size of the sample space, that\u0027s 729."},{"Start":"03:59.540 ","End":"04:01.315","Text":"What about the numerator?"},{"Start":"04:01.315 ","End":"04:04.920","Text":"Let\u0027s take a look at the drawer with the 2 pens."},{"Start":"04:04.920 ","End":"04:06.675","Text":"Out of the 3 drawers,"},{"Start":"04:06.675 ","End":"04:11.115","Text":"I choose 1 that has the 2 pens."},{"Start":"04:11.115 ","End":"04:13.725","Text":"Now I have to choose the 2 pens."},{"Start":"04:13.725 ","End":"04:15.180","Text":"Out of the 6 pens,"},{"Start":"04:15.180 ","End":"04:17.130","Text":"I choose 2 of them."},{"Start":"04:17.130 ","End":"04:21.690","Text":"Now, out of the 2 remaining drawers,"},{"Start":"04:21.690 ","End":"04:28.695","Text":"I choose 1 drawer and out of the remaining 4 pens,"},{"Start":"04:28.695 ","End":"04:30.945","Text":"I choose 3 pens."},{"Start":"04:30.945 ","End":"04:36.770","Text":"Now, this takes care of this configuration right here."},{"Start":"04:36.770 ","End":"04:39.980","Text":"Now, what about this configuration right here?"},{"Start":"04:39.980 ","End":"04:46.260","Text":"I have to add to that this configuration. Let\u0027s do that."},{"Start":"04:46.260 ","End":"04:49.520","Text":"Out of the 3 drawers that I have,"},{"Start":"04:49.520 ","End":"04:52.060","Text":"I have to choose 1 drawer,"},{"Start":"04:52.060 ","End":"04:55.035","Text":"and in that 1 drawer,"},{"Start":"04:55.035 ","End":"04:59.380","Text":"I have to choose 2 pens out of the 6 pens."},{"Start":"04:59.380 ","End":"05:04.340","Text":"Now, out of the remaining 2 drawers,"},{"Start":"05:04.340 ","End":"05:06.275","Text":"I have to choose 1."},{"Start":"05:06.275 ","End":"05:13.230","Text":"In that drawer, I choose all 4 of the remaining pens."},{"Start":"05:13.230 ","End":"05:15.435","Text":"Here I\u0027m done."},{"Start":"05:15.435 ","End":"05:22.320","Text":"This comes out to 450 divided by 729."},{"Start":"05:22.320 ","End":"05:26.570","Text":"In this section,"},{"Start":"05:26.570 ","End":"05:30.605","Text":"we\u0027re asked what are the chances that in exactly 3 drawers each,"},{"Start":"05:30.605 ","End":"05:32.615","Text":"there are exactly 2 pens?"},{"Start":"05:32.615 ","End":"05:36.500","Text":"This is the same configuration that we had in the first section where we had"},{"Start":"05:36.500 ","End":"05:45.460","Text":"the 3 drawers and we had 2 pens in each drawer."},{"Start":"05:45.460 ","End":"05:53.590","Text":"Now, we really don\u0027t need to differentiate between the various drawers right here."},{"Start":"05:53.590 ","End":"05:59.155","Text":"It really doesn\u0027t matter which 2 pens go into which drawers."},{"Start":"05:59.155 ","End":"06:00.775","Text":"Why am I saying that?"},{"Start":"06:00.775 ","End":"06:03.970","Text":"Because the chances here are like this,"},{"Start":"06:03.970 ","End":"06:07.330","Text":"this is 729 in the denominator."},{"Start":"06:07.330 ","End":"06:10.915","Text":"We don\u0027t have to consider the drawers here,"},{"Start":"06:10.915 ","End":"06:16.765","Text":"only the allocation of the pens. What do we have here?"},{"Start":"06:16.765 ","End":"06:18.151","Text":"In the first drawer,"},{"Start":"06:18.151 ","End":"06:21.210","Text":"out of the 6 pens,"},{"Start":"06:21.210 ","End":"06:24.765","Text":"we have to choose 2 pens for the first drawer."},{"Start":"06:24.765 ","End":"06:27.239","Text":"For the second drawer,"},{"Start":"06:27.239 ","End":"06:28.720","Text":"out of the remaining 4,"},{"Start":"06:28.720 ","End":"06:30.160","Text":"we need to choose 2,"},{"Start":"06:30.160 ","End":"06:33.225","Text":"and out of the remaining 2,"},{"Start":"06:33.225 ","End":"06:34.605","Text":"we need to choose them."},{"Start":"06:34.605 ","End":"06:36.480","Text":"We can disregard this."},{"Start":"06:36.480 ","End":"06:40.135","Text":"This is the probability."},{"Start":"06:40.135 ","End":"06:45.200","Text":"We did not consider the allocation or the internal ordering of the drawers."},{"Start":"06:45.200 ","End":"06:51.240","Text":"All we did was to choose which pens go into the first,"},{"Start":"06:51.240 ","End":"06:53.205","Text":"second, and third drawer."},{"Start":"06:53.205 ","End":"06:55.865","Text":"This is the probability,"},{"Start":"06:55.865 ","End":"07:03.780","Text":"and that comes out to 90 divided by 729."}],"ID":12864},{"Watched":false,"Name":"Exercise 20","Duration":"3m 15s","ChapterTopicVideoID":12386,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12386.jpeg","UploadDate":"2019-01-31T23:25:05.4600000","DurationForVideoObject":"PT3M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"In this question, I have n balls that are randomly placed in"},{"Start":"00:02.745 ","End":"00:06.825","Text":"indifferent boxes where a box may contain more than 1 ball."},{"Start":"00:06.825 ","End":"00:11.385","Text":"We\u0027re asked what are the chances that in the ith box where i is between 1 and n,"},{"Start":"00:11.385 ","End":"00:14.550","Text":"there will be exactly k balls here."},{"Start":"00:14.550 ","End":"00:16.980","Text":"Well, what\u0027s the trial here?"},{"Start":"00:16.980 ","End":"00:20.685","Text":"Well the trial is allocation of n balls in"},{"Start":"00:20.685 ","End":"00:25.005","Text":"n different boxes where a box can contain more than 1 ball."},{"Start":"00:25.005 ","End":"00:30.330","Text":"Well, this is just like sequential sampling with returns."},{"Start":"00:30.330 ","End":"00:33.480","Text":"Whenever you have sequential sampling with returns,"},{"Start":"00:33.480 ","End":"00:42.560","Text":"the size of the sample space would be n to the power of k. Now in our case,"},{"Start":"00:42.560 ","End":"00:46.460","Text":"n, that would be the number of boxes and k,"},{"Start":"00:46.460 ","End":"00:49.470","Text":"that\u0027ll be the number of balls."},{"Start":"00:49.540 ","End":"00:54.365","Text":"We can place the first ball in n different boxes and the second ball in"},{"Start":"00:54.365 ","End":"00:59.720","Text":"n different boxes and so on and so forth n times and we have n balls."},{"Start":"00:59.720 ","End":"01:04.560","Text":"In our case, that\u0027ll be n to the power of n. Why is that?"},{"Start":"01:04.560 ","End":"01:12.480","Text":"Well, because we have n balls and we are allocating them into n boxes right here."},{"Start":"01:12.560 ","End":"01:17.390","Text":"Now, let\u0027s calculate the probability here,"},{"Start":"01:17.390 ","End":"01:19.520","Text":"within the denominator what do we have?"},{"Start":"01:19.520 ","End":"01:21.410","Text":"Well, we have the size of the sample space,"},{"Start":"01:21.410 ","End":"01:24.605","Text":"that\u0027s n to the power of n. Now,"},{"Start":"01:24.605 ","End":"01:26.939","Text":"let\u0027s take the ith box."},{"Start":"01:26.939 ","End":"01:32.460","Text":"We want that the ith box will have exactly k balls,"},{"Start":"01:32.460 ","End":"01:35.314","Text":"so out of the n balls,"},{"Start":"01:35.314 ","End":"01:39.649","Text":"we have to choose k of them to be placed into the ith box."},{"Start":"01:39.649 ","End":"01:48.710","Text":"Now, how many balls remain after we\u0027ve placed k balls into the ith box?"},{"Start":"01:48.710 ","End":"01:51.740","Text":"Well, we have n minus k balls."},{"Start":"01:51.740 ","End":"01:55.380","Text":"How many boxes remain?"},{"Start":"01:55.380 ","End":"01:59.500","Text":"Well, we have n minus 1 boxes that remain."},{"Start":"01:59.570 ","End":"02:07.340","Text":"Now, how many combinations or how many different allocations do"},{"Start":"02:07.340 ","End":"02:14.300","Text":"I have of placing n minus k balls into n minus 1 boxes?"},{"Start":"02:14.300 ","End":"02:17.165","Text":"Well, let\u0027s take 1 of the balls here."},{"Start":"02:17.165 ","End":"02:23.210","Text":"Well, that ball has n minus 1 possible boxes to be placed in,"},{"Start":"02:23.210 ","End":"02:26.270","Text":"and the second ball, the same number of combinations,"},{"Start":"02:26.270 ","End":"02:28.565","Text":"the third ball, the same number of combinations."},{"Start":"02:28.565 ","End":"02:34.690","Text":"All in all, I have n minus 1 to the power of n minus k"},{"Start":"02:34.690 ","End":"02:42.155","Text":"possible combinations or allocations of placing n minus k balls into n minus 1 boxes."},{"Start":"02:42.155 ","End":"02:45.340","Text":"This expression goes right here,"},{"Start":"02:45.340 ","End":"02:51.200","Text":"n over k has to be multiplied by this n minus 1 to"},{"Start":"02:51.200 ","End":"02:57.620","Text":"the power of n minus k. This expression right here,"},{"Start":"02:57.620 ","End":"03:02.345","Text":"n over k, is choosing k balls to be placed in 1 box,"},{"Start":"03:02.345 ","End":"03:03.605","Text":"the ith box,"},{"Start":"03:03.605 ","End":"03:07.700","Text":"and this would be the number of combinations of placing"},{"Start":"03:07.700 ","End":"03:12.395","Text":"the remaining balls into the remaining boxes right here."},{"Start":"03:12.395 ","End":"03:16.020","Text":"This then would be the chances."}],"ID":12865},{"Watched":false,"Name":"Exercise 21 - Parts a-c","Duration":"4m 23s","ChapterTopicVideoID":12387,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12387.jpeg","UploadDate":"2019-01-31T23:25:17.0300000","DurationForVideoObject":"PT4M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.715","Text":"In this question, we have 6 finalists in a race."},{"Start":"00:02.715 ","End":"00:06.015","Text":"Only the top 3 finalists receive medals."},{"Start":"00:06.015 ","End":"00:09.753","Text":"We assume that all finalists finishes the race."},{"Start":"00:09.753 ","End":"00:11.520","Text":"We\u0027re asked, what\u0027s the total number of"},{"Start":"00:11.520 ","End":"00:14.670","Text":"combinations for the finalists to complete the race?"},{"Start":"00:14.670 ","End":"00:16.320","Text":"Here we have to pay attention."},{"Start":"00:16.320 ","End":"00:19.050","Text":"We\u0027re not asking about the chances,"},{"Start":"00:19.050 ","End":"00:20.280","Text":"so the probability,"},{"Start":"00:20.280 ","End":"00:22.950","Text":"but the total number of combinations."},{"Start":"00:22.950 ","End":"00:25.335","Text":"Here we\u0027re just counting."},{"Start":"00:25.335 ","End":"00:27.825","Text":"This is pure combinatorics."},{"Start":"00:27.825 ","End":"00:31.215","Text":"We don\u0027t have to divide all the answers."},{"Start":"00:31.215 ","End":"00:33.150","Text":"but the size of the sample space,"},{"Start":"00:33.150 ","End":"00:36.415","Text":"we don\u0027t even have to calculate the sample space."},{"Start":"00:36.415 ","End":"00:38.070","Text":"The question here is,"},{"Start":"00:38.070 ","End":"00:41.785","Text":"what\u0027s the total number of combinations for the finalist to complete the race?"},{"Start":"00:41.785 ","End":"00:45.030","Text":"We have 6 finalists,"},{"Start":"00:45.030 ","End":"00:47.520","Text":"and they all finished."},{"Start":"00:47.520 ","End":"00:52.358","Text":"All we have to do is just order the 6 finalists."},{"Start":"00:52.358 ","End":"00:56.820","Text":"How many ways can we order 6 people?"},{"Start":"00:56.820 ","End":"01:02.730","Text":"6 factorial, and that comes out to 720."},{"Start":"01:02.730 ","End":"01:09.905","Text":"We have 720 ways for the 6 finalists to finish the race."},{"Start":"01:09.905 ","End":"01:11.420","Text":"In this section, we\u0027re asked,"},{"Start":"01:11.420 ","End":"01:16.135","Text":"what are the number of combinations where finalist number 6 receives a medal?"},{"Start":"01:16.135 ","End":"01:18.120","Text":"It really doesn\u0027t matter which one."},{"Start":"01:18.120 ","End":"01:19.250","Text":"It could be the gold,"},{"Start":"01:19.250 ","End":"01:21.545","Text":"silver, or bronze medal here."},{"Start":"01:21.545 ","End":"01:23.705","Text":"Let\u0027s choose a medal,"},{"Start":"01:23.705 ","End":"01:26.885","Text":"out of the 3 medals we have to choose 1,"},{"Start":"01:26.885 ","End":"01:31.895","Text":"and that medal will be given to finalist number 6."},{"Start":"01:31.895 ","End":"01:36.693","Text":"Apart from that, we still have 5 other people that we have to order."},{"Start":"01:36.693 ","End":"01:39.365","Text":"So that\u0027s 5 factorial."},{"Start":"01:39.365 ","End":"01:43.100","Text":"The total number of combinations where"},{"Start":"01:43.100 ","End":"01:47.755","Text":"finalist number 6 receives a medal really doesn\u0027t matter which one,"},{"Start":"01:47.755 ","End":"01:53.940","Text":"and we have the other 5 people finishing the race in any which order,"},{"Start":"01:53.940 ","End":"01:59.250","Text":"that comes out to 360."},{"Start":"01:59.250 ","End":"02:01.160","Text":"In this section, we\u0027re asked what is the number of"},{"Start":"02:01.160 ","End":"02:03.770","Text":"combinations where finalist number 6 receives"},{"Start":"02:03.770 ","End":"02:08.810","Text":"a medal or a finalist number 2 receives the gold medal?"},{"Start":"02:08.810 ","End":"02:13.285","Text":"In this question, we need to define 2 events."},{"Start":"02:13.285 ","End":"02:17.720","Text":"Let\u0027s define A as finalist number 6"},{"Start":"02:17.720 ","End":"02:22.795","Text":"receiving a medal and B would be the event that number 2 receives the gold medal."},{"Start":"02:22.795 ","End":"02:28.265","Text":"We\u0027re looking for the number of combinations of A union B,"},{"Start":"02:28.265 ","End":"02:31.125","Text":"that\u0027s the size of A union B."},{"Start":"02:31.125 ","End":"02:33.715","Text":"That number 6 receives the medal,"},{"Start":"02:33.715 ","End":"02:36.650","Text":"or that number 2 receives the gold medal."},{"Start":"02:36.650 ","End":"02:45.615","Text":"But this equals to the size of A plus the size of B minus the size of A intersect B."},{"Start":"02:45.615 ","End":"02:47.660","Text":"Well, the size of A,"},{"Start":"02:47.660 ","End":"02:50.075","Text":"we\u0027ve calculated that in the last section,"},{"Start":"02:50.075 ","End":"02:52.910","Text":"that\u0027s 3 times 5 factorial,"},{"Start":"02:52.910 ","End":"02:58.055","Text":"that\u0027s the number of combinations that number 6 receives a medal."},{"Start":"02:58.055 ","End":"03:03.575","Text":"Now, what about the number of combinations that number 2 receives the gold medal?"},{"Start":"03:03.575 ","End":"03:08.780","Text":"Well, if number 2 receives the gold medal then there\u0027s 1 medal to choose from,"},{"Start":"03:08.780 ","End":"03:14.250","Text":"that\u0027s the gold times now we have to order 5 people in a row,"},{"Start":"03:14.250 ","End":"03:16.590","Text":"so that\u0027ll be 5 factorial."},{"Start":"03:16.590 ","End":"03:21.590","Text":"Minus, well, what\u0027s the number of combinations of"},{"Start":"03:21.590 ","End":"03:26.810","Text":"the intersect where number 6 receives the medal and that number 2 receives the gold."},{"Start":"03:26.810 ","End":"03:28.760","Text":"Well, if number 2 receives the gold,"},{"Start":"03:28.760 ","End":"03:33.750","Text":"that\u0027s the 1 out of the 3 medals."},{"Start":"03:33.750 ","End":"03:41.190","Text":"Number 6 receives 1 of the remaining 2 combinations, 2 medals."},{"Start":"03:41.190 ","End":"03:44.220","Text":"Times now we have to order 4 people,"},{"Start":"03:44.220 ","End":"03:46.455","Text":"so that\u0027s 4 factorial."},{"Start":"03:46.455 ","End":"03:51.150","Text":"This then would be the size of the 3 events here,"},{"Start":"03:51.150 ","End":"03:54.500","Text":"A, B and A intersect B."},{"Start":"03:54.500 ","End":"03:58.520","Text":"Well, that comes out to 360,"},{"Start":"03:58.520 ","End":"04:02.360","Text":"that\u0027s 3 times 5 factorial plus 1 times 5 factorial,"},{"Start":"04:02.360 ","End":"04:08.090","Text":"that\u0027s a 120 minus 1 times 2 times 4 factorial, that\u0027s 48."},{"Start":"04:08.090 ","End":"04:12.050","Text":"Now that comes out to 432."},{"Start":"04:12.050 ","End":"04:17.480","Text":"This would be the number of combinations where finalist number 6"},{"Start":"04:17.480 ","End":"04:23.620","Text":"receives a medal or finalist number 2 receives the gold medal."}],"ID":12866},{"Watched":false,"Name":"Exercise 22 - Part a","Duration":"5m 13s","ChapterTopicVideoID":12390,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12390.jpeg","UploadDate":"2019-01-31T23:26:08.2030000","DurationForVideoObject":"PT5M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this question, we have a fair die that\u0027s rolled k times and we\u0027re asked,"},{"Start":"00:03.930 ","End":"00:07.760","Text":"what are the odds that the highest value is j?"},{"Start":"00:07.760 ","End":"00:10.940","Text":"We\u0027re rolling the die k times,"},{"Start":"00:10.940 ","End":"00:12.150","Text":"so this is the first roll,"},{"Start":"00:12.150 ","End":"00:16.440","Text":"second roll, and so on and so forth, k times."},{"Start":"00:16.440 ","End":"00:18.930","Text":"Now, when we\u0027re asked about the odds,"},{"Start":"00:18.930 ","End":"00:23.100","Text":"well then we have to calculate the size of the sample space."},{"Start":"00:23.100 ","End":"00:25.890","Text":"Now if it\u0027s a die, I have 6 possible results,"},{"Start":"00:25.890 ","End":"00:28.065","Text":"and by rolling the die k times,"},{"Start":"00:28.065 ","End":"00:31.365","Text":"well I have 6^k possible results."},{"Start":"00:31.365 ","End":"00:35.445","Text":"The first roll I have 6 possible results to choose from."},{"Start":"00:35.445 ","End":"00:39.480","Text":"The second roll I have 6 possible results to choose from and so on and so forth,"},{"Start":"00:39.480 ","End":"00:44.715","Text":"and so it\u0027ll be 6 times 6 times 6 k times, so that\u0027s 6^k."},{"Start":"00:44.715 ","End":"00:51.230","Text":"Now, here we\u0027re asked what are the odds that the highest value is j?"},{"Start":"00:51.230 ","End":"00:53.615","Text":"Well, if the highest values j,"},{"Start":"00:53.615 ","End":"00:59.445","Text":"then I have 1-j possible results for each roll."},{"Start":"00:59.445 ","End":"01:02.410","Text":"That means that I have"},{"Start":"01:02.420 ","End":"01:12.200","Text":"j^k possible combinations of getting 1-j results in k rolls here."},{"Start":"01:12.200 ","End":"01:17.315","Text":"Just like we had here when we had 6^k,"},{"Start":"01:17.315 ","End":"01:20.570","Text":"where 6 was the total number of results that I have,"},{"Start":"01:20.570 ","End":"01:23.180","Text":"or here if j\u0027s the highest value,"},{"Start":"01:23.180 ","End":"01:27.800","Text":"then I have only j possible results to choose from."},{"Start":"01:27.800 ","End":"01:30.080","Text":"I\u0027m rolling the die k times,"},{"Start":"01:30.080 ","End":"01:31.580","Text":"so I have j^k."},{"Start":"01:31.580 ","End":"01:35.195","Text":"The probability here would be in the denominator,"},{"Start":"01:35.195 ","End":"01:38.006","Text":"where there\u0027ll be 6^k,"},{"Start":"01:38.006 ","End":"01:41.815","Text":"and in the numerator, there\u0027ll be j^k."},{"Start":"01:41.815 ","End":"01:47.550","Text":"The thing here is that since j has to be the highest value,"},{"Start":"01:47.550 ","End":"01:54.650","Text":"then j has to appear in the k rolls at least once for j to be the highest value."},{"Start":"01:54.650 ","End":"01:58.610","Text":"But what happens if within all the k rolls,"},{"Start":"01:58.610 ","End":"02:02.890","Text":"I get numbers which are less than j,"},{"Start":"02:02.890 ","End":"02:05.660","Text":"where all the numbers would be less than j."},{"Start":"02:05.660 ","End":"02:10.010","Text":"Well, I have to take away all the possible combinations of"},{"Start":"02:10.010 ","End":"02:14.930","Text":"k rolls where all the values here are less than j."},{"Start":"02:14.930 ","End":"02:18.980","Text":"So that means that I have to take away from"},{"Start":"02:18.980 ","End":"02:26.960","Text":"that j minus 1^k."},{"Start":"02:26.960 ","End":"02:30.065","Text":"Now, that\u0027s the probability and that\u0027s the answer."},{"Start":"02:30.065 ","End":"02:35.465","Text":"But let\u0027s just take a look at an example to get things a little bit clear here."},{"Start":"02:35.465 ","End":"02:39.585","Text":"Assume that I\u0027m rolling the die 10 times,"},{"Start":"02:39.585 ","End":"02:41.595","Text":"so k here would be 10,"},{"Start":"02:41.595 ","End":"02:45.540","Text":"and I want that the highest value j,"},{"Start":"02:45.540 ","End":"02:48.480","Text":"I want that to be 4."},{"Start":"02:48.480 ","End":"02:52.745","Text":"Now. That means that in the denominator,"},{"Start":"02:52.745 ","End":"02:56.330","Text":"I have 6^10,"},{"Start":"02:56.330 ","End":"03:00.255","Text":"because I\u0027m rolling a die and I\u0027m rolling it 10 times."},{"Start":"03:00.255 ","End":"03:02.160","Text":"J here that\u0027s 4,"},{"Start":"03:02.160 ","End":"03:07.550","Text":"what\u0027s the odds that the highest value\u0027s j. I have 1-4,"},{"Start":"03:07.550 ","End":"03:12.140","Text":"that\u0027s the number of results here that I can get for the first roll,"},{"Start":"03:12.140 ","End":"03:13.880","Text":"the second roll, and so on and so forth."},{"Start":"03:13.880 ","End":"03:17.765","Text":"How many possibilities do I have in the first roll to choose from?"},{"Start":"03:17.765 ","End":"03:20.750","Text":"Well, if the number here is from 1-4,"},{"Start":"03:20.750 ","End":"03:25.250","Text":"then I have 4 possible results to choose from here."},{"Start":"03:25.250 ","End":"03:29.495","Text":"Same thing here and so on and so forth for the 10 rolls."},{"Start":"03:29.495 ","End":"03:32.540","Text":"So that\u0027ll be 4^10."},{"Start":"03:32.540 ","End":"03:37.190","Text":"But I have to take away from that the following combinations."},{"Start":"03:37.190 ","End":"03:41.645","Text":"What happens if within the 10 rolls here,"},{"Start":"03:41.645 ","End":"03:44.750","Text":"I get numbers which are less than 4?"},{"Start":"03:44.750 ","End":"03:49.250","Text":"In order for 4 to be the maximum number,"},{"Start":"03:49.250 ","End":"03:53.049","Text":"then I need for 4 to be here at least once."},{"Start":"03:53.049 ","End":"04:00.045","Text":"What happens, let\u0027s say if I get all 1s in the 10 rolls or 1s and 2s and 3s?"},{"Start":"04:00.045 ","End":"04:07.460","Text":"How many possibilities do I have of getting results in which are less than 4?"},{"Start":"04:07.460 ","End":"04:12.495","Text":"Well, here then the possibilities would be 1, 2, and 3?"},{"Start":"04:12.495 ","End":"04:14.720","Text":"I have 3 possibilities here,"},{"Start":"04:14.720 ","End":"04:16.310","Text":"and here I\u0027d have 1, 2,"},{"Start":"04:16.310 ","End":"04:18.095","Text":"and 3, 3 possibilities here."},{"Start":"04:18.095 ","End":"04:20.730","Text":"I have to take away"},{"Start":"04:21.090 ","End":"04:28.695","Text":"3^10 possible combinations where all the numbers here are less than 4."},{"Start":"04:28.695 ","End":"04:31.425","Text":"Well, that\u0027s the same rationale here."},{"Start":"04:31.425 ","End":"04:35.660","Text":"This is just the general expression where this is an example,"},{"Start":"04:35.660 ","End":"04:39.125","Text":"j would be the maximum number."},{"Start":"04:39.125 ","End":"04:44.160","Text":"Here it was 4 here was j. I have 10 rolls here,"},{"Start":"04:44.160 ","End":"04:47.731","Text":"well here that\u0027s k rolls and 3,"},{"Start":"04:47.731 ","End":"04:50.450","Text":"that\u0027s j minus 1."},{"Start":"04:50.450 ","End":"04:57.440","Text":"I\u0027m taking away all the possible combinations where 4 isn\u0027t the highest value,"},{"Start":"04:57.440 ","End":"05:01.200","Text":"but a lower value than 4 is the highest value."},{"Start":"05:04.520 ","End":"05:08.900","Text":"So that\u0027s the expression for the probability that"},{"Start":"05:08.900 ","End":"05:14.279","Text":"the highest value is j when I roll the die, k times."}],"ID":12867},{"Watched":false,"Name":"Exercise 22 - Part b","Duration":"3m 23s","ChapterTopicVideoID":12388,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12388.jpeg","UploadDate":"2019-01-31T23:25:25.3970000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.200","Text":"In this question we\u0027re asked,"},{"Start":"00:01.200 ","End":"00:04.605","Text":"what are the odds that the lowest value is i?"},{"Start":"00:04.605 ","End":"00:08.340","Text":"Here we have the size of the sample space,"},{"Start":"00:08.340 ","End":"00:09.555","Text":"here we have the rows,"},{"Start":"00:09.555 ","End":"00:10.680","Text":"this is row number 1,"},{"Start":"00:10.680 ","End":"00:13.665","Text":"row number 2 and this is row number k. Now,"},{"Start":"00:13.665 ","End":"00:16.290","Text":"instead of going straight to the general equation here,"},{"Start":"00:16.290 ","End":"00:19.020","Text":"let\u0027s first look at an example."},{"Start":"00:19.020 ","End":"00:25.019","Text":"Now, assume that we roll the die 10 times and we want that the lowest value,"},{"Start":"00:25.019 ","End":"00:28.035","Text":"i, is equal to 3."},{"Start":"00:28.035 ","End":"00:30.495","Text":"What\u0027s the probability here?"},{"Start":"00:30.495 ","End":"00:33.960","Text":"Well, in the denominator we have the size of the sample space,"},{"Start":"00:33.960 ","End":"00:37.575","Text":"so that\u0027d be 6 to the power of 10, k equals 10."},{"Start":"00:37.575 ","End":"00:39.580","Text":"Now, what about the numerator?"},{"Start":"00:39.580 ","End":"00:43.719","Text":"Well, if 3 is the lowest number,"},{"Start":"00:43.719 ","End":"00:46.010","Text":"then in row number 1,"},{"Start":"00:46.010 ","End":"00:48.380","Text":"we have 4 numbers to choose from,"},{"Start":"00:48.380 ","End":"00:51.250","Text":"3, 4, 5 or 6."},{"Start":"00:51.250 ","End":"00:57.170","Text":"We have the same number to choose from in row number 2,"},{"Start":"00:57.170 ","End":"01:00.305","Text":"3, 4, 5 or 6 and so on and so forth."},{"Start":"01:00.305 ","End":"01:06.020","Text":"We have 4 to the power of 10 possibilities of getting 3,"},{"Start":"01:06.020 ","End":"01:08.810","Text":"4, 5 or 6 in 10 rows."},{"Start":"01:08.810 ","End":"01:16.910","Text":"But we have to take away from that the following combinations and what are they?"},{"Start":"01:16.910 ","End":"01:25.490","Text":"Well, for 3 to be the lowest value in the 10 rows then 3 has to appear at least once."},{"Start":"01:25.490 ","End":"01:31.015","Text":"That means that if 3 does not appear,"},{"Start":"01:31.015 ","End":"01:33.410","Text":"then in the first row,"},{"Start":"01:33.410 ","End":"01:35.330","Text":"then we have the values of 4,"},{"Start":"01:35.330 ","End":"01:37.775","Text":"5 or 6, 3 isn\u0027t there."},{"Start":"01:37.775 ","End":"01:39.980","Text":"In the second row,"},{"Start":"01:39.980 ","End":"01:41.840","Text":"we have the values 4,"},{"Start":"01:41.840 ","End":"01:43.520","Text":"5, or 6 as well."},{"Start":"01:43.520 ","End":"01:48.090","Text":"We have 3 values to the power of 10."},{"Start":"01:48.090 ","End":"01:56.145","Text":"This is the odds where the lowest value is 3."},{"Start":"01:56.145 ","End":"01:57.485","Text":"This is an example."},{"Start":"01:57.485 ","End":"02:01.520","Text":"Now let\u0027s take this to the general case here."},{"Start":"02:01.520 ","End":"02:06.830","Text":"In the denominator we have 6 to the power of k. Now,"},{"Start":"02:06.830 ","End":"02:09.305","Text":"in the numerator, well,"},{"Start":"02:09.305 ","End":"02:17.450","Text":"we have 7 minus i to the power of k. Now why is that?"},{"Start":"02:17.450 ","End":"02:19.205","Text":"Well, what\u0027s 7 minus i?"},{"Start":"02:19.205 ","End":"02:27.035","Text":"Well, that\u0027s the number of results from i to 6."},{"Start":"02:27.035 ","End":"02:31.210","Text":"We have 6 minus i plus 1."},{"Start":"02:31.210 ","End":"02:37.145","Text":"That\u0027s the number of possible results from i to 6."},{"Start":"02:37.145 ","End":"02:39.830","Text":"That\u0027s why we have 7 minus i."},{"Start":"02:39.830 ","End":"02:46.230","Text":"We have to take away from that 6 minus i to the power of"},{"Start":"02:46.230 ","End":"02:52.410","Text":"k and that\u0027s the number of results from i,"},{"Start":"02:52.410 ","End":"02:57.300","Text":"not including i, all the results above i up to 6."},{"Start":"02:57.300 ","End":"03:01.835","Text":"Here would be the number of results greater or equal to"},{"Start":"03:01.835 ","End":"03:10.110","Text":"i from i to 6 and here would be the number of results above i,"},{"Start":"03:10.110 ","End":"03:13.305","Text":"only greater than i up to 6."},{"Start":"03:13.305 ","End":"03:23.680","Text":"That would be the general equation for calculating the odds where the lowest value is i."}],"ID":12868},{"Watched":false,"Name":"Exercise 22 - Part c","Duration":"7m 41s","ChapterTopicVideoID":12389,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12389.jpeg","UploadDate":"2019-01-31T23:25:48.4430000","DurationForVideoObject":"PT7M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"In this section, we\u0027re asked what are the odds that"},{"Start":"00:02.010 ","End":"00:04.410","Text":"the highest values is J and the lowest value is I,"},{"Start":"00:04.410 ","End":"00:07.680","Text":"where I is less than or equal to J."},{"Start":"00:07.680 ","End":"00:10.710","Text":"Let\u0027s just recall that the size of the sample space"},{"Start":"00:10.710 ","End":"00:13.590","Text":"is 6 to power of K. Here are the rows,"},{"Start":"00:13.590 ","End":"00:15.465","Text":"row number 1, row number 2,"},{"Start":"00:15.465 ","End":"00:23.520","Text":"and so on and so forth until row number K. This is a pretty complicated questions."},{"Start":"00:23.520 ","End":"00:26.715","Text":"Let\u0027s first start with an example."},{"Start":"00:26.715 ","End":"00:36.920","Text":"Let\u0027s assume for the sake of the example that I equals to 2 and j equals to 5."},{"Start":"00:36.920 ","End":"00:41.390","Text":"The possible values for each row would be 2,"},{"Start":"00:41.390 ","End":"00:45.510","Text":"3, 4, and 5."},{"Start":"00:46.240 ","End":"00:48.650","Text":"If that\u0027s the case,"},{"Start":"00:48.650 ","End":"00:52.115","Text":"then let\u0027s try to calculate the odds right now."},{"Start":"00:52.115 ","End":"00:56.240","Text":"In the denominator, we have the size of the sample space,"},{"Start":"00:56.240 ","End":"00:59.090","Text":"that\u0027s 6 to the power of k. Now,"},{"Start":"00:59.090 ","End":"01:01.230","Text":"what about the numerator?"},{"Start":"01:01.540 ","End":"01:03.800","Text":"For each of the rows,"},{"Start":"01:03.800 ","End":"01:06.740","Text":"how many values do I have here?"},{"Start":"01:06.740 ","End":"01:08.870","Text":"Well, I have 4 values from 2-5,"},{"Start":"01:08.870 ","End":"01:09.980","Text":"that\u0027s 4 values, right,"},{"Start":"01:09.980 ","End":"01:11.630","Text":"2, 3, 4, and 5."},{"Start":"01:11.630 ","End":"01:16.085","Text":"I have 4 to the power of k. Again, why is that?"},{"Start":"01:16.085 ","End":"01:18.560","Text":"Well, I have 4 values for the first row,"},{"Start":"01:18.560 ","End":"01:20.510","Text":"4 values for the second row,"},{"Start":"01:20.510 ","End":"01:23.885","Text":"and so on and so forth, K times."},{"Start":"01:23.885 ","End":"01:28.180","Text":"That\u0027ll be 4 to the power of k. Now,"},{"Start":"01:28.180 ","End":"01:33.280","Text":"this number of combinations of k rows, well,"},{"Start":"01:33.280 ","End":"01:43.065","Text":"they also contain all the values where 2 is not included in the k rows."},{"Start":"01:43.065 ","End":"01:47.735","Text":"But in order for 2 to be the minimum value,"},{"Start":"01:47.735 ","End":"01:55.190","Text":"then that means that 2 has to be mentioned at least once in the K rows here."},{"Start":"01:55.190 ","End":"01:59.450","Text":"But again, there are rows where 2 does not appear."},{"Start":"01:59.450 ","End":"02:06.060","Text":"Well, we have 3 or 4 or 5 in all of the k rows here."},{"Start":"02:06.060 ","End":"02:12.230","Text":"We have to take away from all the 4 to the power of k possibilities."},{"Start":"02:12.230 ","End":"02:16.415","Text":"3_ power of k combinations,"},{"Start":"02:16.415 ","End":"02:22.695","Text":"where 2 is not in the rows here."},{"Start":"02:22.695 ","End":"02:25.620","Text":"We want 2 to be the minimum value,"},{"Start":"02:25.620 ","End":"02:30.275","Text":"so 2 has to appear at least once in the K rows."},{"Start":"02:30.275 ","End":"02:35.090","Text":"We have 3 to the power of k rows were 2 does not appear,"},{"Start":"02:35.090 ","End":"02:40.340","Text":"so we have to take that away from the 4 to the power of k. Now,"},{"Start":"02:40.340 ","End":"02:50.275","Text":"we also have to take away the number of combinations where 5 is not the maximum value."},{"Start":"02:50.275 ","End":"02:51.750","Text":"Again, why is that?"},{"Start":"02:51.750 ","End":"02:58.340","Text":"Well, 4 to the power of k includes all possible combinations of 2,"},{"Start":"02:58.340 ","End":"03:00.095","Text":"3, 4, and 5."},{"Start":"03:00.095 ","End":"03:08.125","Text":"But it also includes the combinations where 5 is not in any 1 of the k rows here."},{"Start":"03:08.125 ","End":"03:13.115","Text":"That means that 5 is no longer the maximum value."},{"Start":"03:13.115 ","End":"03:20.050","Text":"Again, the thing here is that 5 has to appear at least once in the K combinations here."},{"Start":"03:20.050 ","End":"03:23.450","Text":"If it doesn\u0027t, then we have to take all those combinations."},{"Start":"03:23.450 ","End":"03:29.510","Text":"Now how many combinations do we have where 5 is not the maximum value?"},{"Start":"03:29.510 ","End":"03:35.210","Text":"Well, that\u0027s 3 to the power of k. The thing is that if we"},{"Start":"03:35.210 ","End":"03:41.085","Text":"take out the number of combinations were 2 does not appears the minimum,"},{"Start":"03:41.085 ","End":"03:44.070","Text":"or 5 does not appear as the maximum,"},{"Start":"03:44.070 ","End":"03:54.685","Text":"then we have these 2 combinations right here that are subtracted twice."},{"Start":"03:54.685 ","End":"04:01.760","Text":"What we need to do is we need to add 2 to the power of k combinations,"},{"Start":"04:01.760 ","End":"04:05.580","Text":"where we\u0027ve subtracted these guys twice here."},{"Start":"04:05.580 ","End":"04:09.150","Text":"We\u0027ve subtracted these guys from 2 to 4,"},{"Start":"04:09.150 ","End":"04:12.525","Text":"and we\u0027ve subtracted these guys from 3 to 5."},{"Start":"04:12.525 ","End":"04:16.155","Text":"That means 3 and 4 were subtracted twice."},{"Start":"04:16.155 ","End":"04:20.015","Text":"We\u0027re adding 2to the power of k,"},{"Start":"04:20.015 ","End":"04:27.850","Text":"or the possible combinations where 3 and 4 had been subtracted twice."},{"Start":"04:27.850 ","End":"04:31.865","Text":"This was the example."},{"Start":"04:31.865 ","End":"04:40.030","Text":"Now let\u0027s take a look at the actual general equation for the probability."},{"Start":"04:40.030 ","End":"04:42.680","Text":"In the general case, what do we have?"},{"Start":"04:42.680 ","End":"04:46.915","Text":"Well, I is the lowest value and j is the highest value."},{"Start":"04:46.915 ","End":"04:50.820","Text":"In essence I have I plus 1,"},{"Start":"04:50.820 ","End":"04:52.830","Text":"and so on and so forth until j."},{"Start":"04:52.830 ","End":"04:56.665","Text":"These are the possible values that I have for each row."},{"Start":"04:56.665 ","End":"05:00.305","Text":"Let\u0027s now calculate the general probability here."},{"Start":"05:00.305 ","End":"05:03.620","Text":"In the denominator, we have 6_ power of k,"},{"Start":"05:03.620 ","End":"05:05.720","Text":"that\u0027s the size of the sample space."},{"Start":"05:05.720 ","End":"05:11.390","Text":"Now, how many numbers do I have between I and j?"},{"Start":"05:11.390 ","End":"05:16.380","Text":"Well, I have j minus I plus 1 numbers."},{"Start":"05:16.380 ","End":"05:22.725","Text":"Each 1 of the rows has j minus I plus 1 possibilities."},{"Start":"05:22.725 ","End":"05:27.890","Text":"I have here j minus I plus 1_ power of k. But that"},{"Start":"05:27.890 ","End":"05:33.485","Text":"includes combinations that I don\u0027t want."},{"Start":"05:33.485 ","End":"05:36.815","Text":"What are the combinations that I don\u0027t want?"},{"Start":"05:36.815 ","End":"05:42.660","Text":"Well, 1 of them is where I is not the lowest numbers."},{"Start":"05:42.660 ","End":"05:44.270","Text":"In 1 of the combinations,"},{"Start":"05:44.270 ","End":"05:49.675","Text":"I could have all the numbers from I plus 1 to j."},{"Start":"05:49.675 ","End":"05:54.170","Text":"Now, I want to get rid of all these combinations because"},{"Start":"05:54.170 ","End":"05:59.040","Text":"the lowest number I has to appear at least once."},{"Start":"05:59.040 ","End":"06:03.440","Text":"How many numbers do I have here from I plus 1 to j?"},{"Start":"06:03.440 ","End":"06:09.980","Text":"Well, that\u0027ll be j minus I to the power of k. That\u0027s the number"},{"Start":"06:09.980 ","End":"06:17.885","Text":"of combinations of k rows where I is not the lowest number."},{"Start":"06:17.885 ","End":"06:27.005","Text":"Now, the other family of combinations is where j is not the largest number."},{"Start":"06:27.005 ","End":"06:29.525","Text":"Now how many numbers do I have here?"},{"Start":"06:29.525 ","End":"06:33.050","Text":"Well, I have j minus I numbers,"},{"Start":"06:33.050 ","End":"06:37.280","Text":"from I to j minus 1 to the power of k."},{"Start":"06:37.280 ","End":"06:43.265","Text":"That\u0027s the number of combinations where j does not appear at all."},{"Start":"06:43.265 ","End":"06:47.420","Text":"J needs to appear at least once for it to be the highest number."},{"Start":"06:47.420 ","End":"06:48.800","Text":"If it doesn\u0027t appear,"},{"Start":"06:48.800 ","End":"06:52.895","Text":"then we have to take all the numbers were j does not appear."},{"Start":"06:52.895 ","End":"06:59.555","Text":"We have to take these combinations out of the total number of combinations that we have."},{"Start":"06:59.555 ","End":"07:05.915","Text":"Now, what about all the numbers in the middle where we\u0027ve taken them out twice."},{"Start":"07:05.915 ","End":"07:07.730","Text":"We have to add them on once."},{"Start":"07:07.730 ","End":"07:11.495","Text":"Now how many numbers do I have that are in the middle?"},{"Start":"07:11.495 ","End":"07:19.810","Text":"Well, I have j minus I minus 1 numbers that are in the middle to the power of k."},{"Start":"07:19.810 ","End":"07:24.500","Text":"That\u0027s the number of combinations that I need to add because I\u0027ve"},{"Start":"07:24.500 ","End":"07:29.690","Text":"taken them out twice when I subtracted these guys right here."},{"Start":"07:29.690 ","End":"07:33.980","Text":"This then would be the general formula for"},{"Start":"07:33.980 ","End":"07:38.675","Text":"calculating the odds that the highest value is J and lowest value is I,"},{"Start":"07:38.675 ","End":"07:42.180","Text":"when I row a die k times."}],"ID":12869},{"Watched":false,"Name":"Exercise 23","Duration":"8m 16s","ChapterTopicVideoID":28549,"CourseChapterTopicPlaylistID":264992,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/28549.jpeg","UploadDate":"2022-02-15T19:16:19.0900000","DurationForVideoObject":"PT8M16S","Description":null,"VideoComments":[],"Subtitles":[],"ID":30069}],"Thumbnail":null,"ID":264992},{"Name":"Binomial Expansion","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Binomial Expansion","Duration":"13m 1s","ChapterTopicVideoID":26106,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26106.jpeg","UploadDate":"2021-08-04T16:29:22.0700000","DurationForVideoObject":"PT13M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"I\u0027m beginning a new topic, the binomial theorem."},{"Start":"00:04.380 ","End":"00:11.710","Text":"This is usually due to the famous Sir Isaac Newton."},{"Start":"00:11.980 ","End":"00:16.895","Text":"We\u0027ll start with something called binomial expansion."},{"Start":"00:16.895 ","End":"00:19.310","Text":"You\u0027ve seen this before."},{"Start":"00:19.310 ","End":"00:23.540","Text":"We want to evaluate a plus b to various powers."},{"Start":"00:23.540 ","End":"00:25.385","Text":"I\u0027ll start with one in the middle,"},{"Start":"00:25.385 ","End":"00:27.845","Text":"a plus b squared."},{"Start":"00:27.845 ","End":"00:36.080","Text":"We know that this is equal to a squared plus 2ab plus b squared."},{"Start":"00:36.080 ","End":"00:40.460","Text":"Then there\u0027s a few more, a plus b^1."},{"Start":"00:40.460 ","End":"00:46.120","Text":"This is trivial, this is just a plus b itself."},{"Start":"00:46.120 ","End":"00:48.690","Text":"We\u0027ll actually start from 0,"},{"Start":"00:48.690 ","End":"00:52.695","Text":"so a plus b^0 is just 1."},{"Start":"00:52.695 ","End":"00:55.049","Text":"How about a plus b cubed?"},{"Start":"00:55.049 ","End":"00:57.175","Text":"You may or may not remember this,"},{"Start":"00:57.175 ","End":"01:01.480","Text":"but we can certainly derive it if we just multiply this by this."},{"Start":"01:01.480 ","End":"01:03.425","Text":"Anyway, the answer is"},{"Start":"01:03.425 ","End":"01:13.695","Text":"a cubed plus 3a square b plus 3ab squared plus b cubed."},{"Start":"01:13.695 ","End":"01:21.160","Text":"In general, our main task is to find some formula for a plus b"},{"Start":"01:21.160 ","End":"01:29.200","Text":"raised to any power n. N is a positive whole number or at least non-negative,"},{"Start":"01:29.200 ","End":"01:31.900","Text":"and see if we can find a pattern,"},{"Start":"01:31.900 ","End":"01:34.255","Text":"a formula for doing this."},{"Start":"01:34.255 ","End":"01:37.295","Text":"Now, there are patterns here."},{"Start":"01:37.295 ","End":"01:41.940","Text":"If we just study even these first 4 rows,"},{"Start":"01:41.940 ","End":"01:44.470","Text":"we can say various things."},{"Start":"01:44.470 ","End":"01:48.310","Text":"First of all, that if this is n, n is 0, 1,"},{"Start":"01:48.310 ","End":"01:49.630","Text":"2, 3,"},{"Start":"01:49.630 ","End":"01:52.675","Text":"we can see that there are n plus 1 terms."},{"Start":"01:52.675 ","End":"01:56.230","Text":"For example, here with a plus b squared,"},{"Start":"01:56.230 ","End":"01:58.540","Text":"there are 1, 2, 3 terms."},{"Start":"01:58.540 ","End":"02:01.900","Text":"Terms are what are separated by plus or minus."},{"Start":"02:01.900 ","End":"02:05.590","Text":"Here there\u0027s 1 term."},{"Start":"02:07.490 ","End":"02:09.780","Text":"Here we have 2,"},{"Start":"02:09.780 ","End":"02:11.175","Text":"here we have 3,"},{"Start":"02:11.175 ","End":"02:13.170","Text":"here we have 4."},{"Start":"02:13.170 ","End":"02:16.795","Text":"If we write what n is, n here is,"},{"Start":"02:16.795 ","End":"02:18.955","Text":"which is the coefficient is 0,"},{"Start":"02:18.955 ","End":"02:22.125","Text":"1, 2, 3."},{"Start":"02:22.125 ","End":"02:26.120","Text":"We want to generalize afterwards to a general n,"},{"Start":"02:26.120 ","End":"02:32.315","Text":"but we see that there are n plus 1 terms."},{"Start":"02:32.315 ","End":"02:38.570","Text":"We can also see that they all begin and end with a 1."},{"Start":"02:38.570 ","End":"02:42.485","Text":"At least the coefficients do because I could put 1 here."},{"Start":"02:42.485 ","End":"02:49.200","Text":"We don\u0027t do that, but they would all start and end in a 1 something."},{"Start":"02:51.380 ","End":"02:57.905","Text":"Well, let\u0027s take this here it\u0027s easier to see, the as decrease."},{"Start":"02:57.905 ","End":"02:59.750","Text":"If we look here at the powers of a,"},{"Start":"02:59.750 ","End":"03:01.445","Text":"here we have a squared."},{"Start":"03:01.445 ","End":"03:04.390","Text":"Here we have a which is a^1,"},{"Start":"03:04.390 ","End":"03:06.750","Text":"and here we don\u0027t have any a at all."},{"Start":"03:06.750 ","End":"03:11.460","Text":"You\u0027d always imagine there\u0027s like an a^0."},{"Start":"03:11.460 ","End":"03:16.830","Text":"Powers of a, they decrease by 1 each"},{"Start":"03:16.830 ","End":"03:23.340","Text":"time from n down to 0."},{"Start":"03:23.340 ","End":"03:26.805","Text":"Here if n is 2, we start off with a squared, a^1, a^0."},{"Start":"03:26.805 ","End":"03:29.100","Text":"Here, a cubed, a squared,"},{"Start":"03:29.100 ","End":"03:32.085","Text":"a^1, a^0. There is no a."},{"Start":"03:32.085 ","End":"03:38.880","Text":"The powers of b increase from 0"},{"Start":"03:38.880 ","End":"03:47.070","Text":"up to n. Here we have a b^0 here,"},{"Start":"03:47.070 ","End":"03:49.215","Text":"b^1, b squared,"},{"Start":"03:49.215 ","End":"03:52.805","Text":"b cubed, and n here is 3."},{"Start":"03:52.805 ","End":"03:56.515","Text":"The pattern is that there\u0027s a coefficient and then"},{"Start":"03:56.515 ","End":"04:00.310","Text":"powers of a that decrease and powers of b that increase."},{"Start":"04:00.310 ","End":"04:03.850","Text":"Notice also that if you add the powers of a and b,"},{"Start":"04:03.850 ","End":"04:06.475","Text":"like here if you add the 2 and the 1,"},{"Start":"04:06.475 ","End":"04:09.720","Text":"you get 3 and this is true throughout,"},{"Start":"04:09.720 ","End":"04:11.475","Text":"3 plus 0 is 3,"},{"Start":"04:11.475 ","End":"04:15.065","Text":"2 plus 1 is 3, 1 plus 2 is 3,"},{"Start":"04:15.065 ","End":"04:17.055","Text":"0 plus 3 is 3."},{"Start":"04:17.055 ","End":"04:19.640","Text":"It\u0027s always a to the something,"},{"Start":"04:19.640 ","End":"04:21.760","Text":"call it k,"},{"Start":"04:21.760 ","End":"04:25.530","Text":"or let\u0027s make the power of b,"},{"Start":"04:25.530 ","End":"04:29.405","Text":"let\u0027s call that k. Here we\u0027d have n minus k."},{"Start":"04:29.405 ","End":"04:36.350","Text":"The total exponent is just n. This is k. This will be n minus k. This is 2,"},{"Start":"04:36.350 ","End":"04:38.645","Text":"this will be 3 minus 2 and so on."},{"Start":"04:38.645 ","End":"04:45.290","Text":"That\u0027s basically the pattern we see and we have some coefficient in front."},{"Start":"04:45.290 ","End":"04:47.285","Text":"Each term has a coefficient,"},{"Start":"04:47.285 ","End":"04:49.265","Text":"a to the something, b to the something,"},{"Start":"04:49.265 ","End":"04:51.500","Text":"the as decrease, the bs decrease."},{"Start":"04:51.500 ","End":"04:56.995","Text":"Now, let\u0027s try and formalize this or make it more precise."},{"Start":"04:56.995 ","End":"05:02.060","Text":"I\u0027m going to introduce something called Pascal\u0027s triangle."},{"Start":"05:02.060 ","End":"05:05.345","Text":"I don\u0027t want to lose this."},{"Start":"05:05.345 ","End":"05:09.095","Text":"I copied it over here."},{"Start":"05:09.095 ","End":"05:12.210","Text":"Now, we can scroll a bit."},{"Start":"05:16.990 ","End":"05:20.405","Text":"Let\u0027s look at this one just for example,"},{"Start":"05:20.405 ","End":"05:23.150","Text":"some number, a coefficient,"},{"Start":"05:23.150 ","End":"05:26.560","Text":"a power of a and a power of b,"},{"Start":"05:26.560 ","End":"05:30.110","Text":"and the sum of the powers together is n. In this case,"},{"Start":"05:30.110 ","End":"05:33.590","Text":"n is 3, so it\u0027s a squared b^1."},{"Start":"05:33.590 ","End":"05:37.085","Text":"What we\u0027re interested in is,"},{"Start":"05:37.085 ","End":"05:38.990","Text":"what is this coefficient?"},{"Start":"05:38.990 ","End":"05:42.290","Text":"Everything else can be predictably generated."},{"Start":"05:42.290 ","End":"05:47.660","Text":"I could immediately guess the next row as a^4,"},{"Start":"05:47.660 ","End":"05:50.405","Text":"then something a cubed b,"},{"Start":"05:50.405 ","End":"05:52.580","Text":"and so on but the numbers,"},{"Start":"05:52.580 ","End":"05:56.310","Text":"the coefficients, that\u0027s the hard part."},{"Start":"05:56.530 ","End":"06:03.875","Text":"We give them a term and this is a definition that this coefficient here,"},{"Start":"06:03.875 ","End":"06:05.830","Text":"we call it,"},{"Start":"06:05.830 ","End":"06:09.080","Text":"it\u0027s often pronounced n choose k"},{"Start":"06:09.080 ","End":"06:11.840","Text":"because it comes from the world of combinatorics where we\u0027re"},{"Start":"06:11.840 ","End":"06:17.990","Text":"choosing certain number of items from a set of items."},{"Start":"06:17.990 ","End":"06:20.105","Text":"Anyway, I don\u0027t want to get into the combinatorics."},{"Start":"06:20.105 ","End":"06:25.820","Text":"You can pronounce it n over k or just n k. I\u0027ll say n choose k. That\u0027s"},{"Start":"06:25.820 ","End":"06:34.120","Text":"the coefficient of the term which has b^k,"},{"Start":"06:34.120 ","End":"06:40.940","Text":"where k is the lower number and it\u0027s in the expansion of a plus b to the n,"},{"Start":"06:40.940 ","End":"06:44.270","Text":"so that the power of a would be n minus k,"},{"Start":"06:44.270 ","End":"06:46.970","Text":"because together they make up n. This number is n"},{"Start":"06:46.970 ","End":"06:52.535","Text":"choose k. There\u0027s something which is what I call, off-by-one."},{"Start":"06:52.535 ","End":"06:55.550","Text":"There\u0027s a difference between the k,"},{"Start":"06:55.550 ","End":"06:59.880","Text":"which is the power of b and the position."},{"Start":"06:59.900 ","End":"07:02.585","Text":"If you do computer science,"},{"Start":"07:02.585 ","End":"07:04.670","Text":"you might start counting from 0,"},{"Start":"07:04.670 ","End":"07:06.725","Text":"but if we count from 1,"},{"Start":"07:06.725 ","End":"07:12.635","Text":"then the 1st term has b^0 here."},{"Start":"07:12.635 ","End":"07:15.175","Text":"The 2nd term has b^1."},{"Start":"07:15.175 ","End":"07:17.429","Text":"The 3rd term has b squared."},{"Start":"07:17.429 ","End":"07:19.814","Text":"The 4th term has b cubed."},{"Start":"07:19.814 ","End":"07:21.510","Text":"For the 3rd term,"},{"Start":"07:21.510 ","End":"07:23.535","Text":"k is 2,"},{"Start":"07:23.535 ","End":"07:26.145","Text":"or if you specify k,"},{"Start":"07:26.145 ","End":"07:30.785","Text":"let\u0027s say as 4 to maybe be in a future row,"},{"Start":"07:30.785 ","End":"07:33.035","Text":"then it\u0027s the 5th term."},{"Start":"07:33.035 ","End":"07:39.785","Text":"The k is 1 less than the term number because we start from k equals 0."},{"Start":"07:39.785 ","End":"07:44.015","Text":"This n choose k is the coefficient of,"},{"Start":"07:44.015 ","End":"07:46.010","Text":"instead of saying the coefficient of this,"},{"Start":"07:46.010 ","End":"07:49.720","Text":"the coefficient of the k plus 1st term,"},{"Start":"07:49.720 ","End":"07:52.770","Text":"the plus 1 reflects this off-by-one."},{"Start":"07:52.770 ","End":"07:56.705","Text":"Alternatively, if you want the kth term,"},{"Start":"07:56.705 ","End":"08:03.150","Text":"then the coefficient is n choose k minus 1."},{"Start":"08:03.150 ","End":"08:05.960","Text":"Now I\u0027ll give examples in a moment,"},{"Start":"08:05.960 ","End":"08:08.900","Text":"but each of these n choose k is one of these numbers,"},{"Start":"08:08.900 ","End":"08:10.190","Text":"1, 1, 1, 1,"},{"Start":"08:10.190 ","End":"08:12.680","Text":"2, 1, 1, 3, 3, 1."},{"Start":"08:12.680 ","End":"08:16.100","Text":"Also we\u0027re going to put these in a triangle and get Pascal\u0027s triangle."},{"Start":"08:16.100 ","End":"08:19.890","Text":"Let\u0027s just get a bit more space here."},{"Start":"08:22.070 ","End":"08:24.820","Text":"There\u0027s one other thing before the triangles,"},{"Start":"08:24.820 ","End":"08:27.440","Text":"and that\u0027s the other notation."},{"Start":"08:27.570 ","End":"08:31.885","Text":"Instead of writing n choose k this way,"},{"Start":"08:31.885 ","End":"08:40.284","Text":"it\u0027s sometimes in the world of combinatorics written as a C. C for combinations actually."},{"Start":"08:40.284 ","End":"08:46.000","Text":"Then the n here and the k here and very importantly,"},{"Start":"08:46.000 ","End":"08:50.409","Text":"there is a formula that computes these."},{"Start":"08:50.409 ","End":"08:53.200","Text":"If I want to know what is the value of n choose k,"},{"Start":"08:53.200 ","End":"08:55.180","Text":"you plug it in here,"},{"Start":"08:55.180 ","End":"08:58.180","Text":"we will see examples and how this relates."},{"Start":"08:58.180 ","End":"09:02.310","Text":"It\u0027ll all tie together very shortly."},{"Start":"09:02.310 ","End":"09:05.025","Text":"That\u0027s the notation, that\u0027s the formula."},{"Start":"09:05.025 ","End":"09:06.855","Text":"Now let\u0027s see in practice."},{"Start":"09:06.855 ","End":"09:10.159","Text":"If you go back, we got up to here."},{"Start":"09:10.159 ","End":"09:14.470","Text":"We just take the coefficients without the a to the something,"},{"Start":"09:14.470 ","End":"09:16.040","Text":"b to the something."},{"Start":"09:16.040 ","End":"09:19.190","Text":"What we had here is 1,"},{"Start":"09:19.190 ","End":"09:20.930","Text":"then 1, 1, then 1,"},{"Start":"09:20.930 ","End":"09:22.910","Text":"2, 1 then 1, 3, 3,"},{"Start":"09:22.910 ","End":"09:26.749","Text":"1, which is just like what\u0027s here."},{"Start":"09:26.749 ","End":"09:30.710","Text":"Let\u0027s just go back again, move on."},{"Start":"09:30.710 ","End":"09:34.985","Text":"In terms of this binomial coefficients symbol,"},{"Start":"09:34.985 ","End":"09:37.880","Text":"this 3 here for example,"},{"Start":"09:37.880 ","End":"09:40.525","Text":"is 3 choose 1."},{"Start":"09:40.525 ","End":"09:43.620","Text":"Because we start from 0,"},{"Start":"09:43.620 ","End":"09:45.440","Text":"the k is what\u0027s underneath."},{"Start":"09:45.440 ","End":"09:49.820","Text":"This is a squared b^1."},{"Start":"09:49.820 ","End":"09:52.875","Text":"B is to the power of 1, so it\u0027s 3 choose 1."},{"Start":"09:52.875 ","End":"09:54.090","Text":"I\u0027m just saying for example,"},{"Start":"09:54.090 ","End":"09:57.045","Text":"this one would correspond to this."},{"Start":"09:57.045 ","End":"10:01.950","Text":"Here it\u0027s 1 choose 0 and 1 choose 1."},{"Start":"10:01.950 ","End":"10:09.780","Text":"Here remember it was a squared plus 2ab plus b squared,"},{"Start":"10:09.780 ","End":"10:12.820","Text":"and there\u0027s a 1 here and a 1 here."},{"Start":"10:13.400 ","End":"10:16.260","Text":"This is this 1 here,"},{"Start":"10:16.260 ","End":"10:17.955","Text":"this is this 2 here."},{"Start":"10:17.955 ","End":"10:20.790","Text":"Here it corresponds to 2,"},{"Start":"10:20.790 ","End":"10:24.210","Text":"2, and 2 at the top because it\u0027s all squared."},{"Start":"10:24.210 ","End":"10:32.715","Text":"The 0, 1, 2 is the exponent of b, b^0."},{"Start":"10:32.715 ","End":"10:36.420","Text":"We can add in here a b^0."},{"Start":"10:36.420 ","End":"10:39.720","Text":"Well, anyway that\u0027s the 2 and that\u0027s the 2."},{"Start":"10:39.720 ","End":"10:41.835","Text":"Just study it for a while."},{"Start":"10:41.835 ","End":"10:45.770","Text":"These are the actual results of the coefficients."},{"Start":"10:45.770 ","End":"10:51.410","Text":"Here, this is the same numbers but expressed in this notation."},{"Start":"10:51.410 ","End":"10:54.470","Text":"Now I haven\u0027t shown you that this formula works."},{"Start":"10:54.470 ","End":"10:56.870","Text":"We\u0027ll see many examples of this later."},{"Start":"10:56.870 ","End":"10:59.160","Text":"So far we\u0027ve just computed up to the 3rd row."},{"Start":"10:59.160 ","End":"11:01.085","Text":"Let\u0027s take this one as an example."},{"Start":"11:01.085 ","End":"11:03.020","Text":"Let\u0027s see if we can get this 3."},{"Start":"11:03.020 ","End":"11:12.004","Text":"This 3 comes from the power of b^1 in the expansion of a plus b cubed."},{"Start":"11:12.004 ","End":"11:13.430","Text":"This is what we want."},{"Start":"11:13.430 ","End":"11:18.150","Text":"So 3 choose 1,"},{"Start":"11:18.150 ","End":"11:20.940","Text":"according to this formula n is 3,"},{"Start":"11:20.940 ","End":"11:23.265","Text":"so it\u0027s 3 factorial."},{"Start":"11:23.265 ","End":"11:31.215","Text":"K is 1, so it\u0027s 1 factorial and n minus k is,"},{"Start":"11:31.215 ","End":"11:34.410","Text":"3 minus 1 is 2 factorial."},{"Start":"11:34.410 ","End":"11:38.505","Text":"Again, this 2 is 3 minus 1 from here."},{"Start":"11:38.505 ","End":"11:45.770","Text":"Now, 3 factorial is 6 because it\u0027s 3 times 2 times 1, this is 6."},{"Start":"11:45.770 ","End":"11:47.825","Text":"1 factorial is 1,"},{"Start":"11:47.825 ","End":"11:50.120","Text":"2 factorial is 2."},{"Start":"11:50.120 ","End":"11:53.195","Text":"6 over 1 times 2."},{"Start":"11:53.195 ","End":"12:03.200","Text":"This gives us equals 3 and that\u0027s the 3 that\u0027s here using this computation."},{"Start":"12:03.200 ","End":"12:09.590","Text":"Before we move on, just one other thing we should note is that there is a way of building"},{"Start":"12:09.590 ","End":"12:16.230","Text":"this table without computing a plus b^4,"},{"Start":"12:16.230 ","End":"12:17.665","Text":"then 5, then 6."},{"Start":"12:17.665 ","End":"12:20.360","Text":"Notice that if I have a certain row,"},{"Start":"12:20.360 ","End":"12:23.990","Text":"we\u0027ve got up to the row with 1, 3, 3, 1."},{"Start":"12:23.990 ","End":"12:29.090","Text":"If we add 3 plus 3, we get 6,"},{"Start":"12:29.090 ","End":"12:35.000","Text":"3 plus 1, we get 4, and here also."},{"Start":"12:35.000 ","End":"12:40.550","Text":"Each number in the following row is the sum of the 2 above it."},{"Start":"12:40.550 ","End":"12:42.185","Text":"1 plus 4 is 5,"},{"Start":"12:42.185 ","End":"12:43.820","Text":"4 plus 6 is 10,"},{"Start":"12:43.820 ","End":"12:46.475","Text":"6 plus 4 is 10, 4 plus 1 is 5."},{"Start":"12:46.475 ","End":"12:50.045","Text":"We always begin and end with a 1."},{"Start":"12:50.045 ","End":"12:52.130","Text":"Once we have a given row,"},{"Start":"12:52.130 ","End":"12:53.340","Text":"we put 1 at the beginning,"},{"Start":"12:53.340 ","End":"12:57.050","Text":"1 at the end, and all the other numbers are the sum of the 2 above."},{"Start":"12:57.050 ","End":"13:01.890","Text":"Let\u0027s take a break now and then continue with the binomial theorem."}],"ID":26927},{"Watched":false,"Name":"The Binomial Theorem","Duration":"17m 29s","ChapterTopicVideoID":26105,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26105.jpeg","UploadDate":"2021-08-04T16:25:00.5330000","DurationForVideoObject":"PT17M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.245","Text":"Continuing after the break, the binomial theorem."},{"Start":"00:04.245 ","End":"00:09.795","Text":"Now, it\u0027s time to reveal what this theorem actually is."},{"Start":"00:09.795 ","End":"00:14.520","Text":"Here it is. It\u0027s a formula that says that"},{"Start":"00:14.520 ","End":"00:19.410","Text":"when we take the binomial a plus b and raise it to the power of n,"},{"Start":"00:19.410 ","End":"00:22.965","Text":"where n is a non-negative integer,"},{"Start":"00:22.965 ","End":"00:26.160","Text":"then it\u0027s equal to the sum"},{"Start":"00:26.160 ","End":"00:34.175","Text":"of n choose k a^n minus k b^ k,"},{"Start":"00:34.175 ","End":"00:39.620","Text":"and k goes from 0 to n. This is the Sigma notation."},{"Start":"00:39.620 ","End":"00:43.295","Text":"In the examples, we\u0027ll also write it out in full."},{"Start":"00:43.295 ","End":"00:52.925","Text":"The only thing that\u0027s missing here that we did earlier is the definition."},{"Start":"00:52.925 ","End":"00:55.640","Text":"As I said, we\u0027ve already covered it,"},{"Start":"00:55.640 ","End":"00:57.950","Text":"but let me refresh,"},{"Start":"00:57.950 ","End":"01:01.440","Text":"n choose k,"},{"Start":"01:01.440 ","End":"01:03.860","Text":"or however you pronounce this symbol,"},{"Start":"01:03.860 ","End":"01:08.495","Text":"is equal to n factorial over"},{"Start":"01:08.495 ","End":"01:14.285","Text":"k factorial n minus k factorial."},{"Start":"01:14.285 ","End":"01:20.700","Text":"This together with this tells us how we expand a plus b to any power."},{"Start":"01:21.560 ","End":"01:24.710","Text":"Let\u0027s take n equals 4."},{"Start":"01:24.710 ","End":"01:28.580","Text":"We\u0027ve already done n goes from 0 to 3."},{"Start":"01:28.580 ","End":"01:32.390","Text":"If you go back, we did a plus b cubed in lower powers."},{"Start":"01:32.390 ","End":"01:36.085","Text":"Let\u0027s do a plus b to the 4th from the formula."},{"Start":"01:36.085 ","End":"01:44.605","Text":"First of all, I\u0027ll use the Sigma to say that\u0027s Sigma where k goes from 0-4."},{"Start":"01:44.605 ","End":"01:47.895","Text":"Remember, this means it\u0027s going to be 5 terms;"},{"Start":"01:47.895 ","End":"01:50.175","Text":"0, 1, 2, 3, 4."},{"Start":"01:50.175 ","End":"01:51.960","Text":"Its always 1 more. That\u0027s this."},{"Start":"01:51.960 ","End":"01:54.960","Text":"Off by 1 keeps recurring,"},{"Start":"01:54.960 ","End":"02:00.095","Text":"and n here is 4 choose k,"},{"Start":"02:00.095 ","End":"02:04.385","Text":"K is the index that varies from 0-4,"},{"Start":"02:04.385 ","End":"02:11.930","Text":"a^4 minus k b^k."},{"Start":"02:11.930 ","End":"02:15.890","Text":"The powers of b are the ones that increase from 0-4."},{"Start":"02:15.890 ","End":"02:20.705","Text":"Next step is to open up the Sigma."},{"Start":"02:20.705 ","End":"02:23.390","Text":"We have 5 terms, first one,"},{"Start":"02:23.390 ","End":"02:26.820","Text":"when k is 0, 4 choose 0,"},{"Start":"02:26.820 ","End":"02:34.145","Text":"a^4 minus 0 is 4 and b to the 0 we don\u0027t write it."},{"Start":"02:34.145 ","End":"02:36.710","Text":"I mean, always this comes out to be 1,"},{"Start":"02:36.710 ","End":"02:39.440","Text":"but at this stage you write it 4 choose 0."},{"Start":"02:39.440 ","End":"02:42.485","Text":"Next 4 choose 1,"},{"Start":"02:42.485 ","End":"02:44.735","Text":"a^4 minus 1,"},{"Start":"02:44.735 ","End":"02:47.210","Text":"the a\u0027s decrease and the b\u0027s increase."},{"Start":"02:47.210 ","End":"02:49.805","Text":"Here we have b^1 which is just b."},{"Start":"02:49.805 ","End":"02:52.615","Text":"Then 4 choose 2 a,"},{"Start":"02:52.615 ","End":"02:57.135","Text":"this time is squared and b increase is squared."},{"Start":"02:57.135 ","End":"03:01.275","Text":"Then 4 choose 3 a squared."},{"Start":"03:01.275 ","End":"03:06.765","Text":"It\u0027s down to a b increases the power of 3."},{"Start":"03:06.765 ","End":"03:12.375","Text":"Then the last one 4 choose 4 a^0."},{"Start":"03:12.375 ","End":"03:16.705","Text":"Just don\u0027t write it, and b^4."},{"Start":"03:16.705 ","End":"03:19.805","Text":"Of course, the first and the last they\u0027re 1."},{"Start":"03:19.805 ","End":"03:22.580","Text":"That\u0027s always the case."},{"Start":"03:22.580 ","End":"03:27.605","Text":"Anyway, let\u0027s now compute these numbers."},{"Start":"03:27.605 ","End":"03:30.740","Text":"It\u0027s really 2 ways of going about this."},{"Start":"03:30.740 ","End":"03:34.085","Text":"This one is just computing straight from the formula,"},{"Start":"03:34.085 ","End":"03:36.695","Text":"or we could use Pascal\u0027s triangle."},{"Start":"03:36.695 ","End":"03:41.282","Text":"Since we got the triangle up to the third row,"},{"Start":"03:41.282 ","End":"03:44.630","Text":"fourth row actually with the one with 3 in it."},{"Start":"03:44.630 ","End":"03:45.800","Text":"We had a 1,"},{"Start":"03:45.800 ","End":"03:48.760","Text":"3, 3, 1."},{"Start":"03:48.760 ","End":"03:50.990","Text":"Above that, we had 1,"},{"Start":"03:50.990 ","End":"03:54.430","Text":"2, 1, and so on."},{"Start":"03:54.430 ","End":"03:56.660","Text":"When we do the next row,"},{"Start":"03:56.660 ","End":"04:00.125","Text":"we start with a 1 and we end with a 1."},{"Start":"04:00.125 ","End":"04:04.243","Text":"Remember, each term now is the sum of the 2 above it."},{"Start":"04:04.243 ","End":"04:06.245","Text":"So 1 and 3 is 4,"},{"Start":"04:06.245 ","End":"04:08.720","Text":"3 and 3 is 6,"},{"Start":"04:08.720 ","End":"04:11.915","Text":"3 and 1 is 4."},{"Start":"04:11.915 ","End":"04:19.220","Text":"The other way to do this to compute these numbers is to use this binomial,"},{"Start":"04:19.220 ","End":"04:24.025","Text":"I mean, this formula for the binomial coefficient."},{"Start":"04:24.025 ","End":"04:26.330","Text":"We\u0027ll do it this way also,"},{"Start":"04:26.330 ","End":"04:28.910","Text":"but I won\u0027t bother with the first and the last."},{"Start":"04:28.910 ","End":"04:31.640","Text":"It works when k is 0 and when k is n,"},{"Start":"04:31.640 ","End":"04:35.070","Text":"but we\u0027ll just do these 3."},{"Start":"04:35.480 ","End":"04:43.330","Text":"4 choose 1 is 4 factorial over 1 factorial,"},{"Start":"04:43.330 ","End":"04:49.150","Text":"and 4 minus 1 is 3 factorial."},{"Start":"04:49.150 ","End":"04:52.885","Text":"That\u0027s here. I\u0027ll write them and then we\u0027ll compute them."},{"Start":"04:52.885 ","End":"04:58.600","Text":"4 choose 2 is 4 factorial over 2 factorial,"},{"Start":"04:58.600 ","End":"05:00.790","Text":"and then 4 minus 2,"},{"Start":"05:00.790 ","End":"05:03.425","Text":"which is also 2 factorial."},{"Start":"05:03.425 ","End":"05:08.665","Text":"Here 4 factorial over 3 factorial,"},{"Start":"05:08.665 ","End":"05:13.520","Text":"4 minus 3 is 1 factorial."},{"Start":"05:13.520 ","End":"05:17.565","Text":"Now 1 factorial is 1."},{"Start":"05:17.565 ","End":"05:19.805","Text":"Doesn\u0027t matter here and here."},{"Start":"05:19.805 ","End":"05:23.300","Text":"What we have, 4 factorial is 24,"},{"Start":"05:23.300 ","End":"05:25.220","Text":"3 factorial is 6."},{"Start":"05:25.220 ","End":"05:29.240","Text":"24 over 6 is 4."},{"Start":"05:29.240 ","End":"05:34.399","Text":"This one is 24 over 2 times 2."},{"Start":"05:34.399 ","End":"05:37.425","Text":"24 over 4 is 6."},{"Start":"05:37.425 ","End":"05:40.610","Text":"This is the same as this just different order."},{"Start":"05:40.610 ","End":"05:42.455","Text":"It\u0027s also 4."},{"Start":"05:42.455 ","End":"05:45.715","Text":"That\u0027s what we got here, here, and here."},{"Start":"05:45.715 ","End":"05:48.740","Text":"Pascal\u0027s triangle method and"},{"Start":"05:48.740 ","End":"05:54.325","Text":"the formula method give us the same numbers and that\u0027s reassuring."},{"Start":"05:54.325 ","End":"05:57.000","Text":"I cleared some space here."},{"Start":"05:57.000 ","End":"06:02.035","Text":"Now we can write the answer as a^4,"},{"Start":"06:02.035 ","End":"06:04.425","Text":"the 1 we don\u0027t write."},{"Start":"06:04.425 ","End":"06:07.335","Text":"Then from here, we have,"},{"Start":"06:07.335 ","End":"06:12.510","Text":"4a cubed b 4 over 2,"},{"Start":"06:12.510 ","End":"06:21.550","Text":"4 choose 2 is 6 a squared b squared."},{"Start":"06:21.740 ","End":"06:24.600","Text":"Then, again,"},{"Start":"06:24.600 ","End":"06:30.255","Text":"4 this term a b cubed."},{"Start":"06:30.255 ","End":"06:32.210","Text":"The last one, 1,"},{"Start":"06:32.210 ","End":"06:36.330","Text":"we don\u0027t write, is just b^4."},{"Start":"06:36.490 ","End":"06:43.955","Text":"So a plus b^4 is this."},{"Start":"06:43.955 ","End":"06:47.730","Text":"That\u0027s the example."},{"Start":"06:47.930 ","End":"06:54.740","Text":"Now let\u0027s look at some of the properties of the binomial coefficients."},{"Start":"06:54.740 ","End":"06:58.990","Text":"This also relates to Pascal\u0027s triangle."},{"Start":"06:58.990 ","End":"07:03.060","Text":"I\u0027ll show you 3 properties."},{"Start":"07:03.060 ","End":"07:11.255","Text":"The first property says n choose 0 is n choose n is 1."},{"Start":"07:11.255 ","End":"07:17.350","Text":"What this means is that the coefficients begin with 1 and end in 1."},{"Start":"07:17.350 ","End":"07:21.180","Text":"If n was like 4 here,"},{"Start":"07:21.180 ","End":"07:24.645","Text":"4 choose 0 is 1,"},{"Start":"07:24.645 ","End":"07:29.060","Text":"4 choose 4 is also 1."},{"Start":"07:29.060 ","End":"07:33.415","Text":"I mean, what this is saying is that these are all 1s,"},{"Start":"07:33.415 ","End":"07:37.535","Text":"and these are all 1s."},{"Start":"07:37.535 ","End":"07:40.605","Text":"That\u0027s the first property."},{"Start":"07:40.605 ","End":"07:47.230","Text":"The next property only even read it out is what is written."},{"Start":"07:47.230 ","End":"07:50.140","Text":"Essentially, this is the symmetry property."},{"Start":"07:50.140 ","End":"07:56.125","Text":"Notice that if I put a vertical line through the middle, it\u0027s symmetrical."},{"Start":"07:56.125 ","End":"08:00.220","Text":"In our case, for example,"},{"Start":"08:00.220 ","End":"08:03.325","Text":"this came out to be 4,"},{"Start":"08:03.325 ","End":"08:07.512","Text":"and this also came out to be 4."},{"Start":"08:07.512 ","End":"08:13.890","Text":"The middle one is its own mirror image if it\u0027s an odd number of terms,"},{"Start":"08:13.890 ","End":"08:18.150","Text":"the middle one but you notice that if you go from left to right,"},{"Start":"08:18.150 ","End":"08:20.310","Text":"1, 4, 6 4, 1, or from right to left,"},{"Start":"08:20.310 ","End":"08:22.560","Text":"1, 4, 6, 4, 1, it\u0027s the same."},{"Start":"08:22.560 ","End":"08:24.360","Text":"This is what this says."},{"Start":"08:24.360 ","End":"08:29.445","Text":"In particular example I chose n is 4, k is 1."},{"Start":"08:29.445 ","End":"08:36.960","Text":"It says that 4 choose 1 is the same as 4 choose 4 minus 1, which is 3."},{"Start":"08:36.960 ","End":"08:42.750","Text":"Other words, if I subtract the 1 from the 4 and get 3 and put 3 here instead,"},{"Start":"08:42.750 ","End":"08:44.340","Text":"it\u0027s going to be the same."},{"Start":"08:44.340 ","End":"08:49.240","Text":"This is essentially what the symmetry is."},{"Start":"08:49.730 ","End":"08:56.610","Text":"The terms of symmetrical if this plus this is 4 or n in general."},{"Start":"08:56.610 ","End":"09:00.090","Text":"Notice that 0 plus 4 is 4,"},{"Start":"09:00.090 ","End":"09:02.040","Text":"1 plus 3 is 4,"},{"Start":"09:02.040 ","End":"09:03.810","Text":"2 plus itself is 4."},{"Start":"09:03.810 ","End":"09:06.360","Text":"That\u0027s the symmetry condition and that\u0027s why it\u0027s n"},{"Start":"09:06.360 ","End":"09:09.525","Text":"minus k together this and this add up to n,"},{"Start":"09:09.525 ","End":"09:12.555","Text":"which means that they\u0027re symmetrically placed."},{"Start":"09:12.555 ","End":"09:15.600","Text":"As I said in the case of"},{"Start":"09:15.600 ","End":"09:19.710","Text":"Pascal\u0027s triangle you can look all over the place 5 is the same as 5,"},{"Start":"09:19.710 ","End":"09:21.225","Text":"10 is the same as 10,"},{"Start":"09:21.225 ","End":"09:25.530","Text":"28 here 28 here and so on and so on."},{"Start":"09:25.530 ","End":"09:32.250","Text":"The third property, this one won\u0027t even read it out just yet,"},{"Start":"09:32.250 ","End":"09:37.950","Text":"but this is the mathematical form of that property."},{"Start":"09:37.950 ","End":"09:39.660","Text":"Let\u0027s take an example."},{"Start":"09:39.660 ","End":"09:43.980","Text":"Let\u0027s say I put a triangle here,"},{"Start":"09:43.980 ","End":"09:49.005","Text":"notice that we said that each entry is the sum of the 2 above it."},{"Start":"09:49.005 ","End":"09:51.495","Text":"That this is equal to this plus this."},{"Start":"09:51.495 ","End":"09:53.130","Text":"Well, what are these?"},{"Start":"09:53.130 ","End":"09:57.345","Text":"This 21 is in row 7."},{"Start":"09:57.345 ","End":"09:59.670","Text":"It\u0027s the eighth row if you count this row,"},{"Start":"09:59.670 ","End":"10:00.690","Text":"but then I say row 7,"},{"Start":"10:00.690 ","End":"10:02.850","Text":"I mean where the 7 is here."},{"Start":"10:02.850 ","End":"10:05.100","Text":"What this says is that 21,"},{"Start":"10:05.100 ","End":"10:14.280","Text":"which is 7 and the number below is 1 mode,"},{"Start":"10:14.280 ","End":"10:18.720","Text":"or rather 1 less than, you count 1,"},{"Start":"10:18.720 ","End":"10:20.115","Text":"2, 3,"},{"Start":"10:20.115 ","End":"10:23.925","Text":"the third term, so the k is 2."},{"Start":"10:23.925 ","End":"10:25.890","Text":"Or you could start counting from 0."},{"Start":"10:25.890 ","End":"10:27.480","Text":"It\u0027s another way to do it, 0, 1,"},{"Start":"10:27.480 ","End":"10:29.070","Text":"2, so that\u0027s 7,"},{"Start":"10:29.070 ","End":"10:31.840","Text":"2 that\u0027s the 21."},{"Start":"10:32.000 ","End":"10:36.600","Text":"The 35 is the following term is"},{"Start":"10:36.600 ","End":"10:44.580","Text":"7 choose 3 because k is 0,"},{"Start":"10:44.580 ","End":"10:48.130","Text":"1, 2, 3 when it\u0027s just 1 more than the 2."},{"Start":"10:48.500 ","End":"10:58.620","Text":"The 56 is going to be the same index as the 35 because it\u0027s like parallel this is 0,"},{"Start":"10:58.620 ","End":"11:01.665","Text":"1, 2, 3, 0, 1, 2, 3."},{"Start":"11:01.665 ","End":"11:04.530","Text":"So that would be 8,"},{"Start":"11:04.530 ","End":"11:08.175","Text":"choose 3 because we\u0027re on the row below,"},{"Start":"11:08.175 ","End":"11:12.810","Text":"which is you can tell from the 8 here just like from the 7 here."},{"Start":"11:12.810 ","End":"11:17.250","Text":"So it\u0027s 8 from here and then 0,1,2,3 or you count 1,"},{"Start":"11:17.250 ","End":"11:19.766","Text":"2, 3, 4. but remember to subtract 1 so you get 3."},{"Start":"11:19.766 ","End":"11:23.625","Text":"That\u0027s 21, that\u0027s 35,"},{"Start":"11:23.625 ","End":"11:27.135","Text":"and this is 56."},{"Start":"11:27.135 ","End":"11:29.325","Text":"Just for extra practice,"},{"Start":"11:29.325 ","End":"11:30.795","Text":"let\u0027s compute them."},{"Start":"11:30.795 ","End":"11:33.480","Text":"7 choose 2."},{"Start":"11:33.480 ","End":"11:40.650","Text":"This would be 7 factorial over 2 factorial,"},{"Start":"11:40.650 ","End":"11:45.940","Text":"7 minus 2, which is 5 factorial."},{"Start":"11:49.130 ","End":"11:53.370","Text":"You can compute it with the calculator,"},{"Start":"11:53.370 ","End":"11:57.300","Text":"but in these cases it\u0027s best to just multiply out 7 factorial"},{"Start":"11:57.300 ","End":"12:03.150","Text":"7 times 6 times 5 times 4 times 3 times 1 times 1."},{"Start":"12:03.150 ","End":"12:07.920","Text":"Then 2 factorial is 2 times 1,"},{"Start":"12:07.920 ","End":"12:13.380","Text":"and 5 factorial is 5 times 4 times 3 times 2 times 1."},{"Start":"12:13.380 ","End":"12:16.590","Text":"Notice that the 5 factorial and 5 factorial"},{"Start":"12:16.590 ","End":"12:19.935","Text":"cancel so all we\u0027re left with is this bit here,"},{"Start":"12:19.935 ","End":"12:24.315","Text":"7 times 6 over 2, 7 times 3."},{"Start":"12:24.315 ","End":"12:27.615","Text":"That comes out to be 21."},{"Start":"12:27.615 ","End":"12:31.530","Text":"You know what I\u0027ll leave you to check the formula that here"},{"Start":"12:31.530 ","End":"12:37.090","Text":"we get 35 and then here we get 56."},{"Start":"12:37.220 ","End":"12:40.320","Text":"I don\u0027t want to waste time doing that."},{"Start":"12:40.320 ","End":"12:43.110","Text":"It\u0027s just straightforward computation."},{"Start":"12:43.110 ","End":"12:48.690","Text":"Next, there\u0027s one more skill we need and that"},{"Start":"12:48.690 ","End":"12:54.285","Text":"is how to find a particular term of a binomial expansion."},{"Start":"12:54.285 ","End":"12:59.430","Text":"Just like before, we had to figure out a plus b to the 4th."},{"Start":"12:59.430 ","End":"13:01.500","Text":"Sometimes we just want,"},{"Start":"13:01.500 ","End":"13:04.690","Text":"I don\u0027t know the 3rd term."},{"Start":"13:07.490 ","End":"13:11.775","Text":"I want to remind you again of the off by one thing."},{"Start":"13:11.775 ","End":"13:18.240","Text":"Meaning that this binomial coefficient,"},{"Start":"13:18.240 ","End":"13:22.950","Text":"n choose k a to the n minus k b to the k is not the kth term."},{"Start":"13:22.950 ","End":"13:27.750","Text":"It\u0027s the k plus 1st term as this off by one thing."},{"Start":"13:27.750 ","End":"13:37.505","Text":"The example is find the 6th term in the expansion."},{"Start":"13:37.505 ","End":"13:40.351","Text":"Here I made it slightly different,"},{"Start":"13:40.351 ","End":"13:41.825","Text":"instead of a plus b,"},{"Start":"13:41.825 ","End":"13:46.760","Text":"got fed up of a plus b and to show you that it doesn\u0027t have to be those 2 letters,"},{"Start":"13:46.760 ","End":"13:52.335","Text":"let\u0027s do x plus y to the 15th we want the 6th term."},{"Start":"13:52.335 ","End":"13:56.025","Text":"6 is 5 plus one."},{"Start":"13:56.025 ","End":"14:01.680","Text":"What we\u0027re talking about is k equals 5 fixed,"},{"Start":"14:01.680 ","End":"14:04.725","Text":"meaning 0, 1, 2, 3, 4, 5."},{"Start":"14:04.725 ","End":"14:07.260","Text":"You count 6 starting from 0."},{"Start":"14:07.260 ","End":"14:13.860","Text":"The n in this case is 15."},{"Start":"14:13.860 ","End":"14:23.460","Text":"What we want, the term will be 15 choose 5,"},{"Start":"14:23.460 ","End":"14:27.450","Text":"that\u0027s the n choose k. Then we\u0027ll want,"},{"Start":"14:27.450 ","End":"14:31.980","Text":"well, not a, but we\u0027ll have x instead."},{"Start":"14:31.980 ","End":"14:35.400","Text":"Of course, if you have different letters is different here."},{"Start":"14:35.400 ","End":"14:39.615","Text":"So, it\u0027s x to the power of n minus k,"},{"Start":"14:39.615 ","End":"14:44.145","Text":"15 minus 5 is 10."},{"Start":"14:44.145 ","End":"14:50.250","Text":"Here b, which is y to the power of k, is this."},{"Start":"14:50.250 ","End":"14:58.380","Text":"This is what we want and the only thing missing is the actual computation of 15 choose 5."},{"Start":"14:58.380 ","End":"15:05.640","Text":"That would be 15 factorial over 5 factorial,"},{"Start":"15:05.640 ","End":"15:10.500","Text":"and then 15 minus 5 is 10 factorial."},{"Start":"15:10.500 ","End":"15:12.525","Text":"As we saw before,"},{"Start":"15:12.525 ","End":"15:15.945","Text":"a whole chunk will cancel if we write out the factors,"},{"Start":"15:15.945 ","End":"15:23.370","Text":"you can write 15 times 14 times 13 times 12 times 11 times."},{"Start":"15:23.370 ","End":"15:25.590","Text":"Notice what comes after the 11."},{"Start":"15:25.590 ","End":"15:28.515","Text":"We get 10, 9,"},{"Start":"15:28.515 ","End":"15:30.855","Text":"8, and so on,"},{"Start":"15:30.855 ","End":"15:33.405","Text":"right down to 1."},{"Start":"15:33.405 ","End":"15:37.980","Text":"On the denominator, we get 5 factorial,"},{"Start":"15:37.980 ","End":"15:43.215","Text":"which is 5 times 4 times 3 times 2 times 1."},{"Start":"15:43.215 ","End":"15:45.495","Text":"After that we get the 10 factorial,"},{"Start":"15:45.495 ","End":"15:47.340","Text":"which is 10, 9, 8,"},{"Start":"15:47.340 ","End":"15:49.605","Text":"7, 6, down to 1."},{"Start":"15:49.605 ","End":"15:55.020","Text":"That this part cancels and we\u0027re left with just this."},{"Start":"15:55.020 ","End":"15:57.720","Text":"Of course we could do this on the calculator,"},{"Start":"15:57.720 ","End":"16:01.470","Text":"but let\u0027s see if we can cancel a bit, be creative."},{"Start":"16:01.470 ","End":"16:05.580","Text":"Yeah, 15 is 5 times 3."},{"Start":"16:05.580 ","End":"16:13.140","Text":"What else? 4 will go into 12,"},{"Start":"16:13.140 ","End":"16:21.225","Text":"3 times, and 2 will go into 14, 7 times."},{"Start":"16:21.225 ","End":"16:24.470","Text":"What are we left with all the denominators gone?"},{"Start":"16:24.470 ","End":"16:33.750","Text":"We\u0027re left with 7 times 13 times 3 times 11."},{"Start":"16:34.130 ","End":"16:39.560","Text":"I don\u0027t know, 7 times 13 is 91,"},{"Start":"16:39.560 ","End":"16:43.115","Text":"3 times 11 is 33."},{"Start":"16:43.115 ","End":"16:46.115","Text":"I\u0027m going to need a calculator."},{"Start":"16:46.115 ","End":"16:51.110","Text":"It comes out to be 3,003."},{"Start":"16:51.110 ","End":"16:54.915","Text":"All what we have to do is plug that here and"},{"Start":"16:54.915 ","End":"17:00.590","Text":"the answer is, well, I\u0027ll write it over here,"},{"Start":"17:00.590 ","End":"17:06.260","Text":"3,003 times x to the 10th,"},{"Start":"17:06.260 ","End":"17:13.275","Text":"y to the 5th and highlight that."},{"Start":"17:13.275 ","End":"17:18.675","Text":"With this example that concludes the tutorial."},{"Start":"17:18.675 ","End":"17:27.110","Text":"There\u0027s a lot of solved exercises following so you should continue with those."},{"Start":"17:27.110 ","End":"17:29.450","Text":"Here we\u0027re done."}],"ID":26926},{"Watched":false,"Name":"Explanation and Examples - Part 1","Duration":"13m 1s","ChapterTopicVideoID":26103,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26103.jpeg","UploadDate":"2021-08-04T16:14:01.0130000","DurationForVideoObject":"PT13M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.114","Text":"I\u0027m beginning a new topic,"},{"Start":"00:02.114 ","End":"00:04.305","Text":"the binomial theorem,"},{"Start":"00:04.305 ","End":"00:11.710","Text":"and this is usually due to the famous Sir Isaac Newton."},{"Start":"00:11.980 ","End":"00:16.895","Text":"We\u0027ll start with something called binomial expansion."},{"Start":"00:16.895 ","End":"00:19.310","Text":"You\u0027ve seen this before."},{"Start":"00:19.310 ","End":"00:23.540","Text":"We want to evaluate a plus b to various powers."},{"Start":"00:23.540 ","End":"00:25.380","Text":"I\u0027ll start with 1 in the middle,"},{"Start":"00:25.380 ","End":"00:27.795","Text":"a plus b squared,"},{"Start":"00:27.795 ","End":"00:36.080","Text":"and we know that this is equal to a squared plus 2ab plus b squared."},{"Start":"00:36.080 ","End":"00:40.460","Text":"Then there\u0027s a few more, a plus b^1."},{"Start":"00:40.460 ","End":"00:46.115","Text":"This is trivial, this is just a plus b itself."},{"Start":"00:46.115 ","End":"00:48.695","Text":"We\u0027d like to start from 0."},{"Start":"00:48.695 ","End":"00:52.685","Text":"A plus b to the 0 is just 1."},{"Start":"00:52.685 ","End":"00:55.049","Text":"How about a plus b cubed?"},{"Start":"00:55.049 ","End":"00:57.175","Text":"You may or may not remember this,"},{"Start":"00:57.175 ","End":"01:01.480","Text":"but we can certainly derive it if we just multiply this by this."},{"Start":"01:01.480 ","End":"01:03.425","Text":"Anyway, the answer is,"},{"Start":"01:03.425 ","End":"01:07.875","Text":"a cubed plus 3a squared b,"},{"Start":"01:07.875 ","End":"01:13.695","Text":"plus 3ab squared, plus b cubed."},{"Start":"01:13.695 ","End":"01:20.290","Text":"In general, our main task is to find some formula for"},{"Start":"01:20.290 ","End":"01:26.920","Text":"a plus b raised to any power n. Our n is a positive whole number,"},{"Start":"01:26.920 ","End":"01:29.200","Text":"or at least non-negative,"},{"Start":"01:29.200 ","End":"01:34.255","Text":"and see if we can find a pattern and a formula for doing this."},{"Start":"01:34.255 ","End":"01:37.295","Text":"Now, there are patterns here."},{"Start":"01:37.295 ","End":"01:41.940","Text":"If we just study even these first 4 rows,"},{"Start":"01:41.940 ","End":"01:44.470","Text":"we can say various things."},{"Start":"01:44.470 ","End":"01:48.310","Text":"First of all, that if this is n and is 0, 1,"},{"Start":"01:48.310 ","End":"01:49.630","Text":"2, 3,"},{"Start":"01:49.630 ","End":"01:52.675","Text":"we can see that they are n plus 1 terms."},{"Start":"01:52.675 ","End":"01:56.230","Text":"For example, here with a plus b squared,"},{"Start":"01:56.230 ","End":"01:58.540","Text":"there are 1, 2, 3 terms."},{"Start":"01:58.540 ","End":"02:01.900","Text":"Terms are what are separated by plus or minus."},{"Start":"02:01.900 ","End":"02:05.590","Text":"Here there\u0027s 1 term,"},{"Start":"02:06.770 ","End":"02:09.780","Text":"1, here we have 2,"},{"Start":"02:09.780 ","End":"02:11.175","Text":"here we have 3,"},{"Start":"02:11.175 ","End":"02:12.975","Text":"here we have 4."},{"Start":"02:12.975 ","End":"02:15.895","Text":"If we write what n is,"},{"Start":"02:15.895 ","End":"02:18.955","Text":"n here which is the coefficient is 0,"},{"Start":"02:18.955 ","End":"02:21.990","Text":"1, 2, 3,"},{"Start":"02:21.990 ","End":"02:26.120","Text":"and we want to generalize afterwards to a general n"},{"Start":"02:26.120 ","End":"02:32.315","Text":"but we see that there are n plus 1 terms."},{"Start":"02:32.315 ","End":"02:38.570","Text":"We can also see that they all begin and end with a 1."},{"Start":"02:38.570 ","End":"02:42.485","Text":"At least the coefficients do because I could put 1 here."},{"Start":"02:42.485 ","End":"02:49.200","Text":"We don\u0027t do that, but they would all start and end in a 1 something."},{"Start":"02:49.360 ","End":"02:52.560","Text":"Notice that the powers, well,"},{"Start":"02:52.560 ","End":"02:57.905","Text":"let\u0027s take this here it\u0027s easier to see, the a\u0027s decrease."},{"Start":"02:57.905 ","End":"02:59.750","Text":"If we look here at the powers of a,"},{"Start":"02:59.750 ","End":"03:01.445","Text":"here we have a squared,"},{"Start":"03:01.445 ","End":"03:04.390","Text":"here we have a which is a^1,"},{"Start":"03:04.390 ","End":"03:06.740","Text":"and here we don\u0027t have any a at all."},{"Start":"03:06.740 ","End":"03:11.450","Text":"He\u0027d always imagine there\u0027s like an a^0."},{"Start":"03:11.450 ","End":"03:16.830","Text":"Powers of a, they decrease by 1 each"},{"Start":"03:16.830 ","End":"03:23.340","Text":"time from n down to 0."},{"Start":"03:23.340 ","End":"03:26.925","Text":"Here if n is 2, we start off with a squared, a^1, a^0."},{"Start":"03:26.925 ","End":"03:29.100","Text":"Here a cubed, a squared,"},{"Start":"03:29.100 ","End":"03:32.085","Text":"a^1, a^0. There is no a."},{"Start":"03:32.085 ","End":"03:37.485","Text":"The powers of b increase from"},{"Start":"03:37.485 ","End":"03:47.070","Text":"0 up to n. Here we have b^0 here,"},{"Start":"03:47.070 ","End":"03:49.215","Text":"b^1, b squared,"},{"Start":"03:49.215 ","End":"03:52.815","Text":"b cubed, and n here is 3."},{"Start":"03:52.815 ","End":"03:55.620","Text":"The pattern is that there\u0027s a coefficient,"},{"Start":"03:55.620 ","End":"04:00.315","Text":"and then powers of a that decrease and powers of b that increase."},{"Start":"04:00.315 ","End":"04:04.600","Text":"Notice also that if you add the powers of a and b, like here,"},{"Start":"04:04.600 ","End":"04:06.475","Text":"if you add the 2 and the 1,"},{"Start":"04:06.475 ","End":"04:09.640","Text":"you get 3, and this is throughout."},{"Start":"04:09.640 ","End":"04:12.820","Text":"3 plus 0 is 3, 2 plus 1 is 3,"},{"Start":"04:12.820 ","End":"04:15.280","Text":"1 plus 2 is 3,"},{"Start":"04:15.280 ","End":"04:17.055","Text":"0 plus 3 is 3."},{"Start":"04:17.055 ","End":"04:19.640","Text":"It\u0027s always a to the something,"},{"Start":"04:19.640 ","End":"04:21.760","Text":"call it k,"},{"Start":"04:21.760 ","End":"04:25.530","Text":"or let\u0027s make the power of b,"},{"Start":"04:25.530 ","End":"04:34.070","Text":"let\u0027s call that k. Here we\u0027d have n minus k. The total exponent is just n. This is k,"},{"Start":"04:34.070 ","End":"04:36.350","Text":"this will be n minus k. This is 2,"},{"Start":"04:36.350 ","End":"04:38.645","Text":"this will be 3 minus 2, and so on."},{"Start":"04:38.645 ","End":"04:41.875","Text":"That\u0027s basically the pattern we see."},{"Start":"04:41.875 ","End":"04:45.270","Text":"Then we have some coefficient in front."},{"Start":"04:45.270 ","End":"04:47.280","Text":"Each term with a coefficient,"},{"Start":"04:47.280 ","End":"04:49.260","Text":"a to the something, b to the something,"},{"Start":"04:49.260 ","End":"04:51.500","Text":"the a\u0027s decrease, the b\u0027s decrease."},{"Start":"04:51.500 ","End":"04:54.740","Text":"Now let\u0027s try and formalize this,"},{"Start":"04:54.740 ","End":"04:56.995","Text":"or make it more precise."},{"Start":"04:56.995 ","End":"05:02.045","Text":"I\u0027m going to introduce something called Pascal\u0027s triangle."},{"Start":"05:02.045 ","End":"05:05.440","Text":"I don\u0027t want to lose this."},{"Start":"05:05.700 ","End":"05:09.157","Text":"I copied it over here,"},{"Start":"05:09.157 ","End":"05:12.240","Text":"now we can scroll a bit."},{"Start":"05:12.520 ","End":"05:17.150","Text":"As we saw, each term is made up."},{"Start":"05:17.150 ","End":"05:20.405","Text":"Let\u0027s look at this 1 just for example."},{"Start":"05:20.405 ","End":"05:23.150","Text":"Some number, a coefficient,"},{"Start":"05:23.150 ","End":"05:26.560","Text":"a power of a and a power of b,"},{"Start":"05:26.560 ","End":"05:30.110","Text":"and the sum of the powers together is n. In this case,"},{"Start":"05:30.110 ","End":"05:33.590","Text":"n is 3, so it\u0027s a squared, b^1."},{"Start":"05:33.590 ","End":"05:37.085","Text":"What we\u0027re interested in is,"},{"Start":"05:37.085 ","End":"05:38.990","Text":"what is this coefficient?"},{"Start":"05:38.990 ","End":"05:42.290","Text":"Everything else can be predictably generated."},{"Start":"05:42.290 ","End":"05:47.660","Text":"I could immediately guess the next row as a to the fourth,"},{"Start":"05:47.660 ","End":"05:51.140","Text":"then something a cubed b, and so on."},{"Start":"05:51.140 ","End":"05:56.320","Text":"But the coefficients, that\u0027s the hard part."},{"Start":"05:56.540 ","End":"06:00.135","Text":"We give them a term,"},{"Start":"06:00.135 ","End":"06:03.875","Text":"and this is a definition that this coefficient here,"},{"Start":"06:03.875 ","End":"06:10.580","Text":"it\u0027s often pronounced n choose k because it comes from the world of"},{"Start":"06:10.580 ","End":"06:18.170","Text":"combinatorics where we\u0027re choosing certain number of items from a set of items."},{"Start":"06:18.170 ","End":"06:20.090","Text":"Anyway, I don\u0027t want to get into the combinatorics."},{"Start":"06:20.090 ","End":"06:22.040","Text":"Or you can pronounce it n over k,"},{"Start":"06:22.040 ","End":"06:27.400","Text":"or just n k. I\u0027ll say n choose k. That\u0027s the coefficient"},{"Start":"06:27.400 ","End":"06:34.120","Text":"of the term which has b^k,"},{"Start":"06:34.120 ","End":"06:36.345","Text":"where k is the lower number,"},{"Start":"06:36.345 ","End":"06:40.940","Text":"and it\u0027s in the expansion of a plus b^n,"},{"Start":"06:40.940 ","End":"06:44.270","Text":"so that the power of a would be n minus k,"},{"Start":"06:44.270 ","End":"06:47.390","Text":"because together they make up n. This number is n choose"},{"Start":"06:47.390 ","End":"06:52.535","Text":"k. There\u0027s something which is what I call off-by-1."},{"Start":"06:52.535 ","End":"06:55.550","Text":"There\u0027s a difference between the k,"},{"Start":"06:55.550 ","End":"06:59.880","Text":"which is the power of b, and the position."},{"Start":"06:59.890 ","End":"07:02.585","Text":"If you do computer science,"},{"Start":"07:02.585 ","End":"07:04.670","Text":"you might start counting from 0,"},{"Start":"07:04.670 ","End":"07:06.725","Text":"but if we count from 1,"},{"Start":"07:06.725 ","End":"07:12.615","Text":"then the first term has b^0 here."},{"Start":"07:12.615 ","End":"07:15.175","Text":"The second term has b^1,"},{"Start":"07:15.175 ","End":"07:17.430","Text":"the third term has b squared,"},{"Start":"07:17.430 ","End":"07:19.820","Text":"the fourth term has b cubed."},{"Start":"07:19.820 ","End":"07:21.530","Text":"For the third term,"},{"Start":"07:21.530 ","End":"07:23.540","Text":"k is 2,"},{"Start":"07:23.540 ","End":"07:26.155","Text":"or if you specify k,"},{"Start":"07:26.155 ","End":"07:28.500","Text":"let\u0027s say as 4,"},{"Start":"07:28.500 ","End":"07:30.785","Text":"maybe b in a future row,"},{"Start":"07:30.785 ","End":"07:33.035","Text":"then it\u0027s the fifth term."},{"Start":"07:33.035 ","End":"07:39.785","Text":"The k is 1 less than the term number because we start from k equals 0."},{"Start":"07:39.785 ","End":"07:49.715","Text":"This n choose k is the coefficient of the k plus first term."},{"Start":"07:49.715 ","End":"07:52.765","Text":"The plus 1 that reflects this off by 1."},{"Start":"07:52.765 ","End":"07:56.705","Text":"Or alternatively, if you want the kth term,"},{"Start":"07:56.705 ","End":"08:03.160","Text":"then the coefficient is n choose k minus 1."},{"Start":"08:03.160 ","End":"08:05.960","Text":"Now, I\u0027ll give examples in a moment,"},{"Start":"08:05.960 ","End":"08:08.900","Text":"but each of these n choose k is 1 of these numbers,"},{"Start":"08:08.900 ","End":"08:10.190","Text":"1, 1, 1, 1,"},{"Start":"08:10.190 ","End":"08:12.680","Text":"2, 1, 1, 3, 3, 1."},{"Start":"08:12.680 ","End":"08:15.960","Text":"Also we\u0027re going to put these in a triangle and get Pascal\u0027s triangle,"},{"Start":"08:15.960 ","End":"08:19.890","Text":"so let\u0027s just get a bit more space here."},{"Start":"08:22.080 ","End":"08:24.820","Text":"There\u0027s 1 other thing before the triangles,"},{"Start":"08:24.820 ","End":"08:27.440","Text":"and that\u0027s the other notation."},{"Start":"08:27.570 ","End":"08:32.935","Text":"Instead of writing n choose k this way, it\u0027s sometimes,"},{"Start":"08:32.935 ","End":"08:34.705","Text":"in the world of combinatorics,"},{"Start":"08:34.705 ","End":"08:37.330","Text":"written as C,"},{"Start":"08:37.330 ","End":"08:40.210","Text":"C for combinations actually,"},{"Start":"08:40.210 ","End":"08:43.670","Text":"and then the n here and the k here,"},{"Start":"08:43.670 ","End":"08:50.409","Text":"and very importantly, there is a formula that computes these."},{"Start":"08:50.409 ","End":"08:53.200","Text":"If I want to know what is the value of n choose k,"},{"Start":"08:53.200 ","End":"08:55.180","Text":"you plug it in here,"},{"Start":"08:55.180 ","End":"08:58.180","Text":"we will see examples and how this relates."},{"Start":"08:58.180 ","End":"09:02.310","Text":"It\u0027ll look tied together very shortly."},{"Start":"09:02.310 ","End":"09:05.025","Text":"That\u0027s the notation, that\u0027s the formula."},{"Start":"09:05.025 ","End":"09:06.859","Text":"Now let\u0027s see in practice."},{"Start":"09:06.859 ","End":"09:10.160","Text":"If you go back, we got up to here."},{"Start":"09:10.160 ","End":"09:14.470","Text":"We just take the coefficients without the a to the something,"},{"Start":"09:14.470 ","End":"09:16.040","Text":"b to the something."},{"Start":"09:16.040 ","End":"09:19.190","Text":"What we had here is 1,"},{"Start":"09:19.190 ","End":"09:20.930","Text":"then 1, 1, then 1,"},{"Start":"09:20.930 ","End":"09:22.910","Text":"2, 1, then 1, 3, 3,"},{"Start":"09:22.910 ","End":"09:26.740","Text":"1, which is just like what\u0027s here."},{"Start":"09:26.740 ","End":"09:30.710","Text":"Let\u0027s just go back again and move on."},{"Start":"09:30.710 ","End":"09:34.985","Text":"In terms of this binomial coefficients symbol,"},{"Start":"09:34.985 ","End":"09:37.880","Text":"this 3 here for example,"},{"Start":"09:37.880 ","End":"09:40.525","Text":"is 3 choose 1."},{"Start":"09:40.525 ","End":"09:43.620","Text":"Because we start from 0,"},{"Start":"09:43.620 ","End":"09:45.440","Text":"the k is what\u0027s underneath."},{"Start":"09:45.440 ","End":"09:48.080","Text":"This is a squared b^1,"},{"Start":"09:48.080 ","End":"09:50.015","Text":"b is to the power of 1,"},{"Start":"09:50.015 ","End":"09:52.890","Text":"so it\u0027s 3 choose 1."},{"Start":"09:52.890 ","End":"09:57.035","Text":"For example, this 1 would correspond to this."},{"Start":"09:57.035 ","End":"10:01.945","Text":"Here it\u0027s 1 choose 0 and 1 choose 1."},{"Start":"10:01.945 ","End":"10:09.750","Text":"Here remember it was a squared plus 2ab plus b squared,"},{"Start":"10:09.750 ","End":"10:12.820","Text":"and there\u0027s a 1 here and a 1 here."},{"Start":"10:13.400 ","End":"10:16.260","Text":"This is this 1 here,"},{"Start":"10:16.260 ","End":"10:17.835","Text":"this is this 2 here,"},{"Start":"10:17.835 ","End":"10:21.210","Text":"and here it corresponds to 2, 2,"},{"Start":"10:21.210 ","End":"10:24.780","Text":"and 2 at the top because it\u0027s all squared, and the 0,1,"},{"Start":"10:24.780 ","End":"10:32.715","Text":"2 is the exponent of b, b to the 0."},{"Start":"10:32.715 ","End":"10:36.420","Text":"We can add in here a, b^0."},{"Start":"10:36.420 ","End":"10:39.720","Text":"Well anyway, that\u0027s the 2 and that\u0027s the 2."},{"Start":"10:39.720 ","End":"10:41.835","Text":"Just study it for a while."},{"Start":"10:41.835 ","End":"10:45.870","Text":"These are the actual results of the coefficients,"},{"Start":"10:45.870 ","End":"10:51.050","Text":"and here this is the same numbers but expressed in this notation."},{"Start":"10:51.050 ","End":"10:54.470","Text":"Now, I haven\u0027t shown you that this formula works."},{"Start":"10:54.470 ","End":"10:56.870","Text":"We\u0027ll see many examples of this later."},{"Start":"10:56.870 ","End":"10:59.230","Text":"So far we\u0027ve just computed up to the third row."},{"Start":"10:59.230 ","End":"11:01.085","Text":"Let\u0027s take this 1 as an example."},{"Start":"11:01.085 ","End":"11:03.020","Text":"Let\u0027s see if we can get this 3."},{"Start":"11:03.020 ","End":"11:12.000","Text":"This 3 comes from the power of b^1 in the expansion of a plus b cubed."},{"Start":"11:12.000 ","End":"11:13.440","Text":"This is what we want,"},{"Start":"11:13.440 ","End":"11:18.150","Text":"so 3 choose 1."},{"Start":"11:18.150 ","End":"11:20.940","Text":"According to this formula, n is 3,"},{"Start":"11:20.940 ","End":"11:24.870","Text":"so it\u0027s 3 factorial, k is 1,"},{"Start":"11:24.870 ","End":"11:27.690","Text":"so it\u0027s 1 factorial,"},{"Start":"11:27.690 ","End":"11:34.400","Text":"and n minus k is 3 minus 1, is 2 factorial."},{"Start":"11:34.400 ","End":"11:38.480","Text":"Again, this 2 is 3 minus 1 from here."},{"Start":"11:38.480 ","End":"11:43.085","Text":"Now, 3 factorial is 6,"},{"Start":"11:43.085 ","End":"11:45.020","Text":"because it\u0027s 3 times 2 times 1."},{"Start":"11:45.020 ","End":"11:47.825","Text":"This is 6. 1 factorial is 1."},{"Start":"11:47.825 ","End":"11:50.120","Text":"2 factorial is 2."},{"Start":"11:50.120 ","End":"11:52.910","Text":"6 over 1 times 2,"},{"Start":"11:52.910 ","End":"11:57.060","Text":"so this gives us equals 3,"},{"Start":"11:57.060 ","End":"12:03.200","Text":"and that\u0027s the 3 that\u0027s here, using this computation."},{"Start":"12:03.200 ","End":"12:09.080","Text":"Before we move on, just 1 other thing we should note is that there is a way of"},{"Start":"12:09.080 ","End":"12:15.520","Text":"building this table without computing a plus b^4,"},{"Start":"12:15.520 ","End":"12:17.665","Text":"then 5, then 6."},{"Start":"12:17.665 ","End":"12:20.360","Text":"Notice that if I have a certain row,"},{"Start":"12:20.360 ","End":"12:23.990","Text":"we\u0027ve got up to the row with 1, 3, 3, 1."},{"Start":"12:23.990 ","End":"12:29.090","Text":"If we add 3 plus 3, we get 6."},{"Start":"12:29.090 ","End":"12:35.000","Text":"3 plus 1 we get 4, and here also."},{"Start":"12:35.000 ","End":"12:40.550","Text":"Each number in the following row is the sum of the 2 above it."},{"Start":"12:40.550 ","End":"12:42.185","Text":"1 plus 4 is 5,"},{"Start":"12:42.185 ","End":"12:43.820","Text":"4 plus 6 is 10,"},{"Start":"12:43.820 ","End":"12:45.200","Text":"6 plus 4 is 10,"},{"Start":"12:45.200 ","End":"12:46.430","Text":"4 plus 1 is 5,"},{"Start":"12:46.430 ","End":"12:49.985","Text":"and we always begin and end with a 1."},{"Start":"12:49.985 ","End":"12:52.130","Text":"Once we have a given row,"},{"Start":"12:52.130 ","End":"12:53.340","Text":"we put 1 at the beginning,"},{"Start":"12:53.340 ","End":"12:57.050","Text":"1 at the end, and all the other numbers of the sum of the 2 above."},{"Start":"12:57.050 ","End":"13:01.890","Text":"Let\u0027s take a break now and then continue with the binomial theorem."}],"ID":26924},{"Watched":false,"Name":"Explanation and Examples - Part 2","Duration":"17m 29s","ChapterTopicVideoID":26104,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26104.jpeg","UploadDate":"2021-08-04T16:19:30.9700000","DurationForVideoObject":"PT17M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"Continuing after the break, the binomial theorem."},{"Start":"00:03.840 ","End":"00:09.780","Text":"Now it\u0027s time to reveal what this theorem actually is."},{"Start":"00:09.780 ","End":"00:14.520","Text":"Here it is. It\u0027s a formula that says that"},{"Start":"00:14.520 ","End":"00:19.410","Text":"when we take the binomial a plus b and raise it to the power of n,"},{"Start":"00:19.410 ","End":"00:22.965","Text":"where n is a non-negative integer,"},{"Start":"00:22.965 ","End":"00:31.330","Text":"then it\u0027s equal to the sum of n choose k,"},{"Start":"00:31.330 ","End":"00:34.175","Text":"a to the n minus k b to the k,"},{"Start":"00:34.175 ","End":"00:40.460","Text":"and k goes from 0 to n. This is the Sigma notation and the examples,"},{"Start":"00:40.460 ","End":"00:43.295","Text":"we\u0027ll also write it out in full."},{"Start":"00:43.295 ","End":"00:52.925","Text":"The only thing that\u0027s missing here that we did earlier is the definition."},{"Start":"00:52.925 ","End":"00:55.640","Text":"As I said, we\u0027ve already covered it,"},{"Start":"00:55.640 ","End":"01:04.670","Text":"but let me refresh n choose k or have you pronounce this symbol is equal"},{"Start":"01:04.670 ","End":"01:14.285","Text":"to n factorial over k factorial n minus k factorial."},{"Start":"01:14.285 ","End":"01:20.700","Text":"This together with this tells us how we expand a plus b to the power."},{"Start":"01:21.560 ","End":"01:24.710","Text":"Let\u0027s take n equals 4."},{"Start":"01:24.710 ","End":"01:28.580","Text":"We\u0027ve already done n goes from 0-3."},{"Start":"01:28.580 ","End":"01:32.390","Text":"If you go back, we did a plus b cubed and lower powers."},{"Start":"01:32.390 ","End":"01:36.085","Text":"Let\u0027s do a plus b^4 from the formula."},{"Start":"01:36.085 ","End":"01:44.605","Text":"First of all, I\u0027ll use Sigma to say that it\u0027s Sigma where k goes from 0-4."},{"Start":"01:44.605 ","End":"01:47.895","Text":"Remember this means it\u0027s going to be 5 terms,"},{"Start":"01:47.895 ","End":"01:50.190","Text":"0, 1, 2, 3, 4."},{"Start":"01:50.190 ","End":"01:51.960","Text":"There is 1 more that\u0027s this."},{"Start":"01:51.960 ","End":"01:55.170","Text":"Off by 1 keeps recurring."},{"Start":"01:55.170 ","End":"02:04.385","Text":"N here is 4 choose k. K is the index that varies from 0-4."},{"Start":"02:04.385 ","End":"02:09.590","Text":"A to the power of 4 minus k,"},{"Start":"02:09.590 ","End":"02:15.791","Text":"b to the power of k. The powers of b are the ones that increase from 0-4."},{"Start":"02:15.791 ","End":"02:20.699","Text":"The next step is to open up the Sigma."},{"Start":"02:20.699 ","End":"02:22.450","Text":"We have 5 terms."},{"Start":"02:22.450 ","End":"02:25.350","Text":"First 1, when k is 0, 4,"},{"Start":"02:25.350 ","End":"02:32.025","Text":"choose 0, a to the 4 minus 0 is 4, and b to the 0."},{"Start":"02:32.025 ","End":"02:34.380","Text":"We don\u0027t write it."},{"Start":"02:34.380 ","End":"02:36.690","Text":"I mean, always this comes out to be 1,"},{"Start":"02:36.690 ","End":"02:39.450","Text":"but this day should write it 4 choose 0."},{"Start":"02:39.450 ","End":"02:42.475","Text":"Next 4 choose 1,"},{"Start":"02:42.475 ","End":"02:44.735","Text":"a to the 4 minus 1,"},{"Start":"02:44.735 ","End":"02:49.805","Text":"the a\u0027s decrease in the b\u0027s increase so here we have b to the 1 which is just b."},{"Start":"02:49.805 ","End":"02:53.015","Text":"Then 4 choose 2, a,"},{"Start":"02:53.015 ","End":"02:57.125","Text":"this time is squared and b increases squared."},{"Start":"02:57.125 ","End":"02:59.930","Text":"Then 4 choose 3,"},{"Start":"02:59.930 ","End":"03:02.810","Text":"a is squared it\u0027s down to a,"},{"Start":"03:02.810 ","End":"03:06.775","Text":"b increases the power of 3."},{"Start":"03:06.775 ","End":"03:09.120","Text":"Then the last 1,"},{"Start":"03:09.120 ","End":"03:12.375","Text":"4 choose 4, a to the power of 0."},{"Start":"03:12.375 ","End":"03:16.710","Text":"Just don\u0027t write it and b^4."},{"Start":"03:16.710 ","End":"03:19.810","Text":"Of course, the first and the last are 1."},{"Start":"03:19.810 ","End":"03:22.580","Text":"That\u0027s always the case."},{"Start":"03:22.580 ","End":"03:27.605","Text":"Anyway, let\u0027s now compute these numbers."},{"Start":"03:27.605 ","End":"03:30.740","Text":"It\u0027s really 2 ways of going about this."},{"Start":"03:30.740 ","End":"03:36.694","Text":"1 is just computing straight from the formula or we could use Pascal\u0027s triangle."},{"Start":"03:36.694 ","End":"03:41.559","Text":"Since we got the triangle up to the third row,"},{"Start":"03:41.559 ","End":"03:44.630","Text":"the fourth row actually with the 1 with 3 in it,"},{"Start":"03:44.630 ","End":"03:50.978","Text":"we had a 1,3,3,1 above that we had 1,"},{"Start":"03:50.978 ","End":"03:54.430","Text":"2,1 so on and so on."},{"Start":"03:54.430 ","End":"03:56.660","Text":"When we do the next row,"},{"Start":"03:56.660 ","End":"04:00.125","Text":"we start with a 1 and we end with a 1."},{"Start":"04:00.125 ","End":"04:06.245","Text":"Remember, each term now is the sum of the 2 above it so 1 and 3 is 4,"},{"Start":"04:06.245 ","End":"04:08.735","Text":"3 and 3 is 6,"},{"Start":"04:08.735 ","End":"04:11.915","Text":"3 and 1 is 4."},{"Start":"04:11.915 ","End":"04:14.465","Text":"The other way to do this,"},{"Start":"04:14.465 ","End":"04:19.640","Text":"to compute these numbers is to use this binomial,"},{"Start":"04:19.640 ","End":"04:24.025","Text":"this formula for the binomial coefficient."},{"Start":"04:24.025 ","End":"04:26.330","Text":"We\u0027ll do it this way also,"},{"Start":"04:26.330 ","End":"04:28.910","Text":"but I won\u0027t bother with the first and the last."},{"Start":"04:28.910 ","End":"04:31.640","Text":"It works when k is 0 and when k is n,"},{"Start":"04:31.640 ","End":"04:35.070","Text":"but we\u0027ll just do these 3."},{"Start":"04:35.480 ","End":"04:43.330","Text":"4 choose 1 is 4 factorial over 1 factorial,"},{"Start":"04:43.330 ","End":"04:49.150","Text":"and 4 minus 1 is 3 factorial."},{"Start":"04:49.150 ","End":"04:50.580","Text":"That\u0027s here. Anyway,"},{"Start":"04:50.580 ","End":"04:52.885","Text":"I\u0027ll write them, and then we\u0027ll compute them."},{"Start":"04:52.885 ","End":"04:58.600","Text":"4 choose 2 is 4 factorial over 2 factorial,"},{"Start":"04:58.600 ","End":"05:00.790","Text":"and then 4 minus 2,"},{"Start":"05:00.790 ","End":"05:03.425","Text":"which is also 2 factorial."},{"Start":"05:03.425 ","End":"05:08.665","Text":"Here 4 factorial over 3 factorial,"},{"Start":"05:08.665 ","End":"05:13.520","Text":"4 minus 3 is 1 factorial."},{"Start":"05:13.520 ","End":"05:17.280","Text":"Now, 1 factorial is 1,"},{"Start":"05:17.280 ","End":"05:19.805","Text":"so doesn\u0027t matter here and here."},{"Start":"05:19.805 ","End":"05:23.300","Text":"What we have, 4 factorial is 24,"},{"Start":"05:23.300 ","End":"05:25.235","Text":"3 factorial is 6,"},{"Start":"05:25.235 ","End":"05:29.250","Text":"24 over 6 is 4."},{"Start":"05:29.250 ","End":"05:34.395","Text":"This 1 is 24 over 2 times 2,"},{"Start":"05:34.395 ","End":"05:37.425","Text":"24 over 4 is 6."},{"Start":"05:37.425 ","End":"05:42.450","Text":"This is the same as this just in different order so it\u0027s also 4."},{"Start":"05:42.450 ","End":"05:45.720","Text":"That\u0027s what we got here, here, and here."},{"Start":"05:45.720 ","End":"05:48.740","Text":"Pascal\u0027s triangle method and"},{"Start":"05:48.740 ","End":"05:54.325","Text":"the formula method give us the same numbers and that\u0027s reassuring."},{"Start":"05:54.325 ","End":"05:57.000","Text":"I cleared some space here."},{"Start":"05:57.000 ","End":"06:02.030","Text":"Now we can write the answer as a^4,"},{"Start":"06:02.030 ","End":"06:04.425","Text":"the 1 we don\u0027t write."},{"Start":"06:04.425 ","End":"06:14.610","Text":"Then from here, we have 4a cubed b 4/2 or choose 2 is 6,"},{"Start":"06:14.610 ","End":"06:21.550","Text":"a squared, b squared."},{"Start":"06:21.740 ","End":"06:26.970","Text":"Then again 4 this time,"},{"Start":"06:26.970 ","End":"06:30.165","Text":"a b cubed,"},{"Start":"06:30.165 ","End":"06:36.310","Text":"and the last 1, 1 if we don\u0027t write is just b^4."},{"Start":"06:36.500 ","End":"06:43.965","Text":"A plus b^4 is this."},{"Start":"06:43.965 ","End":"06:47.740","Text":"Okay, that\u0027s the example."},{"Start":"06:47.930 ","End":"06:54.725","Text":"Now let\u0027s look at some of the properties of the binomial coefficients."},{"Start":"06:54.725 ","End":"06:59.000","Text":"This also relates to Pascal\u0027s triangle."},{"Start":"06:59.000 ","End":"07:03.055","Text":"I\u0027ll show you 3 properties."},{"Start":"07:03.055 ","End":"07:11.260","Text":"The first property says n choose 0 is n choose n is 1."},{"Start":"07:11.260 ","End":"07:17.350","Text":"What this means is that the coefficients begin with 1 and end in 1."},{"Start":"07:17.350 ","End":"07:21.180","Text":"If n was 4 here,"},{"Start":"07:21.180 ","End":"07:24.645","Text":"4 choose 0 is 1,"},{"Start":"07:24.645 ","End":"07:29.415","Text":"4 choose 4 is also 1."},{"Start":"07:29.415 ","End":"07:37.525","Text":"What this is saying is that these are all ones and these are all ones."},{"Start":"07:37.525 ","End":"07:40.610","Text":"That\u0027s the first property."},{"Start":"07:40.610 ","End":"07:47.230","Text":"The next property I won\u0027t even read it out is what is written."},{"Start":"07:47.230 ","End":"07:50.140","Text":"Essentially this is the symmetry property."},{"Start":"07:50.140 ","End":"07:54.505","Text":"Notice that if I put a vertical line through the middle,"},{"Start":"07:54.505 ","End":"07:56.125","Text":"it\u0027s symmetrical."},{"Start":"07:56.125 ","End":"07:58.165","Text":"In our case."},{"Start":"07:58.165 ","End":"08:03.325","Text":"For example, this came out to be 4,"},{"Start":"08:03.325 ","End":"08:06.340","Text":"and this also came out to be"},{"Start":"08:06.340 ","End":"08:13.890","Text":"4.The middle one is it\u0027s own mirror image if it\u0027s an odd number of terms,"},{"Start":"08:13.890 ","End":"08:18.150","Text":"the middle one but you notice that if you go from left to right,"},{"Start":"08:18.150 ","End":"08:20.310","Text":"1, 4, 6 4, 1, or from right to left,"},{"Start":"08:20.310 ","End":"08:22.560","Text":"1, 4, 6, 4, 1, it\u0027s the same."},{"Start":"08:22.560 ","End":"08:24.360","Text":"This is what this says."},{"Start":"08:24.360 ","End":"08:29.445","Text":"In particular example, I chose n is 4, k is 1."},{"Start":"08:29.445 ","End":"08:36.960","Text":"It says that 4 choose 1 is the same as 4 choose 4 minus 1, which is 3."},{"Start":"08:36.960 ","End":"08:40.110","Text":"Other words, if I subtract the 1 from the 4,"},{"Start":"08:40.110 ","End":"08:42.750","Text":"I get 3 and put 3 here instead,"},{"Start":"08:42.750 ","End":"08:44.340","Text":"it\u0027s going to be the same."},{"Start":"08:44.340 ","End":"08:49.215","Text":"This is essentially what the symmetry is."},{"Start":"08:49.215 ","End":"08:57.030","Text":"I mean, the terms of symmetrical if this plus this is 4 or n in general."},{"Start":"08:57.030 ","End":"09:00.090","Text":"Notice that 0 plus 4 is 4."},{"Start":"09:00.090 ","End":"09:02.040","Text":"1 plus 3 is 4."},{"Start":"09:02.040 ","End":"09:03.810","Text":"2 plus itself is 4."},{"Start":"09:03.810 ","End":"09:06.360","Text":"That\u0027s the symmetry condition and that\u0027s why it\u0027s n"},{"Start":"09:06.360 ","End":"09:09.525","Text":"minus k. Together this in this add up to n,"},{"Start":"09:09.525 ","End":"09:13.180","Text":"which means that they\u0027re symmetrically placed."},{"Start":"09:13.460 ","End":"09:16.620","Text":"As I said in the case of Pascal\u0027s triangle,"},{"Start":"09:16.620 ","End":"09:18.390","Text":"I mean you can look all over the place."},{"Start":"09:18.390 ","End":"09:19.710","Text":"5 is the same as 5."},{"Start":"09:19.710 ","End":"09:21.225","Text":"10 is the same as 10."},{"Start":"09:21.225 ","End":"09:23.310","Text":"28 here, 28 here,"},{"Start":"09:23.310 ","End":"09:25.530","Text":"and so on and so on."},{"Start":"09:25.530 ","End":"09:32.250","Text":"The third property, this one won\u0027t even read it out just yet,"},{"Start":"09:32.250 ","End":"09:37.950","Text":"but this is the mathematical form of that property."},{"Start":"09:37.950 ","End":"09:39.660","Text":"Let\u0027s take an example."},{"Start":"09:39.660 ","End":"09:43.980","Text":"Let\u0027s say I put a triangle here."},{"Start":"09:43.980 ","End":"09:49.005","Text":"Notice that we said that each entry is the sum of the 2 above it."},{"Start":"09:49.005 ","End":"09:51.495","Text":"That this is equal to this plus this."},{"Start":"09:51.495 ","End":"09:53.130","Text":"Well, what are these?"},{"Start":"09:53.130 ","End":"09:57.345","Text":"This 21 is in row 7."},{"Start":"09:57.345 ","End":"09:59.670","Text":"It\u0027s the eighth row if you count this row,"},{"Start":"09:59.670 ","End":"10:00.690","Text":"but then I say row 7,"},{"Start":"10:00.690 ","End":"10:02.850","Text":"I mean where the 7 is here."},{"Start":"10:02.850 ","End":"10:05.100","Text":"What this says is at 21,"},{"Start":"10:05.100 ","End":"10:11.115","Text":"which is 7 and the number"},{"Start":"10:11.115 ","End":"10:17.145","Text":"below is 1 mode or rather 1 less than,"},{"Start":"10:17.145 ","End":"10:19.080","Text":"you count 1, 2,"},{"Start":"10:19.080 ","End":"10:21.570","Text":"3, the third term,"},{"Start":"10:21.570 ","End":"10:25.935","Text":"so the k is 2 or you could start counting from 0."},{"Start":"10:25.935 ","End":"10:27.480","Text":"Another way to do it, 0, 1,"},{"Start":"10:27.480 ","End":"10:29.070","Text":"2, so that\u0027s 7,"},{"Start":"10:29.070 ","End":"10:31.840","Text":"2, that\u0027s the 21."},{"Start":"10:32.000 ","End":"10:38.470","Text":"The 35 is the following term is 7."},{"Start":"10:38.600 ","End":"10:45.510","Text":"Choose 3 because k is 0, 1, 2, 3."},{"Start":"10:45.510 ","End":"10:48.130","Text":"Well, it\u0027s just 1 more than the 2."},{"Start":"10:48.500 ","End":"10:57.570","Text":"The 56 is going to be the same index as the 35 because it\u0027s like parallel."},{"Start":"10:57.570 ","End":"10:59.130","Text":"This is 0, 1, 2,"},{"Start":"10:59.130 ","End":"11:01.500","Text":"3, 0, 1, 2, 3,"},{"Start":"11:01.500 ","End":"11:08.175","Text":"so that would be 8 choose 3 because we\u0027re on the row below,"},{"Start":"11:08.175 ","End":"11:12.900","Text":"which you can tell from the 8 here just like from the 7 here."},{"Start":"11:12.900 ","End":"11:15.570","Text":"It\u0027s 8 from here and then 0, 1, 2,"},{"Start":"11:15.570 ","End":"11:17.700","Text":"3 or you count 1, 2, 3,"},{"Start":"11:17.700 ","End":"11:20.265","Text":"4 but remember to subtract 1 so you get 3."},{"Start":"11:20.265 ","End":"11:23.625","Text":"That\u0027s 21, that\u0027s 35,"},{"Start":"11:23.625 ","End":"11:27.135","Text":"and this is 56."},{"Start":"11:27.135 ","End":"11:29.325","Text":"Just for extra practice,"},{"Start":"11:29.325 ","End":"11:30.795","Text":"let\u0027s compute them."},{"Start":"11:30.795 ","End":"11:33.480","Text":"7 choose 2."},{"Start":"11:33.480 ","End":"11:40.650","Text":"This would be 7 factorial over 2 factorial."},{"Start":"11:40.650 ","End":"11:45.940","Text":"7 minus 2, which is 5 factorial."},{"Start":"11:49.130 ","End":"11:53.370","Text":"You can compute it with the calculator,"},{"Start":"11:53.370 ","End":"11:57.299","Text":"but in these case, it\u0027s best to just multiply out 7 factorial."},{"Start":"11:57.299 ","End":"12:03.150","Text":"7 times 6 times 5 times 4 times 3 times 2 times 1."},{"Start":"12:03.150 ","End":"12:07.920","Text":"Then 2 factorial is 2 times 1,"},{"Start":"12:07.920 ","End":"12:13.380","Text":"and 5 factorial is 5 times 4 times 3 times 2 times 1."},{"Start":"12:13.380 ","End":"12:17.100","Text":"Notice that the 5 factorial and 5 factorial cancel,"},{"Start":"12:17.100 ","End":"12:19.935","Text":"so all we\u0027re left with is this bit here,"},{"Start":"12:19.935 ","End":"12:24.315","Text":"7 times 6 over 2, 7 times 3."},{"Start":"12:24.315 ","End":"12:27.720","Text":"That comes out to be 21."},{"Start":"12:27.720 ","End":"12:37.090","Text":"Well, I\u0027ll leave you to check the formula that here we get 35 and then here we get 56."},{"Start":"12:37.220 ","End":"12:40.320","Text":"I don\u0027t want to waste time doing that."},{"Start":"12:40.320 ","End":"12:43.110","Text":"It\u0027s just straightforward computation."},{"Start":"12:43.110 ","End":"12:47.790","Text":"Next is one more skill we need."},{"Start":"12:47.790 ","End":"12:54.285","Text":"That is how to find a particular term of a binomial expansion."},{"Start":"12:54.285 ","End":"12:59.430","Text":"Just like before, we had to figure out a plus b to the fourth."},{"Start":"12:59.430 ","End":"13:04.690","Text":"Sometimes we just want the third term."},{"Start":"13:07.490 ","End":"13:11.775","Text":"I want to remind you again of the off by 1 thing."},{"Start":"13:11.775 ","End":"13:18.240","Text":"Meaning that this binomial coefficient,"},{"Start":"13:18.240 ","End":"13:19.380","Text":"n choose k,"},{"Start":"13:19.380 ","End":"13:22.950","Text":"a^n minus k b^k is not the kth term."},{"Start":"13:22.950 ","End":"13:27.750","Text":"It\u0027s the k plus first term as this off by 1 thing."},{"Start":"13:27.750 ","End":"13:37.505","Text":"The exercise, the example is find the 6th term in the expansion."},{"Start":"13:37.505 ","End":"13:40.351","Text":"Here I made it slightly different."},{"Start":"13:40.351 ","End":"13:42.710","Text":"Instead of a plus b got fed up of"},{"Start":"13:42.710 ","End":"13:46.760","Text":"a plus b. I want to show you that it doesn\u0027t have to be those 2 letters."},{"Start":"13:46.760 ","End":"13:49.724","Text":"Let\u0027s do x plus y^15th."},{"Start":"13:49.724 ","End":"13:52.335","Text":"We want the 6th term."},{"Start":"13:52.335 ","End":"13:55.500","Text":"6 is 5 plus 1,"},{"Start":"13:55.500 ","End":"14:00.900","Text":"so what we\u0027re talking about is k equals 5."},{"Start":"14:00.900 ","End":"14:02.970","Text":"6 meaning 0, 1,"},{"Start":"14:02.970 ","End":"14:04.725","Text":"2, 3, 4, 5."},{"Start":"14:04.725 ","End":"14:07.260","Text":"You count 6 starting from 0."},{"Start":"14:07.260 ","End":"14:13.860","Text":"The n in this case is 15."},{"Start":"14:13.860 ","End":"14:23.460","Text":"What we want, the term will be 15 choose 5,"},{"Start":"14:23.460 ","End":"14:29.865","Text":"that\u0027s the n choose k. Then we\u0027ll want not a,"},{"Start":"14:29.865 ","End":"14:31.980","Text":"but we\u0027ll have x instead."},{"Start":"14:31.980 ","End":"14:35.535","Text":"Of course, if you have different letters is different here."},{"Start":"14:35.535 ","End":"14:44.145","Text":"It\u0027s x to the power of n minus k. 15 minus 5 is 10."},{"Start":"14:44.145 ","End":"14:50.250","Text":"Here b, which is y to the power of k, is this."},{"Start":"14:50.250 ","End":"14:52.125","Text":"This is what we want."},{"Start":"14:52.125 ","End":"14:58.379","Text":"The only thing missing is the actual computation of 15 choose 5."},{"Start":"14:58.379 ","End":"15:05.640","Text":"That would be 15 factorial over 5 factorial,"},{"Start":"15:05.640 ","End":"15:10.500","Text":"and then 15 minus 5 is 10 factorial."},{"Start":"15:10.500 ","End":"15:12.525","Text":"As we saw before,"},{"Start":"15:12.525 ","End":"15:15.945","Text":"a whole chunk will cancel if we write out the factors."},{"Start":"15:15.945 ","End":"15:23.370","Text":"We can write 15 times 14 times 13 times 12 times 11 times."},{"Start":"15:23.370 ","End":"15:25.590","Text":"Notice what comes after the 11,"},{"Start":"15:25.590 ","End":"15:27.120","Text":"we get 10,"},{"Start":"15:27.120 ","End":"15:29.820","Text":"9, 8,"},{"Start":"15:29.820 ","End":"15:33.405","Text":"and so on, right down to 1."},{"Start":"15:33.405 ","End":"15:37.980","Text":"On the denominator, we get 5 factorial,"},{"Start":"15:37.980 ","End":"15:43.215","Text":"which is 5 times 4 times 3 times 2 times 1."},{"Start":"15:43.215 ","End":"15:45.495","Text":"After that we get the term factorial,"},{"Start":"15:45.495 ","End":"15:48.150","Text":"which is 10, 9, 8, 7, 6,"},{"Start":"15:48.150 ","End":"15:55.020","Text":"down to 1, that this part cancels and we\u0027re left with just this."},{"Start":"15:55.020 ","End":"15:57.720","Text":"Of course, we could do this on the calculator,"},{"Start":"15:57.720 ","End":"16:01.470","Text":"but let\u0027s see if we can cancel a bit, like be creative."},{"Start":"16:01.470 ","End":"16:05.580","Text":"15 is 5 times 3."},{"Start":"16:05.580 ","End":"16:13.140","Text":"What else? 4 will go into 12,"},{"Start":"16:13.140 ","End":"16:21.225","Text":"3 times, and 2 will go into 14, 7 times."},{"Start":"16:21.225 ","End":"16:24.680","Text":"What are we left with? All the denominators gone."},{"Start":"16:24.680 ","End":"16:33.750","Text":"We left with 7 times 13 times 3 times 11."},{"Start":"16:34.580 ","End":"16:39.570","Text":"7 times 13 is 91,"},{"Start":"16:39.570 ","End":"16:43.245","Text":"3 times 11 is 33."},{"Start":"16:43.245 ","End":"16:46.120","Text":"I\u0027m going to need a calculator."},{"Start":"16:46.120 ","End":"16:51.530","Text":"It comes out to be 3,003."},{"Start":"16:51.530 ","End":"16:54.420","Text":"We\u0027ll have to do is plug that here."},{"Start":"16:54.420 ","End":"16:56.320","Text":"The answer"},{"Start":"16:56.320 ","End":"17:06.245","Text":"is 3,003 times x^10th,"},{"Start":"17:06.245 ","End":"17:08.970","Text":"y^5th and highlight that."},{"Start":"17:13.100 ","End":"17:15.470","Text":"With this example,"},{"Start":"17:15.470 ","End":"17:18.410","Text":"that concludes the tutorial."},{"Start":"17:18.410 ","End":"17:24.865","Text":"There\u0027s a lot of solved exercises following."},{"Start":"17:24.865 ","End":"17:29.470","Text":"You should continue with those here. We\u0027re done."}],"ID":26925},{"Watched":false,"Name":"Exercise 1","Duration":"5m 53s","ChapterTopicVideoID":26107,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26107.jpeg","UploadDate":"2021-08-04T16:31:10.6400000","DurationForVideoObject":"PT5M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, which is really 4 exercises in 1,"},{"Start":"00:03.570 ","End":"00:06.225","Text":"we have to calculate the binomial coefficients."},{"Start":"00:06.225 ","End":"00:08.595","Text":"Let\u0027s remember the formula,"},{"Start":"00:08.595 ","End":"00:12.192","Text":"which is really a definition."},{"Start":"00:12.192 ","End":"00:15.975","Text":"If we have n here and k here,"},{"Start":"00:15.975 ","End":"00:20.670","Text":"then it\u0027s equal to n factorial over k"},{"Start":"00:20.670 ","End":"00:27.310","Text":"factorial times n minus k factorial."},{"Start":"00:30.650 ","End":"00:34.560","Text":"Straightforward application of the formula."},{"Start":"00:34.560 ","End":"00:39.195","Text":"In A, we have 5, 3."},{"Start":"00:39.195 ","End":"00:45.332","Text":"So it\u0027s 5 factorial over 3 factorial,"},{"Start":"00:45.332 ","End":"00:49.580","Text":"and usually mentally do the 5 minus 3,"},{"Start":"00:49.580 ","End":"00:51.560","Text":"which is 2 factorial."},{"Start":"00:51.560 ","End":"00:59.720","Text":"This gives us 5 factorial,"},{"Start":"00:59.720 ","End":"01:04.955","Text":"5 times 4 times 3 times 2 times 1."},{"Start":"01:04.955 ","End":"01:07.850","Text":"I know it\u0027s silly to write the times 1,"},{"Start":"01:07.850 ","End":"01:13.415","Text":"but it just the way we do it because it\u0027s a pattern from 5 down to 1."},{"Start":"01:13.415 ","End":"01:18.140","Text":"Then 3 factorial, 3 times 2 times 1,"},{"Start":"01:18.140 ","End":"01:21.770","Text":"and 2 factorial, 2 times 1."},{"Start":"01:21.770 ","End":"01:25.890","Text":"Now, some things cancel,"},{"Start":"01:25.890 ","End":"01:28.890","Text":"2 and 1, 2 and 1,"},{"Start":"01:28.890 ","End":"01:31.785","Text":"this 3 with this 3."},{"Start":"01:31.785 ","End":"01:38.170","Text":"I guess 2 goes into 4 twice,"},{"Start":"01:38.170 ","End":"01:41.310","Text":"and so the answer is 10."},{"Start":"01:41.310 ","End":"01:48.150","Text":"Now part B, 8, 3,"},{"Start":"01:48.150 ","End":"01:52.085","Text":"8 factorial, 3 factorial,"},{"Start":"01:52.085 ","End":"01:53.885","Text":"and the 8 minus 3,"},{"Start":"01:53.885 ","End":"01:58.535","Text":"do in our heads, is 5 factorial."},{"Start":"01:58.535 ","End":"02:00.680","Text":"If we were really spelling it out,"},{"Start":"02:00.680 ","End":"02:03.725","Text":"you\u0027d have an extra step where instead of 5 factorial,"},{"Start":"02:03.725 ","End":"02:07.385","Text":"we\u0027d have 8 minus 3 factorial."},{"Start":"02:07.385 ","End":"02:08.450","Text":"But as I said,"},{"Start":"02:08.450 ","End":"02:14.530","Text":"why waste the step because the subtraction part is the easy part."},{"Start":"02:15.560 ","End":"02:27.230","Text":"So 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1,"},{"Start":"02:27.230 ","End":"02:32.140","Text":"5,4, 3, 2, 1."},{"Start":"02:32.140 ","End":"02:36.278","Text":"This whole bit cancels with this,"},{"Start":"02:36.278 ","End":"02:41.250","Text":"and look, 3 times 2 is 6."},{"Start":"02:41.250 ","End":"02:47.570","Text":"What we\u0027re left with is 8 times 7, which is 56."},{"Start":"02:47.570 ","End":"02:50.825","Text":"Now the next 1,"},{"Start":"02:50.825 ","End":"02:59.305","Text":"10/1, used to saying 10 choose 1,"},{"Start":"02:59.305 ","End":"03:03.200","Text":"because these are also from combinatorics."},{"Start":"03:03.200 ","End":"03:06.830","Text":"Anyway, it doesn\u0027t matter how you pronounce it, that\u0027s how it\u0027s written."},{"Start":"03:06.830 ","End":"03:14.025","Text":"That is equal to 10 factorial over 1 factorial,"},{"Start":"03:14.025 ","End":"03:15.450","Text":"10 minus 1,"},{"Start":"03:15.450 ","End":"03:19.005","Text":"do it mentally, 9 factorial."},{"Start":"03:19.005 ","End":"03:20.625","Text":"We get 10,"},{"Start":"03:20.625 ","End":"03:22.500","Text":"9, 8,"},{"Start":"03:22.500 ","End":"03:26.415","Text":"7, 6, 5, 4, 3, 2,"},{"Start":"03:26.415 ","End":"03:31.440","Text":"1 over 1 factorial is just 1,"},{"Start":"03:31.440 ","End":"03:33.855","Text":"and then 9,"},{"Start":"03:33.855 ","End":"03:35.340","Text":"8, 7,"},{"Start":"03:35.340 ","End":"03:37.020","Text":"6, 5,"},{"Start":"03:37.020 ","End":"03:40.560","Text":"4, 3, 2, 1."},{"Start":"03:40.560 ","End":"03:44.250","Text":"This whole bit cancels with this,"},{"Start":"03:44.250 ","End":"03:47.265","Text":"and what we\u0027re left with is 10/1,"},{"Start":"03:47.265 ","End":"03:50.400","Text":"which is simply 10."},{"Start":"03:50.400 ","End":"03:52.340","Text":"Now the last part,"},{"Start":"03:52.340 ","End":"03:58.340","Text":"which is the only part which is abstract or general,"},{"Start":"03:58.340 ","End":"04:00.365","Text":"not specific numbers,"},{"Start":"04:00.365 ","End":"04:04.020","Text":"n over n minus 1."},{"Start":"04:04.520 ","End":"04:07.140","Text":"That\u0027s part D,"},{"Start":"04:07.140 ","End":"04:10.185","Text":"and here, n minus 1 here."},{"Start":"04:10.185 ","End":"04:12.335","Text":"Now we\u0027ve lost the formula,"},{"Start":"04:12.335 ","End":"04:13.760","Text":"but it\u0027s back there,"},{"Start":"04:13.760 ","End":"04:15.994","Text":"we take the factorial of this."},{"Start":"04:15.994 ","End":"04:17.840","Text":"You should memorize that formula."},{"Start":"04:17.840 ","End":"04:19.220","Text":"That\u0027s the definition,"},{"Start":"04:19.220 ","End":"04:22.460","Text":"you can\u0027t go very far without that formula."},{"Start":"04:22.460 ","End":"04:26.705","Text":"Over this factorial, which is n minus 1 factorial."},{"Start":"04:26.705 ","End":"04:31.710","Text":"This time, I will write out the subtraction before I actually do the subtraction."},{"Start":"04:31.710 ","End":"04:36.480","Text":"It\u0027s n, the top minus the bottom, n minus 1."},{"Start":"04:36.480 ","End":"04:42.270","Text":"I\u0027m going to need a pair of brackets here and factorial, that\u0027s equal 2."},{"Start":"04:42.270 ","End":"04:47.040","Text":"N minus n minus 1 is just 1 factorial."},{"Start":"04:47.040 ","End":"04:48.920","Text":"We have n factorial,"},{"Start":"04:48.920 ","End":"04:52.175","Text":"which is n, n minus 1."},{"Start":"04:52.175 ","End":"04:55.545","Text":"You have to put dot, dot, dot because we don\u0027t know what n is."},{"Start":"04:55.545 ","End":"04:58.580","Text":"Down to 2 times 1,"},{"Start":"04:58.580 ","End":"05:03.950","Text":"and then n minus 1 factorial is just the same thing,"},{"Start":"05:03.950 ","End":"05:05.975","Text":"but beginning with n minus 1,"},{"Start":"05:05.975 ","End":"05:07.960","Text":"n minus 2,"},{"Start":"05:07.960 ","End":"05:11.835","Text":"and so on down to 2 times 1."},{"Start":"05:11.835 ","End":"05:15.445","Text":"Then 1 factorial, which is just 1."},{"Start":"05:15.445 ","End":"05:20.550","Text":"Now this whole bit cancels with this whole bit."},{"Start":"05:21.010 ","End":"05:25.445","Text":"What we\u0027re left with is n over 1,"},{"Start":"05:25.445 ","End":"05:31.590","Text":"which is just n. Just give you a concrete example,"},{"Start":"05:31.590 ","End":"05:38.765","Text":"suppose I had 15/14."},{"Start":"05:38.765 ","End":"05:41.135","Text":"This what this pattern means."},{"Start":"05:41.135 ","End":"05:43.486","Text":"What\u0027s below is 1 less than what\u0027s above,"},{"Start":"05:43.486 ","End":"05:46.250","Text":"then we know that the answer is the 1 on the top,"},{"Start":"05:46.250 ","End":"05:47.690","Text":"that is equal to 15."},{"Start":"05:47.690 ","End":"05:49.505","Text":"That\u0027s just an example."},{"Start":"05:49.505 ","End":"05:54.120","Text":"We\u0027re done with this exercise."}],"ID":26928},{"Watched":false,"Name":"Exercise 2","Duration":"1m 36s","ChapterTopicVideoID":26108,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26108.jpeg","UploadDate":"2021-08-04T16:31:44.3730000","DurationForVideoObject":"PT1M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we have another proof question,"},{"Start":"00:03.510 ","End":"00:06.975","Text":"but not really as tough as a proof question."},{"Start":"00:06.975 ","End":"00:10.155","Text":"Just a matter of substituting in the formula."},{"Start":"00:10.155 ","End":"00:14.985","Text":"Perhaps a lighter reminder if we have n over k,"},{"Start":"00:14.985 ","End":"00:16.620","Text":"not really sure how to pronounce this,"},{"Start":"00:16.620 ","End":"00:19.442","Text":"I say n choose k from combinatorics,"},{"Start":"00:19.442 ","End":"00:23.730","Text":"anyway it\u0027s equal by definition to n factorial"},{"Start":"00:23.730 ","End":"00:30.293","Text":"over k factorial times n minus k factorial."},{"Start":"00:30.293 ","End":"00:40.230","Text":"Applying that here, we would get n is equal to n factorial over."},{"Start":"00:40.230 ","End":"00:44.015","Text":"Now n is k, so that\u0027s n factorial,"},{"Start":"00:44.015 ","End":"00:49.205","Text":"then n minus k factorial is n minus n factorial."},{"Start":"00:49.205 ","End":"00:55.280","Text":"Now, n factorial cancels with n factorial and"},{"Start":"00:55.280 ","End":"01:02.200","Text":"0 factorial is just 1 and so we end up with 1."},{"Start":"01:02.200 ","End":"01:06.650","Text":"This basically means that when we take a plus b to the power of n,"},{"Start":"01:06.650 ","End":"01:09.650","Text":"the first term is a to the n and its coefficient is 1."},{"Start":"01:09.650 ","End":"01:12.470","Text":"The last term is b to the n and its coefficient is 1,"},{"Start":"01:12.470 ","End":"01:15.465","Text":"the first and last coefficients are 1."},{"Start":"01:15.465 ","End":"01:19.110","Text":"Let\u0027s do the other 1 though, and 0,"},{"Start":"01:19.110 ","End":"01:27.755","Text":"so n factorial over 0 factorial n minus 0 factorial."},{"Start":"01:27.755 ","End":"01:30.950","Text":"Well, so same thing as what we had above 0 factorial here and here,"},{"Start":"01:30.950 ","End":"01:32.615","Text":"n factorial here and here,"},{"Start":"01:32.615 ","End":"01:36.540","Text":"so this will also equal 1 and that\u0027s it."}],"ID":26929},{"Watched":false,"Name":"Exercise 3","Duration":"7m 29s","ChapterTopicVideoID":26109,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26109.jpeg","UploadDate":"2021-08-04T16:33:48.9570000","DurationForVideoObject":"PT7M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"This exercise is a proof question,"},{"Start":"00:03.390 ","End":"00:10.095","Text":"and it turns out to be very useful for actually computing these binomial coefficients."},{"Start":"00:10.095 ","End":"00:16.410","Text":"What we have to show is that for all n and for all k which is less than n, now,"},{"Start":"00:16.410 ","End":"00:18.900","Text":"typically we won\u0027t take 0 and n,"},{"Start":"00:18.900 ","End":"00:22.455","Text":"we\u0027ll take it between 1 and n minus 1,"},{"Start":"00:22.455 ","End":"00:26.160","Text":"but theoretically it will work for the endpoints also."},{"Start":"00:26.160 ","End":"00:31.680","Text":"We have to show that this binomial coefficient can be described in 2 different ways."},{"Start":"00:31.680 ","End":"00:35.310","Text":"1 is using k factorial in the denominator,"},{"Start":"00:35.310 ","End":"00:38.100","Text":"the other uses n minus k factorial."},{"Start":"00:38.100 ","End":"00:43.240","Text":"In each case, we start from n and work our way down."},{"Start":"00:43.240 ","End":"00:45.030","Text":"You know what?"},{"Start":"00:45.030 ","End":"00:47.165","Text":"An example will illustrate this best."},{"Start":"00:47.165 ","End":"00:49.535","Text":"What I\u0027m going to say is that the same number"},{"Start":"00:49.535 ","End":"00:52.295","Text":"of factors on the top and the bottom, but we\u0027ll see this."},{"Start":"00:52.295 ","End":"00:57.680","Text":"I\u0027ll take the example of 8 choose 3."},{"Start":"00:57.680 ","End":"01:01.565","Text":"Now, if I use the first formula,"},{"Start":"01:01.565 ","End":"01:07.270","Text":"it says 8 times 7."},{"Start":"01:07.270 ","End":"01:12.335","Text":"What we have to do is go down up to n."},{"Start":"01:12.335 ","End":"01:22.110","Text":"N minus k on the bottom will be 5 and we\u0027ll go down to k plus 1 which is 4."},{"Start":"01:22.110 ","End":"01:23.550","Text":"It\u0027s 8, 7,"},{"Start":"01:23.550 ","End":"01:26.580","Text":"6, 5, 4."},{"Start":"01:26.580 ","End":"01:29.540","Text":"On the bottom, it\u0027s 5 factorial;"},{"Start":"01:29.540 ","End":"01:33.530","Text":"5, 4, 3, 2, 1."},{"Start":"01:33.530 ","End":"01:37.760","Text":"Notice that they really are lined up in pairs, the factors."},{"Start":"01:37.760 ","End":"01:42.140","Text":"This will help us to figure out the numerator."},{"Start":"01:42.140 ","End":"01:44.750","Text":"We won\u0027t have to figure out what is k plus 1 and all that,"},{"Start":"01:44.750 ","End":"01:47.419","Text":"we\u0027ll just write 5 factorial."},{"Start":"01:47.419 ","End":"01:52.565","Text":"The other expression is k factorial,"},{"Start":"01:52.565 ","End":"01:54.320","Text":"which is 3 factorial,"},{"Start":"01:54.320 ","End":"01:56.915","Text":"which is 3 times 2 times 1."},{"Start":"01:56.915 ","End":"01:59.540","Text":"On the top, like I said,"},{"Start":"01:59.540 ","End":"02:00.890","Text":"you don\u0027t have to interpret this,"},{"Start":"02:00.890 ","End":"02:07.490","Text":"you could just write starting from 8 just keep going down until you hit the last 1."},{"Start":"02:07.490 ","End":"02:08.929","Text":"But if you use the formula,"},{"Start":"02:08.929 ","End":"02:10.760","Text":"this is n minus k plus 1."},{"Start":"02:10.760 ","End":"02:14.555","Text":"8 minus 3 plus 1 is indeed 6."},{"Start":"02:14.555 ","End":"02:16.955","Text":"Now, why is this so?"},{"Start":"02:16.955 ","End":"02:19.970","Text":"Let\u0027s look at it in the example."},{"Start":"02:19.970 ","End":"02:21.845","Text":"If we\u0027re using the formula,"},{"Start":"02:21.845 ","End":"02:23.540","Text":"then we would say it\u0027s"},{"Start":"02:23.540 ","End":"02:32.870","Text":"8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over 3 factorial,"},{"Start":"02:32.870 ","End":"02:33.980","Text":"5 factorial."},{"Start":"02:33.980 ","End":"02:36.035","Text":"3 times 2 times 1."},{"Start":"02:36.035 ","End":"02:38.915","Text":"I would like to use bracket for emphasis."},{"Start":"02:38.915 ","End":"02:44.120","Text":"5 factorial; 5 times 4 times 3 times 2 times 1."},{"Start":"02:44.120 ","End":"02:51.540","Text":"Now, I can cancel either 1 of these string of factors."},{"Start":"02:51.540 ","End":"02:55.860","Text":"If we cancel the first string,"},{"Start":"02:55.860 ","End":"02:57.840","Text":"3, 2, 1 with 3, 2,"},{"Start":"02:57.840 ","End":"03:03.630","Text":"1, then we get the first expression."},{"Start":"03:03.630 ","End":"03:08.445","Text":"Notice, it\u0027s 8 down to 4 and then 5 factorial."},{"Start":"03:08.445 ","End":"03:12.560","Text":"But, if instead of doing this,"},{"Start":"03:12.560 ","End":"03:14.735","Text":"we cancel this,"},{"Start":"03:14.735 ","End":"03:18.215","Text":"and then we go all the way up to 5,"},{"Start":"03:18.215 ","End":"03:21.275","Text":"then we would get this expression."},{"Start":"03:21.275 ","End":"03:25.955","Text":"We get 8, 7, 6 over 3, 2, 1."},{"Start":"03:25.955 ","End":"03:30.500","Text":"Now, typically we choose the 1 that is smaller,"},{"Start":"03:30.500 ","End":"03:32.059","Text":"I mean has less factors."},{"Start":"03:32.059 ","End":"03:33.830","Text":"I would say, which is smaller?"},{"Start":"03:33.830 ","End":"03:37.950","Text":"Well, k or n minus k. In this case,"},{"Start":"03:37.950 ","End":"03:40.035","Text":"3 is less than 8."},{"Start":"03:40.035 ","End":"03:44.940","Text":"We would go with this 1,"},{"Start":"03:44.940 ","End":"03:48.795","Text":"which is this 1 but if it was say a 5 here,"},{"Start":"03:48.795 ","End":"03:51.975","Text":"then we\u0027d go with this 1."},{"Start":"03:51.975 ","End":"03:57.720","Text":"Now, let\u0027s do the general case and we\u0027ll prove it."},{"Start":"03:57.720 ","End":"04:02.700","Text":"Let\u0027s start off with the definition,"},{"Start":"04:02.700 ","End":"04:09.890","Text":"n over k or n choose k is n times n minus 1,"},{"Start":"04:09.890 ","End":"04:19.700","Text":"and so on, down to 2 times 1 over."},{"Start":"04:19.700 ","End":"04:24.410","Text":"Then we want k factorial,"},{"Start":"04:24.410 ","End":"04:34.620","Text":"which is k times and so on down to 1,"},{"Start":"04:35.450 ","End":"04:38.895","Text":"and that\u0027s like here."},{"Start":"04:38.895 ","End":"04:42.615","Text":"Then the other 1 is n minus k factorial."},{"Start":"04:42.615 ","End":"04:49.280","Text":"It\u0027s n minus k times each time decreasing by 1."},{"Start":"04:49.280 ","End":"04:50.675","Text":"I won\u0027t write the middle bits,"},{"Start":"04:50.675 ","End":"04:53.940","Text":"just from n minus k down to 1."},{"Start":"04:54.170 ","End":"04:56.610","Text":"On the denominator,"},{"Start":"04:56.610 ","End":"04:59.240","Text":"I\u0027ll just leave it as in the formula,"},{"Start":"04:59.240 ","End":"05:04.570","Text":"k factorial n minus k factorial."},{"Start":"05:04.570 ","End":"05:12.054","Text":"Now, what I\u0027m going to do is rewrite it just like we did here in 2 different ways."},{"Start":"05:12.054 ","End":"05:14.620","Text":"In the first way,"},{"Start":"05:14.620 ","End":"05:17.245","Text":"I want the k factorial to cancel."},{"Start":"05:17.245 ","End":"05:19.265","Text":"What I can do,"},{"Start":"05:19.265 ","End":"05:22.125","Text":"I want to split it up into 2."},{"Start":"05:22.125 ","End":"05:27.075","Text":"On the 1 hand it equals n, n minus 1."},{"Start":"05:27.075 ","End":"05:29.580","Text":"I want the k factorial to cancel,"},{"Start":"05:29.580 ","End":"05:34.230","Text":"so I\u0027ll stop 1 short of k,"},{"Start":"05:34.230 ","End":"05:39.930","Text":"so we\u0027ll get k times k minus 1 down to 1."},{"Start":"05:39.930 ","End":"05:46.060","Text":"The 1 before k would be k plus 1."},{"Start":"05:49.460 ","End":"05:54.440","Text":"We can see that this part is k factorial."},{"Start":"05:54.440 ","End":"05:56.630","Text":"Now on the denominator,"},{"Start":"05:56.630 ","End":"06:05.230","Text":"we had k factorial and n minus k factorial."},{"Start":"06:05.230 ","End":"06:10.005","Text":"Then this cancels with this."},{"Start":"06:10.005 ","End":"06:18.990","Text":"What we\u0027re left with is the first formula, this 1 here."},{"Start":"06:19.000 ","End":"06:22.485","Text":"When we compare it, it\u0027s the same."},{"Start":"06:22.485 ","End":"06:26.310","Text":"On the other hand, we could split this numerator differently."},{"Start":"06:26.310 ","End":"06:29.445","Text":"We could go n, n minus 1."},{"Start":"06:29.445 ","End":"06:35.150","Text":"This time I want the n minus k factorial to cancel."},{"Start":"06:35.150 ","End":"06:36.980","Text":"Let\u0027s see where we\u0027ll continue to."},{"Start":"06:36.980 ","End":"06:40.670","Text":"I need to have n minus k factorial,"},{"Start":"06:40.670 ","End":"06:46.575","Text":"n minus k times n minus k minus 1,"},{"Start":"06:46.575 ","End":"06:50.385","Text":"and so on down to 1."},{"Start":"06:50.385 ","End":"06:58.135","Text":"The 1 before that would be n minus k plus 1."},{"Start":"06:58.135 ","End":"07:01.880","Text":"On the denominator, k factorial,"},{"Start":"07:01.880 ","End":"07:05.880","Text":"n minus k factorial."},{"Start":"07:06.320 ","End":"07:09.870","Text":"This time we\u0027ll cancel differently."},{"Start":"07:09.870 ","End":"07:13.335","Text":"This bit will go with this,"},{"Start":"07:13.335 ","End":"07:15.690","Text":"and what we\u0027re left with,"},{"Start":"07:15.690 ","End":"07:23.780","Text":"n down to n minus k plus 1 over k factorial is exactly this bit."},{"Start":"07:23.780 ","End":"07:30.150","Text":"That proves the second half. That\u0027s it. We\u0027re done."}],"ID":26930},{"Watched":false,"Name":"Exercise 4","Duration":"6m 48s","ChapterTopicVideoID":26110,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26110.jpeg","UploadDate":"2021-08-04T16:35:49.2100000","DurationForVideoObject":"PT6M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.175","Text":"In this exercise, we have to use"},{"Start":"00:02.175 ","End":"00:07.320","Text":"Newton\u0027s binomial expansion to write out x plus y to the 6."},{"Start":"00:07.320 ","End":"00:09.090","Text":"I\u0027ll just write out the formula,"},{"Start":"00:09.090 ","End":"00:10.393","Text":"at least in 1 version."},{"Start":"00:10.393 ","End":"00:19.675","Text":"It\u0027s written as a plus b to the power of n is the sum k goes from 0 to n,"},{"Start":"00:19.675 ","End":"00:28.145","Text":"and choose k a to the n minus k b^k."},{"Start":"00:28.145 ","End":"00:33.515","Text":"Replace a and b by x and y and replace n by 6."},{"Start":"00:33.515 ","End":"00:44.645","Text":"What we\u0027ll get is that this is equal to the sum k goes from 0 to 6."},{"Start":"00:44.645 ","End":"00:50.100","Text":"6 choose k,"},{"Start":"00:50.360 ","End":"00:56.220","Text":"x to the power of 6 minus k,"},{"Start":"00:56.220 ","End":"01:05.260","Text":"that\u0027s this bit, y^k."},{"Start":"01:05.260 ","End":"01:12.935","Text":"Now let\u0027s see. I want to drop the Sigma and we\u0027ll get actually 7 terms from 0 to 6."},{"Start":"01:12.935 ","End":"01:16.310","Text":"If k is 0, we get 6,"},{"Start":"01:16.310 ","End":"01:22.205","Text":"0 x^6, y^0."},{"Start":"01:22.205 ","End":"01:28.925","Text":"Next 6, 1, x^5, y^1."},{"Start":"01:28.925 ","End":"01:33.110","Text":"Each time, we\u0027re reducing something by 1 or increasing by 1."},{"Start":"01:33.110 ","End":"01:37.975","Text":"Here it\u0027s 6, 2 x^4, y squared."},{"Start":"01:37.975 ","End":"01:40.605","Text":"There we go, plus,"},{"Start":"01:40.605 ","End":"01:42.045","Text":"then we need 6,"},{"Start":"01:42.045 ","End":"01:45.330","Text":"3, x cubed,"},{"Start":"01:45.330 ","End":"01:47.940","Text":"y cubed; 6,"},{"Start":"01:47.940 ","End":"01:55.198","Text":"4, x squared, y^4."},{"Start":"01:55.198 ","End":"01:57.080","Text":"Here the powers of y are increasing."},{"Start":"01:57.080 ","End":"02:03.040","Text":"Then 6, 5,"},{"Start":"02:03.040 ","End":"02:06.500","Text":"x^1, y^5, and finally 6,"},{"Start":"02:06.500 ","End":"02:15.180","Text":"6, x^0,"},{"Start":"02:15.180 ","End":"02:16.820","Text":"y^6."},{"Start":"02:16.820 ","End":"02:18.650","Text":"Now to calculate these coefficients,"},{"Start":"02:18.650 ","End":"02:25.190","Text":"I would like to use 1 of the previous problems which gave an alternate form for n choose"},{"Start":"02:25.190 ","End":"02:32.690","Text":"k as on the denominator n minus k factorial,"},{"Start":"02:32.690 ","End":"02:34.280","Text":"and on the numerator,"},{"Start":"02:34.280 ","End":"02:37.880","Text":"n times n minus 1 down,"},{"Start":"02:37.880 ","End":"02:41.450","Text":"up to dot, dot, dot."},{"Start":"02:41.450 ","End":"02:45.710","Text":"The last one would be k plus 1."},{"Start":"02:45.710 ","End":"02:48.755","Text":"It turns out the same number of terms here and here,"},{"Start":"02:48.755 ","End":"02:51.070","Text":"so let\u0027s use that."},{"Start":"02:51.070 ","End":"02:53.400","Text":"The first and the last coefficients are always 1."},{"Start":"02:53.400 ","End":"02:55.850","Text":"There\u0027s no point wasting time with that."},{"Start":"02:55.850 ","End":"03:00.410","Text":"This is 1. This is x^6, y^0 is 1."},{"Start":"03:00.410 ","End":"03:09.245","Text":"The next one, 6 choose 1 means that we get using this formula,"},{"Start":"03:09.245 ","End":"03:17.440","Text":"I should have written the other form of this."},{"Start":"03:17.440 ","End":"03:24.410","Text":"The other form is we\u0027re on the bottom we put k factorial instead of n minus k factorial,"},{"Start":"03:24.410 ","End":"03:25.835","Text":"and here we have n,"},{"Start":"03:25.835 ","End":"03:30.390","Text":"n minus 1 and it goes down to instead of k plus 1,"},{"Start":"03:30.390 ","End":"03:33.756","Text":"this time we have n minus k plus 1."},{"Start":"03:33.756 ","End":"03:42.170","Text":"The one I would suggest using is the one where the denominator is the smallest,"},{"Start":"03:42.170 ","End":"03:48.000","Text":"whatever smaller, k or n minus k. Back here."},{"Start":"03:48.000 ","End":"03:50.185","Text":"For the first and the last,"},{"Start":"03:50.185 ","End":"03:52.105","Text":"we always know it\u0027s going to be 1,"},{"Start":"03:52.105 ","End":"03:53.320","Text":"so this one and this one,"},{"Start":"03:53.320 ","End":"03:55.195","Text":"we don\u0027t use the formula for,"},{"Start":"03:55.195 ","End":"03:56.545","Text":"to say this is 1,"},{"Start":"03:56.545 ","End":"03:58.280","Text":"this is x^6,"},{"Start":"03:58.280 ","End":"04:00.240","Text":"and y^0 is 1,"},{"Start":"04:00.240 ","End":"04:02.925","Text":"and this one is y^6."},{"Start":"04:02.925 ","End":"04:06.730","Text":"That\u0027s a good start. Now and the rest we can use the formula."},{"Start":"04:06.730 ","End":"04:09.730","Text":"Now here, k is smaller than n minus k,"},{"Start":"04:09.730 ","End":"04:12.925","Text":"so we\u0027ll just put k factorial,"},{"Start":"04:12.925 ","End":"04:14.610","Text":"which is just 1,"},{"Start":"04:14.610 ","End":"04:18.250","Text":"and on the numerator we"},{"Start":"04:18.250 ","End":"04:22.240","Text":"just start from 6 and keep filling with the same number of elements."},{"Start":"04:22.240 ","End":"04:27.055","Text":"It\u0027s just 6 over 1, x^5, y."},{"Start":"04:27.055 ","End":"04:30.440","Text":"The pattern will be clearer when k is bigger than 1."},{"Start":"04:30.440 ","End":"04:33.680","Text":"Here, for example, 2 is less than 6 minus 2,"},{"Start":"04:33.680 ","End":"04:35.600","Text":"so we start off with 2 factorial,"},{"Start":"04:35.600 ","End":"04:36.950","Text":"which is 2 times 1,"},{"Start":"04:36.950 ","End":"04:38.375","Text":"and on the numerator,"},{"Start":"04:38.375 ","End":"04:44.765","Text":"we start from 6 and we count down and match factor for factor."},{"Start":"04:44.765 ","End":"04:48.180","Text":"Term is when you add, factor is when you multiply."},{"Start":"04:48.190 ","End":"04:52.275","Text":"x^4, y squared."},{"Start":"04:52.275 ","End":"04:55.916","Text":"Here, 3 or 6 minus 3, same thing,"},{"Start":"04:55.916 ","End":"04:58.160","Text":"it\u0027s going to be 3 factorial,"},{"Start":"04:58.160 ","End":"04:59.900","Text":"which is 3 times 2 times 1."},{"Start":"04:59.900 ","End":"05:02.600","Text":"Here we start with 6 and count down 6,"},{"Start":"05:02.600 ","End":"05:03.920","Text":"5, 4,"},{"Start":"05:03.920 ","End":"05:09.050","Text":"and it will be x cubed, y cubed."},{"Start":"05:09.050 ","End":"05:13.730","Text":"Then 6 minus 4 is smaller than 4,"},{"Start":"05:13.730 ","End":"05:16.355","Text":"so we\u0027ll use 2 factorial here."},{"Start":"05:16.355 ","End":"05:21.730","Text":"We\u0027ll get 2 times 1,"},{"Start":"05:21.730 ","End":"05:25.110","Text":"then from here 6 times 5."},{"Start":"05:25.110 ","End":"05:28.080","Text":"It\u0027s x squared, y^4."},{"Start":"05:28.080 ","End":"05:31.170","Text":"Here 6 minus 5 is smaller than 5,"},{"Start":"05:31.170 ","End":"05:36.600","Text":"so we use the 6 minus 5 is 1 factorial."},{"Start":"05:36.600 ","End":"05:39.270","Text":"Here we just put the 6."},{"Start":"05:39.270 ","End":"05:47.320","Text":"We need x, y^5 here."},{"Start":"05:47.320 ","End":"05:50.585","Text":"Now it\u0027s just the computational part."},{"Start":"05:50.585 ","End":"05:56.045","Text":"Let\u0027s see if it got the equals here and the equals here."},{"Start":"05:56.045 ","End":"06:02.485","Text":"x^6, 6 over 1 is 6, 6x^5 y."},{"Start":"06:02.485 ","End":"06:05.330","Text":"6 times 5 over 2 times 1, 2 goes into 6,"},{"Start":"06:05.330 ","End":"06:10.830","Text":"3 times 5 is 15x^4 y squared."},{"Start":"06:10.830 ","End":"06:12.960","Text":"Then here 3 times 2 goes with 6,"},{"Start":"06:12.960 ","End":"06:19.225","Text":"so that gives me 20x cubed, y cubed."},{"Start":"06:19.225 ","End":"06:22.250","Text":"We get the same things."},{"Start":"06:22.250 ","End":"06:24.860","Text":"Look, this is the same as this and this is the same as this."},{"Start":"06:24.860 ","End":"06:27.335","Text":"In fact, it always comes out symmetrical."},{"Start":"06:27.335 ","End":"06:30.231","Text":"Let me just copy the coefficients 15,"},{"Start":"06:30.231 ","End":"06:33.345","Text":"this time it\u0027s x squared y^4."},{"Start":"06:33.345 ","End":"06:40.470","Text":"Then copy there\u0027s a 6xy^5 plus y^6."},{"Start":"06:40.470 ","End":"06:43.785","Text":"That\u0027s the answer for x plus y^6,"},{"Start":"06:43.785 ","End":"06:48.610","Text":"7 terms, and we\u0027re done."}],"ID":26931},{"Watched":false,"Name":"Exercise 5","Duration":"8m 1s","ChapterTopicVideoID":26111,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26111.jpeg","UploadDate":"2021-08-04T16:38:01.8770000","DurationForVideoObject":"PT8M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.275","Text":"In this exercise, we have to compute the binomial expansion of 3a minus 4b^5."},{"Start":"00:07.275 ","End":"00:09.630","Text":"I\u0027ll write the formula,"},{"Start":"00:09.630 ","End":"00:12.210","Text":"I\u0027ll copy it from a previous exercise."},{"Start":"00:12.210 ","End":"00:13.890","Text":"Here we are,"},{"Start":"00:13.890 ","End":"00:18.930","Text":"so n will be 5, and instead of a,"},{"Start":"00:18.930 ","End":"00:21.765","Text":"we will have 3a,"},{"Start":"00:21.765 ","End":"00:23.670","Text":"and wherever we see b,"},{"Start":"00:23.670 ","End":"00:27.315","Text":"we\u0027ll replace that by minus 4b."},{"Start":"00:27.315 ","End":"00:36.195","Text":"What we\u0027ll get is that this will equal the sum k goes from"},{"Start":"00:36.195 ","End":"00:45.960","Text":"0 to 5 of 5 choose k,"},{"Start":"00:45.960 ","End":"00:52.250","Text":"3a, we need the brackets here, of course,"},{"Start":"00:52.250 ","End":"00:58.070","Text":"to the power of 5 minus"},{"Start":"00:58.070 ","End":"01:05.090","Text":"k and minus 4b^k."},{"Start":"01:05.090 ","End":"01:08.330","Text":"This is going to cause an alternation in signs, of course,"},{"Start":"01:08.330 ","End":"01:14.285","Text":"because when you have a minus to an even power it\u0027s plus,"},{"Start":"01:14.285 ","End":"01:17.510","Text":"and to an odd power it\u0027s going to stay minus."},{"Start":"01:17.510 ","End":"01:20.255","Text":"Let\u0027s see what this comes out."},{"Start":"01:20.255 ","End":"01:22.510","Text":"We\u0027ll break up the Sigma,"},{"Start":"01:22.510 ","End":"01:26.010","Text":"so we have 5 over 0,"},{"Start":"01:26.010 ","End":"01:29.560","Text":"will be 6 terms from 0 to 5."},{"Start":"01:30.470 ","End":"01:37.185","Text":"3a^5 minus 4b^0."},{"Start":"01:37.185 ","End":"01:40.170","Text":"Then that k equals 1,"},{"Start":"01:40.170 ","End":"01:42.300","Text":"5, 1, 3a,"},{"Start":"01:42.300 ","End":"01:45.570","Text":"this is going to be 4,"},{"Start":"01:45.570 ","End":"01:51.220","Text":"and this decrease and this increase to the power of 1."},{"Start":"01:51.890 ","End":"01:57.000","Text":"Next, 5 choose 2,"},{"Start":"01:57.000 ","End":"02:07.005","Text":"3a cubed minus 4b squared plus,"},{"Start":"02:07.005 ","End":"02:10.260","Text":"continuing the pattern, 5,"},{"Start":"02:10.260 ","End":"02:11.880","Text":"0, 1, 2, it\u0027s 5,"},{"Start":"02:11.880 ","End":"02:17.655","Text":"3, it\u0027s 5 over k, 3a."},{"Start":"02:17.655 ","End":"02:20.050","Text":"Now where are we up to?"},{"Start":"02:20.410 ","End":"02:23.235","Text":"This goes down 1,"},{"Start":"02:23.235 ","End":"02:28.650","Text":"the other 1 goes up 1 minus 4b^3."},{"Start":"02:28.650 ","End":"02:32.250","Text":"Next, 5 choose 4,"},{"Start":"02:32.250 ","End":"02:39.190","Text":"3a continues to decrease down to 1 minus 4b^4."},{"Start":"02:39.190 ","End":"02:42.490","Text":"Finally, 5 choose 5,"},{"Start":"02:42.490 ","End":"02:50.750","Text":"3a^0, and the minus 4b^5."},{"Start":"02:52.910 ","End":"02:55.400","Text":"Let\u0027s do some computations."},{"Start":"02:55.400 ","End":"02:56.540","Text":"Well, the first and the last,"},{"Start":"02:56.540 ","End":"02:58.504","Text":"we don\u0027t need to compute."},{"Start":"02:58.504 ","End":"03:04.255","Text":"These coefficients are always 1 and to the power of 0 we can ignore."},{"Start":"03:04.255 ","End":"03:11.300","Text":"Here we get 3a^5 and I\u0027ll just leave it like that for the moment."},{"Start":"03:11.300 ","End":"03:17.595","Text":"The next 1 will be 5 choose 1,"},{"Start":"03:17.595 ","End":"03:20.020","Text":"we\u0027ll use the formula."},{"Start":"03:22.190 ","End":"03:29.300","Text":"You could just get used to the fact that when it\u0027s 1 on the number here,"},{"Start":"03:29.300 ","End":"03:31.550","Text":"it\u0027s just equal to this number above."},{"Start":"03:31.550 ","End":"03:33.080","Text":"If I write it as a product,"},{"Start":"03:33.080 ","End":"03:35.000","Text":"it\u0027s just 5 over 1."},{"Start":"03:35.000 ","End":"03:38.955","Text":"We just go from 5 down to just 5."},{"Start":"03:38.955 ","End":"03:45.810","Text":"Well, I\u0027ll write it as 5 over 1 factorial and then we"},{"Start":"03:45.810 ","End":"03:54.000","Text":"have 3a^4 minus 4b^1,"},{"Start":"03:54.000 ","End":"03:56.385","Text":"so I won\u0027t bother writing the 1."},{"Start":"03:56.385 ","End":"03:58.590","Text":"Next, now 5,"},{"Start":"03:58.590 ","End":"04:03.210","Text":"2 means we go 2 terms, 2 times 1,"},{"Start":"04:03.210 ","End":"04:06.765","Text":"and then we just go down from 5,"},{"Start":"04:06.765 ","End":"04:13.455","Text":"3a cubed minus 4b squared."},{"Start":"04:13.455 ","End":"04:17.775","Text":"Now here, 5 choose 3,"},{"Start":"04:17.775 ","End":"04:23.250","Text":"we can use the formula where there\u0027s a variation instead of 3,"},{"Start":"04:23.250 ","End":"04:24.900","Text":"we put 5 minus 3,"},{"Start":"04:24.900 ","End":"04:26.040","Text":"which is 2,"},{"Start":"04:26.040 ","End":"04:28.650","Text":"so we\u0027ll get the same thing as this."},{"Start":"04:28.650 ","End":"04:31.170","Text":"It\u0027s also 5, 4,"},{"Start":"04:31.170 ","End":"04:36.090","Text":"2 times 1 and then"},{"Start":"04:36.090 ","End":"04:44.355","Text":"3a squared minus 4b cubed."},{"Start":"04:44.355 ","End":"04:48.480","Text":"Then 5, 4 is the same as 5,"},{"Start":"04:48.480 ","End":"04:51.945","Text":"1, we can always do the 5 minus 4,"},{"Start":"04:51.945 ","End":"04:56.105","Text":"but these things always come out to be symmetric so we\u0027ll get again,"},{"Start":"04:56.105 ","End":"04:58.435","Text":"5 over 1,"},{"Start":"04:58.435 ","End":"05:08.130","Text":"this time 3a and then minus 4b^4 from here."},{"Start":"05:10.120 ","End":"05:13.070","Text":"Let me squeeze it in."},{"Start":"05:13.070 ","End":"05:15.305","Text":"Now I have room."},{"Start":"05:15.305 ","End":"05:18.940","Text":"The last 1, the coefficient is always 1,"},{"Start":"05:18.940 ","End":"05:21.390","Text":"and this to the power of 0 we ignore,"},{"Start":"05:21.390 ","End":"05:25.210","Text":"so it\u0027s just minus 4b^5."},{"Start":"05:25.370 ","End":"05:28.500","Text":"Continuing,"},{"Start":"05:28.500 ","End":"05:39.030","Text":"let\u0027s compute 3^5 is 243a^5."},{"Start":"05:39.030 ","End":"05:45.480","Text":"Next, 5 times 3^4 times minus 4,"},{"Start":"05:45.480 ","End":"05:51.480","Text":"it will be minus 3^4 is 81,"},{"Start":"05:51.480 ","End":"05:53.850","Text":"5 times 4 is 20,"},{"Start":"05:53.850 ","End":"05:55.570","Text":"so"},{"Start":"05:58.670 ","End":"06:05.610","Text":"1,620a^4b."},{"Start":"06:05.610 ","End":"06:12.755","Text":"Next, these are 5 times 4 over 2 times 1 is 10,"},{"Start":"06:12.755 ","End":"06:17.550","Text":"and that will make this 1 10 also,"},{"Start":"06:17.630 ","End":"06:21.220","Text":"and the rest of them are straightforward."},{"Start":"06:22.820 ","End":"06:25.680","Text":"3 cubed is 27,"},{"Start":"06:25.680 ","End":"06:28.905","Text":"4 squared is 16,"},{"Start":"06:28.905 ","End":"06:33.509","Text":"27 times 16 is 432 times 10,"},{"Start":"06:33.509 ","End":"06:36.810","Text":"makes it 4,320,"},{"Start":"06:36.810 ","End":"06:39.860","Text":"it\u0027s a plus because of the even power here,"},{"Start":"06:39.860 ","End":"06:43.990","Text":"a cubed b squared."},{"Start":"06:43.990 ","End":"06:49.815","Text":"Then here we have 4 cubed is"},{"Start":"06:49.815 ","End":"06:56.790","Text":"64 times 9 is 576,"},{"Start":"06:56.790 ","End":"06:58.350","Text":"but it\u0027s also times 10,"},{"Start":"06:58.350 ","End":"07:00.370","Text":"so let\u0027s stick another 0 on,"},{"Start":"07:00.370 ","End":"07:05.400","Text":"and that\u0027s a squared b cubed."},{"Start":"07:05.400 ","End":"07:13.045","Text":"Then 4^4 is 256,"},{"Start":"07:13.045 ","End":"07:19.055","Text":"256 times 15,"},{"Start":"07:19.055 ","End":"07:25.050","Text":"I make that 3,840 and this time it will"},{"Start":"07:25.050 ","End":"07:30.855","Text":"be minus, and it\u0027s ab^4."},{"Start":"07:30.855 ","End":"07:36.510","Text":"Finally, 4^5 is 1,024b^5."},{"Start":"07:36.510 ","End":"07:41.010","Text":"Let\u0027s see if the sign is alternate, minus, plus."},{"Start":"07:41.010 ","End":"07:47.410","Text":"I did something wrong here, minus, plus."},{"Start":"07:47.840 ","End":"07:51.840","Text":"Sorry, the sign should alternate of course."},{"Start":"07:51.840 ","End":"07:55.740","Text":"Minus, plus, minus, plus, minus."},{"Start":"07:55.740 ","End":"07:58.340","Text":"That\u0027s the answer, a bit messy,"},{"Start":"07:58.340 ","End":"08:01.830","Text":"but there we go, we\u0027re done."}],"ID":26932},{"Watched":false,"Name":"Exercise 6","Duration":"3m 26s","ChapterTopicVideoID":26112,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26112.jpeg","UploadDate":"2021-08-04T16:38:58.4130000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.760","Text":"In this exercise, we have to write the first 4 terms in"},{"Start":"00:03.760 ","End":"00:08.065","Text":"the binomial expansion of x plus y to the power of 16."},{"Start":"00:08.065 ","End":"00:10.360","Text":"Now, you\u0027re supposed to know the formula,"},{"Start":"00:10.360 ","End":"00:12.040","Text":"but in case you\u0027ve forgotten it,"},{"Start":"00:12.040 ","End":"00:13.375","Text":"I\u0027ve written it here."},{"Start":"00:13.375 ","End":"00:20.855","Text":"What we have now is x plus y to the power of 16."},{"Start":"00:20.855 ","End":"00:25.020","Text":"N is 16 and a and b are x and y."},{"Start":"00:25.020 ","End":"00:31.620","Text":"This is going to equal 16 choose 0,"},{"Start":"00:31.620 ","End":"00:34.510","Text":"x to the power of,"},{"Start":"00:34.510 ","End":"00:38.270","Text":"we start off with the n minus k,"},{"Start":"00:38.270 ","End":"00:40.520","Text":"so that\u0027s 16 minus 0."},{"Start":"00:40.520 ","End":"00:45.005","Text":"That\u0027s 16, y to the 0."},{"Start":"00:45.005 ","End":"00:50.505","Text":"Next term, 16, 1,"},{"Start":"00:50.505 ","End":"00:54.105","Text":"x to the 16 minus 1,"},{"Start":"00:54.105 ","End":"00:56.970","Text":"y to the power of 1."},{"Start":"00:56.970 ","End":"00:59.430","Text":"Then just keep going."},{"Start":"00:59.430 ","End":"01:00.885","Text":"16, 2,"},{"Start":"01:00.885 ","End":"01:06.885","Text":"x to the 14, y squared,"},{"Start":"01:06.885 ","End":"01:11.745","Text":"and 16 choose 3,"},{"Start":"01:11.745 ","End":"01:17.715","Text":"x to the 13, y cubed."},{"Start":"01:17.715 ","End":"01:20.220","Text":"Now, the first 1 is always 1,"},{"Start":"01:20.220 ","End":"01:21.690","Text":"and so that\u0027s the easiest."},{"Start":"01:21.690 ","End":"01:22.920","Text":"Y to the 0 is 1,"},{"Start":"01:22.920 ","End":"01:24.975","Text":"so it\u0027s x to the 16th."},{"Start":"01:24.975 ","End":"01:28.760","Text":"Now, this we\u0027ve computed enough and it\u0027s over 1, it\u0027s just 16."},{"Start":"01:28.760 ","End":"01:32.700","Text":"Don\u0027t bother with expanding it."},{"Start":"01:32.700 ","End":"01:35.265","Text":"X to the 15th,"},{"Start":"01:35.265 ","End":"01:37.380","Text":"y to the 1 is y."},{"Start":"01:37.380 ","End":"01:41.940","Text":"Now 16 choose 2 is 16."},{"Start":"01:41.940 ","End":"01:43.200","Text":"Well, do the denominator first,"},{"Start":"01:43.200 ","End":"01:47.330","Text":"is 2 factorial, 16 times 15,"},{"Start":"01:47.330 ","End":"01:49.430","Text":"same number of factors,"},{"Start":"01:49.430 ","End":"01:53.390","Text":"x to the 14th, y squared."},{"Start":"01:53.390 ","End":"01:57.250","Text":"Then here we have 3 factorial on the bottom,"},{"Start":"01:57.250 ","End":"02:01.330","Text":"or we could have done 13 factorial, whichever is smaller."},{"Start":"02:01.330 ","End":"02:05.750","Text":"Then 16, 15, 14."},{"Start":"02:05.750 ","End":"02:07.099","Text":"You don\u0027t have to do any computation."},{"Start":"02:07.099 ","End":"02:09.440","Text":"See, you have to make sure that 3, 2,"},{"Start":"02:09.440 ","End":"02:12.770","Text":"1 to the 3 factors here."},{"Start":"02:12.770 ","End":"02:15.500","Text":"X to the 13th,"},{"Start":"02:15.500 ","End":"02:19.675","Text":"y cubed, and then just numerical computation."},{"Start":"02:19.675 ","End":"02:21.660","Text":"X to the 16th,"},{"Start":"02:21.660 ","End":"02:25.365","Text":"here I\u0027m copying,16x to the 15th,"},{"Start":"02:25.365 ","End":"02:29.040","Text":"y, 2 into 16 goes 8 times."},{"Start":"02:29.040 ","End":"02:37.290","Text":"8 times 15 is120x to the 14th, y squared."},{"Start":"02:37.290 ","End":"02:39.600","Text":"Then, let\u0027s see."},{"Start":"02:39.600 ","End":"02:42.989","Text":"3 goes into 15,"},{"Start":"02:42.989 ","End":"02:45.135","Text":"so that\u0027s going to be 5."},{"Start":"02:45.135 ","End":"02:47.925","Text":"Then 2 into 16,"},{"Start":"02:47.925 ","End":"02:50.340","Text":"or into 14, doesn\u0027t matter,"},{"Start":"02:50.340 ","End":"02:53.250","Text":"is going to 16, 8 times."},{"Start":"02:53.250 ","End":"02:56.400","Text":"Now, we need to compute 8 times 5 times 14,"},{"Start":"02:56.400 ","End":"02:58.425","Text":"what\u0027s 40 times 14?"},{"Start":"02:58.425 ","End":"03:00.105","Text":"4 times 14 is 56,"},{"Start":"03:00.105 ","End":"03:08.760","Text":"so that\u0027s 560x to the 13th, y cubed."},{"Start":"03:08.760 ","End":"03:11.570","Text":"That\u0027s the first 4 terms."},{"Start":"03:11.570 ","End":"03:15.935","Text":"I should really indicate that this goes on and so on."},{"Start":"03:15.935 ","End":"03:18.260","Text":"In fact, we even know the last term."},{"Start":"03:18.260 ","End":"03:21.260","Text":"The last term will be y to the 16th."},{"Start":"03:21.260 ","End":"03:26.400","Text":"You don\u0027t have to write that, I\u0027m putting it in there. That\u0027s it."}],"ID":26933},{"Watched":false,"Name":"Exercise 7","Duration":"4m 2s","ChapterTopicVideoID":26113,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26113.jpeg","UploadDate":"2021-08-04T16:40:06.0170000","DurationForVideoObject":"PT4M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to write just the first 3 terms in"},{"Start":"00:03.270 ","End":"00:07.685","Text":"the binomial expansion of this plus this to the 10th,"},{"Start":"00:07.685 ","End":"00:10.025","Text":"and here\u0027s the formula."},{"Start":"00:10.025 ","End":"00:11.705","Text":"This will be a,"},{"Start":"00:11.705 ","End":"00:14.345","Text":"this will be b,"},{"Start":"00:14.345 ","End":"00:20.740","Text":"and this will be n. Let\u0027s see."},{"Start":"00:20.740 ","End":"00:23.775","Text":"What we get for this,"},{"Start":"00:23.775 ","End":"00:32.565","Text":"I\u0027ll copy it and what we get is 10 choose 0."},{"Start":"00:32.565 ","End":"00:38.445","Text":"The first 1, 3u squared to the power"},{"Start":"00:38.445 ","End":"00:45.570","Text":"of 10 and 2v cubed to the power of 0 and keep going."},{"Start":"00:45.570 ","End":"00:48.180","Text":"It\u0027s 10. This time it\u0027s 1,"},{"Start":"00:48.180 ","End":"00:52.090","Text":"3u squared, the 10 minus 1,"},{"Start":"00:52.090 ","End":"00:54.140","Text":"and to the cube,"},{"Start":"00:54.140 ","End":"00:59.359","Text":"to the power of 1 and the third term, 10 choose 2,"},{"Start":"00:59.359 ","End":"01:03.860","Text":"3u squared to the 10 minus 2,"},{"Start":"01:03.860 ","End":"01:08.380","Text":"2v cubed to the power of 2."},{"Start":"01:08.440 ","End":"01:10.970","Text":"Well, the first 3 terms,"},{"Start":"01:10.970 ","End":"01:13.190","Text":"so I\u0027ll write plus dot, dot, dot."},{"Start":"01:13.190 ","End":"01:14.960","Text":"I don\u0027t mean that this is equal to this."},{"Start":"01:14.960 ","End":"01:18.665","Text":"Yeah. What is this equal to?"},{"Start":"01:18.665 ","End":"01:20.690","Text":"Well, the first term is always easiest."},{"Start":"01:20.690 ","End":"01:24.630","Text":"It\u0027s just 3u squared to the 10th."},{"Start":"01:26.260 ","End":"01:32.030","Text":"It\u0027s in general, a to the n plus 10 choose 1."},{"Start":"01:32.030 ","End":"01:34.910","Text":"We know don\u0027t need to use the formula,"},{"Start":"01:34.910 ","End":"01:37.010","Text":"n over 1 is just n,"},{"Start":"01:37.010 ","End":"01:45.060","Text":"so that\u0027s 10 times 3u squared to the 1/9 to the cubed,"},{"Start":"01:45.060 ","End":"01:48.345","Text":"and then here we will use the formula,"},{"Start":"01:48.345 ","End":"01:53.389","Text":"so we can put on the bottom either k factorial or n minus k factorial,"},{"Start":"01:53.389 ","End":"01:55.670","Text":"whichever is smaller, so as 2 factorial,"},{"Start":"01:55.670 ","End":"01:56.920","Text":"2 times 1,"},{"Start":"01:56.920 ","End":"02:02.030","Text":"then the rule is you start with 10 and go down matching factor for factor."},{"Start":"02:02.030 ","End":"02:06.410","Text":"Although technically this would be up to 10 minus 2 plus 1,"},{"Start":"02:06.410 ","End":"02:11.585","Text":"which is 9 and then 3u squared to the 8,"},{"Start":"02:11.585 ","End":"02:17.760","Text":"2v cubed squared plus dot, dot, dot equals."},{"Start":"02:18.430 ","End":"02:22.400","Text":"I don\u0027t know that we have to compute 3 to the power of 10."},{"Start":"02:22.400 ","End":"02:23.675","Text":"If we don\u0027t have a calculator,"},{"Start":"02:23.675 ","End":"02:26.585","Text":"you could leave it as 3 to the 10th,"},{"Start":"02:26.585 ","End":"02:33.340","Text":"and then u squared to the 10th would be u to the 20th plus."},{"Start":"02:33.830 ","End":"02:35.850","Text":"Now 10 times 2,"},{"Start":"02:35.850 ","End":"02:45.355","Text":"I can multiply without a calculator and we\u0027ll write 20 and the 3 to the 9th."},{"Start":"02:45.355 ","End":"02:48.170","Text":"If you want to actually compute the number,"},{"Start":"02:48.170 ","End":"02:49.415","Text":"I\u0027ll leave it like that."},{"Start":"02:49.415 ","End":"02:53.855","Text":"Then we have u squared to the 9th is u to the 18th,"},{"Start":"02:53.855 ","End":"02:55.925","Text":"and then v cubed."},{"Start":"02:55.925 ","End":"03:01.020","Text":"The next one will be let\u0027s see what we can"},{"Start":"03:01.020 ","End":"03:05.945","Text":"compute this or do that at the side, or maybe here,"},{"Start":"03:05.945 ","End":"03:09.770","Text":"10 times 9 over 2 times 1 the"},{"Start":"03:09.770 ","End":"03:16.520","Text":"coefficient and then we\u0027ll need 3 to the 8th and then 2 squared."},{"Start":"03:16.520 ","End":"03:20.710","Text":"Well, some of it we can do without the calculator."},{"Start":"03:20.710 ","End":"03:26.945","Text":"This 2 can cancel with the channel or better still cancel it with this 2,"},{"Start":"03:26.945 ","End":"03:28.685","Text":"so what do we have here,"},{"Start":"03:28.685 ","End":"03:31.625","Text":"10 times 9 times 2,"},{"Start":"03:31.625 ","End":"03:39.120","Text":"so write it here, a 180."},{"Start":"03:39.120 ","End":"03:44.810","Text":"Then 3 to the 8th is a bit big to compute without a calculator,"},{"Start":"03:44.810 ","End":"03:47.675","Text":"so I\u0027ll just write it as 3 to the 8th."},{"Start":"03:47.675 ","End":"03:52.340","Text":"Now, u squared to the 8th is u to the 16th,"},{"Start":"03:52.340 ","End":"03:57.020","Text":"and v cubed squared is v to the 6 plus dot,"},{"Start":"03:57.020 ","End":"04:02.520","Text":"dot, dot, because this is just the first 3 terms and that\u0027s it."}],"ID":26934},{"Watched":false,"Name":"Exercise 8","Duration":"3m 1s","ChapterTopicVideoID":26114,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26114.jpeg","UploadDate":"2021-08-04T16:40:56.3770000","DurationForVideoObject":"PT3M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"In this exercise, we\u0027re asked to prove a formula relating to the binomial coefficients."},{"Start":"00:07.050 ","End":"00:10.695","Text":"The formula basically says what\u0027s written here,"},{"Start":"00:10.695 ","End":"00:17.945","Text":"that n choose k is the same as n choose n minus k. Before we get into proof,"},{"Start":"00:17.945 ","End":"00:23.015","Text":"I\u0027d like to give an example from 1 of the earlier exercises where we computed"},{"Start":"00:23.015 ","End":"00:28.040","Text":"all the binomial coefficients relating to n equals 6."},{"Start":"00:28.040 ","End":"00:32.150","Text":"We computed this or just write them in"},{"Start":"00:32.150 ","End":"00:37.560","Text":"this form and then all give the numerical results that we got,"},{"Start":"00:37.560 ","End":"00:42.420","Text":"6, 5, and finally 6, 6."},{"Start":"00:42.420 ","End":"00:47.226","Text":"Now the answers we got were 1 here, 6,"},{"Start":"00:47.226 ","End":"00:51.705","Text":"15, this was 20,"},{"Start":"00:51.705 ","End":"00:55.215","Text":"this was 15, this was 6,"},{"Start":"00:55.215 ","End":"00:56.860","Text":"and this was 1."},{"Start":"00:56.860 ","End":"01:03.710","Text":"Now, notice that there\u0027s a symmetry that this is equal to this,"},{"Start":"01:03.710 ","End":"01:06.335","Text":"and this is equal to this,"},{"Start":"01:06.335 ","End":"01:08.135","Text":"and this is equal to this."},{"Start":"01:08.135 ","End":"01:10.115","Text":"Well, this is equal to itself."},{"Start":"01:10.115 ","End":"01:11.720","Text":"Now, if you look at them,"},{"Start":"01:11.720 ","End":"01:13.835","Text":"this means that 6,0,"},{"Start":"01:13.835 ","End":"01:17.600","Text":"and 6,6 are the same."},{"Start":"01:17.600 ","End":"01:21.335","Text":"If I take 6 and subtract the 0,"},{"Start":"01:21.335 ","End":"01:22.645","Text":"I get the 6 here."},{"Start":"01:22.645 ","End":"01:25.920","Text":"6 minus 1 is 5 or 6 minus 5 is 1."},{"Start":"01:25.920 ","End":"01:31.185","Text":"In other words, if I subtract the lower 1 from the upper 1 and put it here,"},{"Start":"01:31.185 ","End":"01:32.655","Text":"you get the same thing,"},{"Start":"01:32.655 ","End":"01:36.150","Text":"6,2 and 6,4 of the same."},{"Start":"01:36.150 ","End":"01:38.280","Text":"Maybe I\u0027ll just mark that here."},{"Start":"01:38.280 ","End":"01:40.125","Text":"This is the same as this,"},{"Start":"01:40.125 ","End":"01:46.030","Text":"this is the same as this and this is itself because 6 minus 3 is 3."},{"Start":"01:46.790 ","End":"01:50.070","Text":"Next, 6 minus 4 is 2, and so on."},{"Start":"01:50.070 ","End":"01:55.885","Text":"You get the idea, now let\u0027s prove it. Here goes."},{"Start":"01:55.885 ","End":"01:58.200","Text":"I\u0027ll start with this,"},{"Start":"01:58.200 ","End":"02:00.000","Text":"and I\u0027ll reach this."},{"Start":"02:00.000 ","End":"02:04.841","Text":"N choose n minus k is equal to,"},{"Start":"02:04.841 ","End":"02:08.489","Text":"what\u0027s upstairs, factorial over,"},{"Start":"02:08.489 ","End":"02:15.215","Text":"what\u0027s downstairs, factorial times upstairs minus downstairs,"},{"Start":"02:15.215 ","End":"02:21.080","Text":"n minus n minus k factorial."},{"Start":"02:21.690 ","End":"02:25.755","Text":"Now, n minus n minus k is just k,"},{"Start":"02:25.755 ","End":"02:30.755","Text":"so what we get is n factorial over n"},{"Start":"02:30.755 ","End":"02:37.820","Text":"minus k factorial and then k factorial."},{"Start":"02:37.820 ","End":"02:42.560","Text":"Now, I can change the order of things."},{"Start":"02:42.560 ","End":"02:43.925","Text":"We\u0027re almost there,"},{"Start":"02:43.925 ","End":"02:50.570","Text":"all I have to do is write k factorial and then n minus k factorial."},{"Start":"02:50.570 ","End":"02:52.715","Text":"If you go look at the definition,"},{"Start":"02:52.715 ","End":"02:57.530","Text":"this is exactly the definition of n choose k. This equals this,"},{"Start":"02:57.530 ","End":"03:01.440","Text":"which is what we had to prove. We\u0027re done."}],"ID":26935},{"Watched":false,"Name":"Exercise 9","Duration":"7m 54s","ChapterTopicVideoID":26115,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26115.jpeg","UploadDate":"2021-08-04T16:43:07.4970000","DurationForVideoObject":"PT7M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.175","Text":"In this exercise, we have to prove an identity involving the binomial coefficients."},{"Start":"00:08.175 ","End":"00:10.650","Text":"I won\u0027t read it out, it\u0027s as written."},{"Start":"00:10.650 ","End":"00:12.960","Text":"But let\u0027s take an example first."},{"Start":"00:12.960 ","End":"00:20.685","Text":"Suppose I take n equals 5 and k equals 2,"},{"Start":"00:20.685 ","End":"00:26.025","Text":"then what I would get is 5 choose"},{"Start":"00:26.025 ","End":"00:32.070","Text":"k minus 1 is 1 and then k itself is 2."},{"Start":"00:32.070 ","End":"00:34.553","Text":"This, well we don\u0027t know,"},{"Start":"00:34.553 ","End":"00:35.790","Text":"this is what the claim is,"},{"Start":"00:35.790 ","End":"00:41.100","Text":"will equal n plus 1 which is 6 choose 2."},{"Start":"00:41.100 ","End":"00:49.485","Text":"Now, let\u0027s verify that 5 choose 1 is 5,"},{"Start":"00:49.485 ","End":"00:52.472","Text":"5 choose 2,"},{"Start":"00:52.472 ","End":"00:55.340","Text":"well, it\u0027s 2 factorial in the denominator,"},{"Start":"00:55.340 ","End":"01:02.250","Text":"and then 5 times 4 equals question mark."},{"Start":"01:02.250 ","End":"01:04.533","Text":"2 times 1."},{"Start":"01:04.533 ","End":"01:06.725","Text":"6 times 5."},{"Start":"01:06.725 ","End":"01:08.630","Text":"So what are we saying?"},{"Start":"01:08.630 ","End":"01:16.760","Text":"That 5 plus this comes out to be 10 and this comes out to be 2 into 6,3."},{"Start":"01:16.760 ","End":"01:18.875","Text":"3 times 5 is 15."},{"Start":"01:18.875 ","End":"01:21.620","Text":"Well, that\u0027s certainly true."},{"Start":"01:21.620 ","End":"01:24.590","Text":"Let\u0027s go and prove it."},{"Start":"01:24.590 ","End":"01:26.945","Text":"We\u0027ll use the definition."},{"Start":"01:26.945 ","End":"01:35.630","Text":"Well, start with the left-hand side and what we have n choose"},{"Start":"01:35.630 ","End":"01:45.290","Text":"k minus 1 is n factorial over now this thing factorial,"},{"Start":"01:45.290 ","End":"01:51.920","Text":"which is k minus 1 factorial and then n minus k minus 1 factorial,"},{"Start":"01:51.920 ","End":"01:54.342","Text":"n subtract k minus 1."},{"Start":"01:54.342 ","End":"01:58.565","Text":"I\u0027ll do it is n minus k plus 1 factorial."},{"Start":"01:58.565 ","End":"02:05.915","Text":"The other one is n factorial over k factorial,"},{"Start":"02:05.915 ","End":"02:09.935","Text":"n minus k factorial straight from the definition."},{"Start":"02:09.935 ","End":"02:12.499","Text":"Now that\u0027s the left-hand side."},{"Start":"02:12.499 ","End":"02:19.505","Text":"What we\u0027ll do is we\u0027ll start with this and we\u0027ll see if we can reach the right-hand side,"},{"Start":"02:19.505 ","End":"02:23.080","Text":"left-hand side, right-hand side."},{"Start":"02:23.080 ","End":"02:26.090","Text":"That\u0027s one way of showing equality stuff on"},{"Start":"02:26.090 ","End":"02:29.735","Text":"one side and keep developing it until we get to the other side."},{"Start":"02:29.735 ","End":"02:33.710","Text":"Let\u0027s see what we can do to simplify this."},{"Start":"02:33.710 ","End":"02:39.270","Text":"Now here we have k minus 1 factorial and here we have k factorial."},{"Start":"02:39.820 ","End":"02:42.680","Text":"I could say in general,"},{"Start":"02:42.680 ","End":"02:48.395","Text":"that some number r factorial"},{"Start":"02:48.395 ","End":"02:55.520","Text":"is equal to r times r minus 1 factorial."},{"Start":"02:55.520 ","End":"03:01.250","Text":"Because here it\u0027s all the numbers from r down to 1 multiplied together."},{"Start":"03:01.250 ","End":"03:06.765","Text":"If I take the first factor as r then I\u0027ll get from r minus 1 down to 1."},{"Start":"03:06.765 ","End":"03:11.750","Text":"For example, 6 factorial is 6 times 5 factorial."},{"Start":"03:11.750 ","End":"03:14.450","Text":"If you write down from 6 down to 1,"},{"Start":"03:14.450 ","End":"03:17.285","Text":"just take the 6 and we go from 5 down to 1."},{"Start":"03:17.285 ","End":"03:21.475","Text":"I\u0027m going to work on both of these simultaneously."},{"Start":"03:21.475 ","End":"03:25.255","Text":"Here n factorial, here n factorial."},{"Start":"03:25.255 ","End":"03:28.040","Text":"Now, if I take r to be k,"},{"Start":"03:28.040 ","End":"03:30.080","Text":"which is what I set out to do,"},{"Start":"03:30.080 ","End":"03:33.920","Text":"then here we\u0027ll have k minus 1 factorial."},{"Start":"03:33.920 ","End":"03:41.487","Text":"But here instead of k factorial I\u0027ll write k times k minus 1 factorial."},{"Start":"03:41.487 ","End":"03:44.270","Text":"Now these 2 are starting to look closer"},{"Start":"03:44.270 ","End":"03:47.494","Text":"to each other and I\u0027m going to use this rule again."},{"Start":"03:47.494 ","End":"03:50.375","Text":"So I can write this as"},{"Start":"03:50.375 ","End":"03:59.695","Text":"n minus k plus 1 times this thing without the plus 1."},{"Start":"03:59.695 ","End":"04:09.230","Text":"Because this is like r and I\u0027ll take r minus 1 will be just n minus k factorial."},{"Start":"04:09.230 ","End":"04:15.530","Text":"Here, just copy the n minus k factorial."},{"Start":"04:15.530 ","End":"04:19.685","Text":"Now, these are very similar."},{"Start":"04:19.685 ","End":"04:28.745","Text":"The differences being that here we have n minus k plus 1,"},{"Start":"04:28.745 ","End":"04:32.855","Text":"and here we have k but other than that,"},{"Start":"04:32.855 ","End":"04:36.275","Text":"both these expressions are the same."},{"Start":"04:36.275 ","End":"04:40.955","Text":"What we get if I take the common part out,"},{"Start":"04:40.955 ","End":"04:49.890","Text":"which is n factorial over k minus 1 factorial,"},{"Start":"04:49.990 ","End":"04:55.620","Text":"n minus k factorial."},{"Start":"04:58.070 ","End":"05:06.480","Text":"Then we have 1 over n minus k plus"},{"Start":"05:06.480 ","End":"05:14.835","Text":"1 plus 1 over k. So what we get,"},{"Start":"05:14.835 ","End":"05:22.570","Text":"now copy this n factorial over k minus 1 factorial,"},{"Start":"05:22.570 ","End":"05:25.195","Text":"n minus k factorial."},{"Start":"05:25.195 ","End":"05:26.994","Text":"Now if we add these,"},{"Start":"05:26.994 ","End":"05:30.445","Text":"if you put it over a common denominator,"},{"Start":"05:30.445 ","End":"05:32.920","Text":"which is this times this,"},{"Start":"05:32.920 ","End":"05:40.330","Text":"what we\u0027ll get is n minus k plus 1 and then"},{"Start":"05:40.330 ","End":"05:48.690","Text":"k and then we\u0027ll get k plus n minus k plus 1 and if you think about it,"},{"Start":"05:48.690 ","End":"05:53.800","Text":"k plus n minus k plus 1 is just n plus 1."},{"Start":"05:53.800 ","End":"05:56.755","Text":"So here we have n plus 1."},{"Start":"05:56.755 ","End":"05:59.350","Text":"Now, if we use this rule again,"},{"Start":"05:59.350 ","End":"06:04.900","Text":"in fact, we\u0027ll use this rule 3 times."},{"Start":"06:04.900 ","End":"06:10.120","Text":"First of all, if I take r to be n plus 1,"},{"Start":"06:10.120 ","End":"06:18.340","Text":"then these 2 the numerator combines to equal n plus 1 factorial."},{"Start":"06:18.340 ","End":"06:20.480","Text":"That\u0027s with this and this."},{"Start":"06:20.480 ","End":"06:27.530","Text":"Then if I take the k minus 1 factorial with the k that will give me"},{"Start":"06:27.530 ","End":"06:36.685","Text":"k factorial that\u0027s on the denominator and pattern should indicate."},{"Start":"06:36.685 ","End":"06:40.385","Text":"So we\u0027re taking this with this, with this rule,"},{"Start":"06:40.385 ","End":"06:42.628","Text":"this with this,"},{"Start":"06:42.628 ","End":"06:47.685","Text":"and then again, this with this."},{"Start":"06:47.685 ","End":"06:49.550","Text":"The one we haven\u0027t done is this with this,"},{"Start":"06:49.550 ","End":"06:54.330","Text":"and that just becomes n minus k plus 1 factorial."},{"Start":"06:56.800 ","End":"07:00.380","Text":"Now instead of n minus k plus 1,"},{"Start":"07:00.380 ","End":"07:03.665","Text":"I could write it, well,"},{"Start":"07:03.665 ","End":"07:08.180","Text":"I\u0027ll just indicate this is obviously the same as if I"},{"Start":"07:08.180 ","End":"07:12.890","Text":"wrote n plus 1 minus k and the reason I want to"},{"Start":"07:12.890 ","End":"07:23.075","Text":"do that is that then this comes out exactly to be the definition of n plus 1 choose k,"},{"Start":"07:23.075 ","End":"07:25.963","Text":"because here\u0027s the n plus 1 factorial,"},{"Start":"07:25.963 ","End":"07:27.950","Text":"here\u0027s the k factorial,"},{"Start":"07:27.950 ","End":"07:32.150","Text":"and this minus this is n plus 1 minus k."},{"Start":"07:32.150 ","End":"07:42.469","Text":"So we have exactly arrived from the left-hand side to the right-hand side."},{"Start":"07:42.469 ","End":"07:45.230","Text":"We started from here, we got this and this,"},{"Start":"07:45.230 ","End":"07:47.285","Text":"developed it and we go here."},{"Start":"07:47.285 ","End":"07:51.094","Text":"As they say in Latin,"},{"Start":"07:51.094 ","End":"07:55.230","Text":"QED, we proved what had to be proved."}],"ID":26936},{"Watched":false,"Name":"Exercise 10","Duration":"5m 51s","ChapterTopicVideoID":26116,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26116.jpeg","UploadDate":"2021-08-04T16:44:54.3330000","DurationForVideoObject":"PT5M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"This exercise is really best explained with diagrams rather than in words."},{"Start":"00:05.970 ","End":"00:09.420","Text":"It\u0027s hard to follow from a few words what it means."},{"Start":"00:09.420 ","End":"00:12.330","Text":"But basically, if you arrange the coefficients,"},{"Start":"00:12.330 ","End":"00:21.750","Text":"the binomial into a triangle form, centered symmetrically."},{"Start":"00:21.750 ","End":"00:26.235","Text":"Then like a plus b squared is a squared plus 2ab plus b squared."},{"Start":"00:26.235 ","End":"00:27.670","Text":"We take the coefficients 1,"},{"Start":"00:27.670 ","End":"00:29.675","Text":"2, 1 and we keep going."},{"Start":"00:29.675 ","End":"00:32.150","Text":"We get what is called Pascal\u0027s triangle,"},{"Start":"00:32.150 ","End":"00:34.880","Text":"which is really infinite. It goes on forever."},{"Start":"00:34.880 ","End":"00:40.715","Text":"Let me explain what we want here with a couple of diagrams."},{"Start":"00:40.715 ","End":"00:43.790","Text":"For example, this last row,"},{"Start":"00:43.790 ","End":"00:45.650","Text":"1, 4, 6, 4, 1."},{"Start":"00:45.650 ","End":"00:51.130","Text":"These are the coefficients of a plus b to the power of 4."},{"Start":"00:51.130 ","End":"00:53.925","Text":"This is row 4."},{"Start":"00:53.925 ","End":"00:59.480","Text":"It\u0027s 4 choose 0 is the coefficient of the first term."},{"Start":"00:59.480 ","End":"01:00.590","Text":"Then 4 choose 1,"},{"Start":"01:00.590 ","End":"01:03.305","Text":"4 choose 2, 4,3, 4,4."},{"Start":"01:03.305 ","End":"01:06.410","Text":"We know that the first and last ones are always 1."},{"Start":"01:06.410 ","End":"01:08.750","Text":"If we make the computations,"},{"Start":"01:08.750 ","End":"01:11.134","Text":"let\u0027s say up to 10,"},{"Start":"01:11.134 ","End":"01:14.930","Text":"then here\u0027s what it comes out numerically."},{"Start":"01:14.930 ","End":"01:19.910","Text":"The point is that there\u0027s a pattern that enables us to compute these very quickly."},{"Start":"01:19.910 ","End":"01:24.140","Text":"But more than that, we don\u0027t even have to compute them if you\u0027d like print out or"},{"Start":"01:24.140 ","End":"01:28.580","Text":"keep as an image up to 10 or up to however many you\u0027ll need."},{"Start":"01:28.580 ","End":"01:30.740","Text":"Then you don\u0027t have to actually do computations."},{"Start":"01:30.740 ","End":"01:32.420","Text":"You can just refer to the diagram."},{"Start":"01:32.420 ","End":"01:34.955","Text":"But if you do want to generate the triangle,"},{"Start":"01:34.955 ","End":"01:36.500","Text":"then there\u0027s a simple rule."},{"Start":"01:36.500 ","End":"01:40.070","Text":"Let\u0027s say we\u0027ve computed it up to 4."},{"Start":"01:40.070 ","End":"01:43.800","Text":"We have up to here."},{"Start":"01:43.880 ","End":"01:47.160","Text":"But then we don\u0027t have 5 onwards."},{"Start":"01:47.160 ","End":"01:49.325","Text":"What we would do is first of all,"},{"Start":"01:49.325 ","End":"01:53.335","Text":"for the next row, we place a 1 at the beginning and at the end."},{"Start":"01:53.335 ","End":"01:56.930","Text":"Then each entry, this will be a blank,"},{"Start":"01:56.930 ","End":"02:00.320","Text":"but I\u0027d say 1 plus 4 and get 5,"},{"Start":"02:00.320 ","End":"02:02.605","Text":"4 plus 6 and get 10,"},{"Start":"02:02.605 ","End":"02:04.365","Text":"6 plus 4, 10,"},{"Start":"02:04.365 ","End":"02:05.955","Text":"4 plus 1, 5,"},{"Start":"02:05.955 ","End":"02:08.600","Text":"and we put a 1 at the beginning and a 1 at the end."},{"Start":"02:08.600 ","End":"02:10.400","Text":"Then we\u0027ve got the 5th row,"},{"Start":"02:10.400 ","End":"02:11.630","Text":"then the 6th row,"},{"Start":"02:11.630 ","End":"02:13.260","Text":"1 at beginning, 1 at the end."},{"Start":"02:13.260 ","End":"02:15.679","Text":"Then we\u0027ve computed by 1 plus 5."},{"Start":"02:15.679 ","End":"02:17.635","Text":"10 plus 5 is 15,"},{"Start":"02:17.635 ","End":"02:21.220","Text":"10 and 10 is 20, and so on."},{"Start":"02:22.270 ","End":"02:24.650","Text":"It looks like this rule works."},{"Start":"02:24.650 ","End":"02:26.674","Text":"How would we describe this in general?"},{"Start":"02:26.674 ","End":"02:29.810","Text":"But let\u0027s look at a specific example."},{"Start":"02:29.810 ","End":"02:33.480","Text":"Let\u0027s go to the next row,"},{"Start":"02:33.480 ","End":"02:38.714","Text":"there\u0027s 35, which we compute from 15 plus 20."},{"Start":"02:38.714 ","End":"02:41.990","Text":"What is that in terms of the binomial coefficients?"},{"Start":"02:41.990 ","End":"02:50.795","Text":"It\u0027s the fact that this plus this is equal to this because this is 7 choose 3,"},{"Start":"02:50.795 ","End":"02:52.910","Text":"0, 1, 2, 3."},{"Start":"02:52.910 ","End":"02:55.280","Text":"What is this? This is 6 choose 0,"},{"Start":"02:55.280 ","End":"02:57.320","Text":"1, 2, and 3."},{"Start":"02:57.320 ","End":"03:00.610","Text":"It\u0027s 6,2 plus 6,3 is 7,3."},{"Start":"03:00.610 ","End":"03:02.685","Text":"I\u0027ll write that example."},{"Start":"03:02.685 ","End":"03:11.310","Text":"6,2 plus 6,3 is equal to 7,3."},{"Start":"03:11.310 ","End":"03:13.580","Text":"If we generalize this,"},{"Start":"03:13.580 ","End":"03:18.230","Text":"notice that when you go up into the right,"},{"Start":"03:18.230 ","End":"03:21.800","Text":"the number below is the same."},{"Start":"03:21.800 ","End":"03:28.065","Text":"Like here, all the entries have a 3 here."},{"Start":"03:28.065 ","End":"03:36.545","Text":"If we go up, we reduce the top number by 1 and keep the bottom number like we did here."},{"Start":"03:36.545 ","End":"03:39.255","Text":"We increase for the right one,"},{"Start":"03:39.255 ","End":"03:41.490","Text":"the 6 we increase by 1 and leave that."},{"Start":"03:41.490 ","End":"03:43.310","Text":"Then for the other entry,"},{"Start":"03:43.310 ","End":"03:45.110","Text":"we just subtract 1 from here."},{"Start":"03:45.110 ","End":"03:47.030","Text":"What we would get in general,"},{"Start":"03:47.030 ","End":"03:51.260","Text":"and what we\u0027re required to prove is the following."},{"Start":"03:51.260 ","End":"03:59.630","Text":"N choose k plus n"},{"Start":"03:59.630 ","End":"04:09.825","Text":"choose k plus 1 will equal n plus 1 choose k plus 1."},{"Start":"04:09.825 ","End":"04:15.830","Text":"But here, k has to be less than n. Bigger or equal to 0 of course,"},{"Start":"04:15.830 ","End":"04:18.319","Text":"and less than n. Otherwise,"},{"Start":"04:18.319 ","End":"04:20.780","Text":"we can\u0027t do the k plus 1."},{"Start":"04:20.780 ","End":"04:24.560","Text":"In any event, k will be usually bigger than 0."},{"Start":"04:24.560 ","End":"04:27.930","Text":"Also, we\u0027re not interested in the 1s at the end."},{"Start":"04:27.930 ","End":"04:29.495","Text":"How do we prove this?"},{"Start":"04:29.495 ","End":"04:31.130","Text":"We use the formula."},{"Start":"04:31.130 ","End":"04:36.230","Text":"But wait, we don\u0027t have to actually prove it because there\u0027s a previous exercise where we"},{"Start":"04:36.230 ","End":"04:41.675","Text":"showed this identity of the formula."},{"Start":"04:41.675 ","End":"04:43.655","Text":"But not quite. If I,"},{"Start":"04:43.655 ","End":"04:46.085","Text":"instead of k, put k minus 1."},{"Start":"04:46.085 ","End":"04:48.750","Text":"I mean, it\u0027s not clear which to choose as k,"},{"Start":"04:48.750 ","End":"04:50.010","Text":"the 3 or the 2 here."},{"Start":"04:50.010 ","End":"04:51.835","Text":"If we chose the 3,"},{"Start":"04:51.835 ","End":"04:54.305","Text":"then we would express this slightly differently."},{"Start":"04:54.305 ","End":"05:03.990","Text":"We would say n choose k minus 1 plus n choose k is equal to n plus 1 choose"},{"Start":"05:03.990 ","End":"05:10.700","Text":"k. In this case the condition would be that k is bigger than 0 and"},{"Start":"05:10.700 ","End":"05:17.705","Text":"of course k has to be less than or equal to n. This,"},{"Start":"05:17.705 ","End":"05:23.410","Text":"if you check back that we solved this in a previous exercise."},{"Start":"05:23.410 ","End":"05:25.205","Text":"Actually we\u0027re done."},{"Start":"05:25.205 ","End":"05:27.720","Text":"There\u0027s nothing more to do."},{"Start":"05:28.120 ","End":"05:31.580","Text":"Those who didn\u0027t know it, this is Pascal\u0027s triangle."},{"Start":"05:31.580 ","End":"05:34.295","Text":"I recommend printing out,"},{"Start":"05:34.295 ","End":"05:37.070","Text":"let\u0027s say up to whatever,"},{"Start":"05:37.070 ","End":"05:39.080","Text":"up to 10, however many."},{"Start":"05:39.080 ","End":"05:41.240","Text":"Then you don\u0027t have to do computations."},{"Start":"05:41.240 ","End":"05:45.620","Text":"Well, you\u0027d have to ask the instructor if you\u0027re allowed to bring in"},{"Start":"05:45.620 ","End":"05:51.270","Text":"Pascal\u0027s triangle or not. That\u0027s it for this clip."}],"ID":26937},{"Watched":false,"Name":"Exercise 11","Duration":"14m 34s","ChapterTopicVideoID":26117,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26117.jpeg","UploadDate":"2021-08-04T16:49:53.2330000","DurationForVideoObject":"PT14M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.465","Text":"This is a more theoretical exercise."},{"Start":"00:03.465 ","End":"00:07.170","Text":"So far we\u0027ve been using Newton\u0027s binomial theorem,"},{"Start":"00:07.170 ","End":"00:08.845","Text":"but we haven\u0027t proved it."},{"Start":"00:08.845 ","End":"00:11.190","Text":"This exercise, we\u0027re asked to prove it using"},{"Start":"00:11.190 ","End":"00:13.950","Text":"mathematical induction and I\u0027m going to assume"},{"Start":"00:13.950 ","End":"00:17.745","Text":"that you know what mathematical induction is."},{"Start":"00:17.745 ","End":"00:24.990","Text":"Essentially, what we have to show with induction is that first of all,"},{"Start":"00:24.990 ","End":"00:31.110","Text":"it\u0027s true for n equals 1, that it\u0027s true."},{"Start":"00:31.110 ","End":"00:39.180","Text":"Then we have to next show that if it\u0027s true for a particular n then it\u0027s"},{"Start":"00:39.180 ","End":"00:47.290","Text":"going to be true for n plus 1 or I\u0027d rather replace n by k. I\u0027ll rephrase that."},{"Start":"00:47.290 ","End":"00:49.270","Text":"If it\u0027s true for a particular n,"},{"Start":"00:49.270 ","End":"00:50.994","Text":"say n equals k,"},{"Start":"00:50.994 ","End":"00:56.380","Text":"we have to show from this that it must also be true for the next k,"},{"Start":"00:56.380 ","End":"00:58.580","Text":"for k plus 1."},{"Start":"00:58.580 ","End":"01:05.275","Text":"This we show just by substituting n equals 1 and seeing that we get a true statement."},{"Start":"01:05.275 ","End":"01:07.480","Text":"Sometimes it\u0027s phrased in terms of proposition."},{"Start":"01:07.480 ","End":"01:09.730","Text":"We say that the proposition which in this case"},{"Start":"01:09.730 ","End":"01:12.310","Text":"is Newton\u0027s binomial theorem for a given n,"},{"Start":"01:12.310 ","End":"01:14.095","Text":"we call it p_n."},{"Start":"01:14.095 ","End":"01:21.740","Text":"Then there are 2 things that we have to show step a that p_1 is true."},{"Start":"01:22.020 ","End":"01:28.250","Text":"Secondly, we show not that p_k is true because that would"},{"Start":"01:28.250 ","End":"01:34.250","Text":"be going in circles but that if p_k is true that means n equals k,"},{"Start":"01:34.250 ","End":"01:39.985","Text":"then it implies that p_k plus 1 is true."},{"Start":"01:39.985 ","End":"01:43.330","Text":"Let\u0027s start with part a,"},{"Start":"01:43.330 ","End":"01:47.270","Text":"which is usually the easy part where n is equal to 1."},{"Start":"01:47.270 ","End":"01:50.765","Text":"What this says is that,"},{"Start":"01:50.765 ","End":"01:54.425","Text":"well, it\u0027s actually true for n equals 0."},{"Start":"01:54.425 ","End":"01:57.410","Text":"We\u0027re not asked to prove if for n equals 0 anyway,"},{"Start":"01:57.410 ","End":"01:59.210","Text":"but it is true for n equals 0."},{"Start":"01:59.210 ","End":"02:00.950","Text":"It just works out."},{"Start":"02:00.950 ","End":"02:04.845","Text":"We only get 1 term which is equal to 1 and a plus b to the 0 is 1."},{"Start":"02:04.845 ","End":"02:06.510","Text":"We\u0027re starting from 1, 2,"},{"Start":"02:06.510 ","End":"02:08.550","Text":"3, from 1 onwards."},{"Start":"02:08.550 ","End":"02:16.010","Text":"When n equals 1, what this says is that a plus b to the power of 1 equals,"},{"Start":"02:16.010 ","End":"02:17.360","Text":"and this is what we\u0027re checking,"},{"Start":"02:17.360 ","End":"02:19.025","Text":"so I put a question mark here."},{"Start":"02:19.025 ","End":"02:24.425","Text":"The sum k goes from 0-1,"},{"Start":"02:24.425 ","End":"02:27.650","Text":"1, choose k,"},{"Start":"02:27.650 ","End":"02:35.220","Text":"a^1 minus kb to the k. Now,"},{"Start":"02:35.220 ","End":"02:38.765","Text":"this is going to have 2 terms when k is 0 and k is 1."},{"Start":"02:38.765 ","End":"02:47.235","Text":"When k is 0, we\u0027ll get 1 choose 0, a^1, b^0."},{"Start":"02:47.235 ","End":"02:49.405","Text":"When k is 1,"},{"Start":"02:49.405 ","End":"02:57.590","Text":"then we will get 1 choose 1, a^0, b^1."},{"Start":"02:57.590 ","End":"03:02.640","Text":"What this says is 1 choose 0 is 1,"},{"Start":"03:02.640 ","End":"03:04.305","Text":"this is just a^1."},{"Start":"03:04.305 ","End":"03:13.965","Text":"This is just a and this 1 choose 1 is 1 times 1 times b^1, it\u0027s just b."},{"Start":"03:13.965 ","End":"03:17.555","Text":"Is a plus b^1 equal to a plus b?"},{"Start":"03:17.555 ","End":"03:20.735","Text":"Question mark, the answer is yes."},{"Start":"03:20.735 ","End":"03:22.550","Text":"That\u0027s that part."},{"Start":"03:22.550 ","End":"03:25.865","Text":"Now, the more difficult part, the induction step."},{"Start":"03:25.865 ","End":"03:35.630","Text":"I\u0027m going to maybe k is also not a good letter because we have k here. You know what?"},{"Start":"03:35.630 ","End":"03:44.375","Text":"I\u0027ll change that k to r. I erased the k\u0027s and now I\u0027ll put r\u0027s in their place."},{"Start":"03:44.375 ","End":"03:46.340","Text":"R plus 1."},{"Start":"03:46.340 ","End":"03:50.285","Text":"I don\u0027t want to reuse the same letter for different purposes."},{"Start":"03:50.285 ","End":"03:55.055","Text":"We\u0027re assuming that it\u0027s true for n equals r,"},{"Start":"03:55.055 ","End":"04:01.520","Text":"which means that a plus b for a specific r but general,"},{"Start":"04:01.520 ","End":"04:04.985","Text":"a is equal to the sum."},{"Start":"04:04.985 ","End":"04:10.515","Text":"K goes from 0 to r. Just replace n by r here."},{"Start":"04:10.515 ","End":"04:19.880","Text":"R choose k, a^r minus k, and b^k."},{"Start":"04:19.880 ","End":"04:23.645","Text":"We\u0027re given this and by a series of steps,"},{"Start":"04:23.645 ","End":"04:27.380","Text":"we have to show that this is true."},{"Start":"04:27.380 ","End":"04:34.820","Text":"Well, I\u0027ll write the endpoint that we\u0027re trying to reach and then by a series of 0,"},{"Start":"04:34.820 ","End":"04:37.650","Text":"however we can, we want to prove."},{"Start":"04:37.650 ","End":"04:40.360","Text":"To save time I did a copy-paste and I\u0027m going to"},{"Start":"04:40.360 ","End":"04:49.300","Text":"replace r by r plus 1,"},{"Start":"04:49.300 ","End":"04:54.705","Text":"r plus 1, r plus 1."},{"Start":"04:54.705 ","End":"04:56.825","Text":"Now the strategy will be,"},{"Start":"04:56.825 ","End":"05:00.125","Text":"we take this statement and we know that it\u0027s true."},{"Start":"05:00.125 ","End":"05:03.005","Text":"That\u0027s the assumption, the induction hypothesis."},{"Start":"05:03.005 ","End":"05:09.050","Text":"What we\u0027re going to do is develop the a plus b to the r plus 1,"},{"Start":"05:09.050 ","End":"05:12.200","Text":"which is the left-hand side of what we want to show."},{"Start":"05:12.200 ","End":"05:14.090","Text":"By a series of steps,"},{"Start":"05:14.090 ","End":"05:16.190","Text":"I want to reach this expression."},{"Start":"05:16.190 ","End":"05:19.044","Text":"This is like, I don\u0027t know,"},{"Start":"05:19.044 ","End":"05:22.460","Text":"I changed the color on this because this is what we\u0027re going to"},{"Start":"05:22.460 ","End":"05:25.940","Text":"prove to emphasize that. Let\u0027s see."},{"Start":"05:25.940 ","End":"05:28.370","Text":"The first thing we can do is something to"},{"Start":"05:28.370 ","End":"05:31.669","Text":"the power of r plus 1 using the rules of exponents,"},{"Start":"05:31.669 ","End":"05:35.820","Text":"well, 1 plus r is the same as r plus 1."},{"Start":"05:35.820 ","End":"05:41.030","Text":"I can say that this is equal to a plus b times"},{"Start":"05:41.030 ","End":"05:47.910","Text":"a plus b to the r which is a plus b."},{"Start":"05:47.910 ","End":"05:49.880","Text":"By the induction hypothesis,"},{"Start":"05:49.880 ","End":"05:51.530","Text":"I know what this is."},{"Start":"05:51.530 ","End":"05:59.580","Text":"This is equal to this which I\u0027ll copy-paste here except that we\u0027ll need brackets now."},{"Start":"05:59.840 ","End":"06:06.020","Text":"Use the distributive law we\u0027ll say that a plus b times this is a times this,"},{"Start":"06:06.020 ","End":"06:07.805","Text":"plus b times this."},{"Start":"06:07.805 ","End":"06:16.540","Text":"We have a times this which is here, plus b times."},{"Start":"06:16.540 ","End":"06:19.340","Text":"The next thing we\u0027re going to do is here,"},{"Start":"06:19.340 ","End":"06:21.140","Text":"if I multiply by a,"},{"Start":"06:21.140 ","End":"06:22.820","Text":"which doesn\u0027t depend on k,"},{"Start":"06:22.820 ","End":"06:25.805","Text":"I can multiply this by a,"},{"Start":"06:25.805 ","End":"06:27.290","Text":"put the a in here."},{"Start":"06:27.290 ","End":"06:31.590","Text":"All have to do is increase this index by 1."},{"Start":"06:32.450 ","End":"06:37.440","Text":"Like I said, we increase the index so I put a plus 1 here."},{"Start":"06:37.440 ","End":"06:39.150","Text":"For the next 1,"},{"Start":"06:39.150 ","End":"06:43.410","Text":"I multiplied by b so I increase this index by 1."},{"Start":"06:43.410 ","End":"06:49.270","Text":"Again I copy-pasted it plus and then I increase this one by 1."},{"Start":"06:51.260 ","End":"06:56.760","Text":"The next step would be to expand the Sigma\u0027s,"},{"Start":"06:56.760 ","End":"07:06.270","Text":"to write all r plus 1 terms for each of these to main terms so a small space,"},{"Start":"07:06.270 ","End":"07:13.144","Text":"the first part, this 1 will be r choose 0."},{"Start":"07:13.144 ","End":"07:16.780","Text":"Now, k is 0 here,"},{"Start":"07:16.780 ","End":"07:18.130","Text":"so it\u0027s just r plus"},{"Start":"07:18.130 ","End":"07:21.500","Text":"1 and"},{"Start":"07:23.580 ","End":"07:28.150","Text":"b^0,"},{"Start":"07:28.150 ","End":"07:32.300","Text":"plus r choose 1."},{"Start":"07:32.670 ","End":"07:34.870","Text":"Then k is 1,"},{"Start":"07:34.870 ","End":"07:42.430","Text":"so we just get a^ r, b^1."},{"Start":"07:42.430 ","End":"07:44.635","Text":"Then I\u0027ll write the 3rd one."},{"Start":"07:44.635 ","End":"07:51.250","Text":"Choose 2, the indices here,"},{"Start":"07:51.250 ","End":"07:53.020","Text":"the exponents are decreasing by 1."},{"Start":"07:53.020 ","End":"07:55.330","Text":"This will be r minus 1,"},{"Start":"07:55.330 ","End":"08:02.020","Text":"b squared and then dot dot dot etc."},{"Start":"08:02.020 ","End":"08:03.595","Text":"The last 2 terms."},{"Start":"08:03.595 ","End":"08:10.150","Text":"We have r minus 1 something and then r r. Let\u0027s see what\u0027s this."},{"Start":"08:10.150 ","End":"08:18.115","Text":"This would be, if k is r minus 1,"},{"Start":"08:18.115 ","End":"08:23.320","Text":"then this computation with k equals r minus 1,"},{"Start":"08:23.320 ","End":"08:32.750","Text":"the r will cancel and the 1 will be plus so it will be a squared b to the minus 1."},{"Start":"08:32.750 ","End":"08:34.890","Text":"The last 1,"},{"Start":"08:34.890 ","End":"08:43.385","Text":"r choose r would be a^1, b^r."},{"Start":"08:43.385 ","End":"08:47.005","Text":"Now, that\u0027s just all this bit."},{"Start":"08:47.005 ","End":"08:55.130","Text":"Plus now we\u0027ll expand this and I\u0027ll put them in a certain way."},{"Start":"08:55.650 ","End":"08:59.680","Text":"I\u0027ll use a bit of color, this offset a bit,"},{"Start":"08:59.680 ","End":"09:01.570","Text":"I write the first term under this 1."},{"Start":"09:01.570 ","End":"09:06.650","Text":"When k is 0, we get r choose 0,"},{"Start":"09:07.560 ","End":"09:13.570","Text":"a^r and then b^1."},{"Start":"09:13.570 ","End":"09:18.070","Text":"Notice that the coefficients are the same and this is what\u0027s going to happen."},{"Start":"09:18.070 ","End":"09:23.780","Text":"When k is 1, then we get r choose 1."},{"Start":"09:26.810 ","End":"09:31.410","Text":"Just safely just take my word for it."},{"Start":"09:31.410 ","End":"09:38.170","Text":"We\u0027ll just quickly do this. It will come out the same when k is r minus 1, then"},{"Start":"09:48.570 ","End":"09:54.020","Text":"here you actually need k to be r minus 2."},{"Start":"09:54.780 ","End":"09:58.810","Text":"You\u0027ll see because then if this r minus 2,"},{"Start":"09:58.810 ","End":"10:00.880","Text":"then we do indeed get"},{"Start":"10:00.880 ","End":"10:05.060","Text":"a squared and r minus"},{"Start":"10:09.840 ","End":"10:15.174","Text":"2 plus 1 is minus 1."},{"Start":"10:15.174 ","End":"10:21.820","Text":"Then we let k be r minus 1 so we get r r minus 1."},{"Start":"10:21.820 ","End":"10:24.400","Text":"Here it\u0027s a^1, b^r."},{"Start":"10:24.400 ","End":"10:26.905","Text":"The last term when k is r,"},{"Start":"10:26.905 ","End":"10:36.310","Text":"is r r a^0, b^r plus 1."},{"Start":"10:36.310 ","End":"10:40.015","Text":"Everything aligns except the first and the last."},{"Start":"10:40.015 ","End":"10:44.800","Text":"What we\u0027ll do is to add pairwise."},{"Start":"10:44.800 ","End":"10:46.795","Text":"We\u0027ll add this with this."},{"Start":"10:46.795 ","End":"10:49.659","Text":"We\u0027ll add this with this,"},{"Start":"10:49.659 ","End":"10:54.325","Text":"we\u0027ll add this with this, this with this."},{"Start":"10:54.325 ","End":"10:59.065","Text":"We have a formula already because we\u0027ve already proven"},{"Start":"10:59.065 ","End":"11:05.560","Text":"previously that we did it with n and k,"},{"Start":"11:05.560 ","End":"11:12.790","Text":"but we could also do it with r and k that let\u0027s say this is k,"},{"Start":"11:12.790 ","End":"11:21.550","Text":"this is k minus 1 r choose k plus r choose k minus"},{"Start":"11:21.550 ","End":"11:30.820","Text":"1 is equal to r plus 1 choose k. If we use that formula throughout,"},{"Start":"11:30.820 ","End":"11:36.880","Text":"it\u0027s going to be r plus 1 and then it\u0027s going to be the top 1 of these."},{"Start":"11:36.880 ","End":"11:42.700","Text":"This pair will come out to be r plus 1 choose 1."},{"Start":"11:42.700 ","End":"11:49.915","Text":"They\u0027re all going to start with r plus 1 and we\u0027ll just take the upper 1 to r plus 1."},{"Start":"11:49.915 ","End":"11:53.560","Text":"This is r minus 1 here,"},{"Start":"11:53.560 ","End":"11:58.908","Text":"r plus 1 r. Here, well,"},{"Start":"11:58.908 ","End":"12:00.235","Text":"the first and the last 1,"},{"Start":"12:00.235 ","End":"12:05.740","Text":"we can just bring them down r choose 0 and"},{"Start":"12:05.740 ","End":"12:12.145","Text":"here r r I think maybe I\u0027ll change the color."},{"Start":"12:12.145 ","End":"12:15.265","Text":"I think that\u0027s better."},{"Start":"12:15.265 ","End":"12:19.930","Text":"Now, I\u0027ll just copy the a to the something,"},{"Start":"12:19.930 ","End":"12:21.820","Text":"b to the something."},{"Start":"12:21.820 ","End":"12:27.820","Text":"Copied that here and similarly,"},{"Start":"12:27.820 ","End":"12:36.870","Text":"a^r b^1, a^r minus 1 b squared a squared."},{"Start":"12:36.870 ","End":"12:40.365","Text":"Just as common factor, I\u0027m just copying it."},{"Start":"12:40.365 ","End":"12:44.290","Text":"B^r minus 1, a^1,"},{"Start":"12:44.290 ","End":"12:52.030","Text":"b^r a^0, b^r plus 1."},{"Start":"12:52.030 ","End":"12:57.490","Text":"Now something\u0027s not quite right because the pattern is broken everything is"},{"Start":"12:57.490 ","End":"13:02.980","Text":"r plus 1 except the first and the last but does that really matter?"},{"Start":"13:02.980 ","End":"13:10.930","Text":"The thing is that this is 1 and so because it\u0027s 1,"},{"Start":"13:10.930 ","End":"13:15.040","Text":"wouldn\u0027t hurt if I replace this by r plus 1,"},{"Start":"13:15.040 ","End":"13:19.435","Text":"r plus 1 like so."},{"Start":"13:19.435 ","End":"13:27.970","Text":"Here r 0 is also 1 so I could replace the r by r plus 1, like so."},{"Start":"13:27.970 ","End":"13:31.540","Text":"Now, if we write this in Sigma form,"},{"Start":"13:31.540 ","End":"13:36.025","Text":"we can see that this is exactly equal to the sum."},{"Start":"13:36.025 ","End":"13:43.030","Text":"The lower index will be k goes from 0 up to r plus 1."},{"Start":"13:43.030 ","End":"13:48.580","Text":"Here we have a^r plus 1,"},{"Start":"13:48.580 ","End":"13:51.280","Text":"minus k, it\u0027s r plus 1,"},{"Start":"13:51.280 ","End":"13:53.335","Text":"r plus 1 minus 1 and so on."},{"Start":"13:53.335 ","End":"13:56.830","Text":"The b to the power of k,"},{"Start":"13:56.830 ","End":"13:59.440","Text":"where k go from 0,"},{"Start":"13:59.440 ","End":"14:01.855","Text":"1, 2 up to r plus 1."},{"Start":"14:01.855 ","End":"14:07.375","Text":"This is right and this is exactly what we were looking for."},{"Start":"14:07.375 ","End":"14:10.000","Text":"Let\u0027s scroll back just take a look at"},{"Start":"14:10.000 ","End":"14:13.810","Text":"this and let\u0027s scroll back and see what we were looking for."},{"Start":"14:13.810 ","End":"14:16.240","Text":"We were looking for this."},{"Start":"14:16.240 ","End":"14:21.520","Text":"I go back down again."},{"Start":"14:21.520 ","End":"14:23.110","Text":"It doesn\u0027t all fit on 1 screen,"},{"Start":"14:23.110 ","End":"14:29.435","Text":"but this is what we had down below and so this is what we have to prove,"},{"Start":"14:29.435 ","End":"14:31.175","Text":"and we\u0027ve proved it."},{"Start":"14:31.175 ","End":"14:34.800","Text":"That concludes this exercise."}],"ID":26938},{"Watched":false,"Name":"Exercise 12","Duration":"4m 40s","ChapterTopicVideoID":26101,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26101.jpeg","UploadDate":"2021-08-04T16:06:39.2330000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"In this exercise, we\u0027re asked to write the 7th term"},{"Start":"00:03.885 ","End":"00:08.370","Text":"in the binomial expansion of this expression."},{"Start":"00:08.370 ","End":"00:12.960","Text":"Of course, we\u0027re going to use the Newton\u0027s binomial formula,"},{"Start":"00:12.960 ","End":"00:16.900","Text":"and here\u0027s a reminder of the formula."},{"Start":"00:16.900 ","End":"00:21.255","Text":"What we have is that this is a this is b,"},{"Start":"00:21.255 ","End":"00:23.715","Text":"and this is n. Now,"},{"Start":"00:23.715 ","End":"00:27.030","Text":"pay attention the term 7th."},{"Start":"00:27.030 ","End":"00:31.785","Text":"You have to be careful here because the terms start from 0."},{"Start":"00:31.785 ","End":"00:34.125","Text":"This is the kth term."},{"Start":"00:34.125 ","End":"00:35.910","Text":"If we\u0027re going to go to 7,"},{"Start":"00:35.910 ","End":"00:41.820","Text":"well, to form 7 terms,"},{"Start":"00:41.820 ","End":"00:43.950","Text":"what I\u0027m saying is it will end up at 6,"},{"Start":"00:43.950 ","End":"00:45.335","Text":"that will be 7 terms."},{"Start":"00:45.335 ","End":"00:50.030","Text":"In other words, we\u0027re going to let n equal 6 in the formula."},{"Start":"00:50.030 ","End":"00:55.490","Text":"Applying it, we just need"},{"Start":"00:55.490 ","End":"01:03.300","Text":"this term, which is this."},{"Start":"01:06.800 ","End":"01:12.040","Text":"In other words, k is going to equal 6."},{"Start":"01:12.040 ","End":"01:14.300","Text":"Now what we want is just the term,"},{"Start":"01:14.300 ","End":"01:15.958","Text":"not the whole thing."},{"Start":"01:15.958 ","End":"01:23.820","Text":"We want just this bit when k equals 6."},{"Start":"01:23.820 ","End":"01:26.180","Text":"We\u0027ve already stated what a and"},{"Start":"01:26.180 ","End":"01:28.190","Text":"what b and what n are, I will just copy this term like so."},{"Start":"01:28.190 ","End":"01:33.715","Text":"Wrong color, there we go."},{"Start":"01:33.715 ","End":"01:42.615","Text":"Now it\u0027s just substitutions n is 12,"},{"Start":"01:42.615 ","End":"01:48.095","Text":"k is 6, 7th term k is 6 a"},{"Start":"01:48.095 ","End":"01:54.860","Text":"is square root of t to the power of n is 12,"},{"Start":"01:54.860 ","End":"01:58.130","Text":"and k is 6,"},{"Start":"01:58.130 ","End":"02:03.835","Text":"and b is 1 over the square root of"},{"Start":"02:03.835 ","End":"02:09.710","Text":"t and this will take to the power of k, which is 6."},{"Start":"02:09.710 ","End":"02:12.515","Text":"So this is what we have to evaluate."},{"Start":"02:12.515 ","End":"02:18.915","Text":"Now the binomial coefficient will do that side."},{"Start":"02:18.915 ","End":"02:21.505","Text":"So 12, choose 6."},{"Start":"02:21.505 ","End":"02:29.945","Text":"On the denominator, we\u0027re going to have 6 times 5 times 4 times 3 times 2 times 1."},{"Start":"02:29.945 ","End":"02:31.535","Text":"Then we start from 12,"},{"Start":"02:31.535 ","End":"02:35.195","Text":"matching the factors 1 on 1, 12,"},{"Start":"02:35.195 ","End":"02:41.885","Text":"11, 10, 9, 8, 7."},{"Start":"02:41.885 ","End":"02:45.145","Text":"Let\u0027s see what cancels."},{"Start":"02:45.145 ","End":"02:47.850","Text":"6 times 2 is 12."},{"Start":"02:47.850 ","End":"02:49.560","Text":"So I could do this, this,"},{"Start":"02:49.560 ","End":"02:57.490","Text":"and this 5 into 10 goes twice."},{"Start":"02:58.610 ","End":"03:03.270","Text":"4 into 8 goes"},{"Start":"03:03.270 ","End":"03:11.060","Text":"twice and 3 into 9 goes 3 times."},{"Start":"03:11.060 ","End":"03:12.935","Text":"There\u0027s no denominator."},{"Start":"03:12.935 ","End":"03:15.455","Text":"Now let\u0027s see what we get."},{"Start":"03:15.455 ","End":"03:23.670","Text":"We have 11 times 2 times 3 times 2 times 7."},{"Start":"03:25.280 ","End":"03:28.345","Text":"I\u0027ll try doing it without the calculator,"},{"Start":"03:28.345 ","End":"03:33.500","Text":"2 times 3 times 2 is 12 times 7 is 84."},{"Start":"03:33.500 ","End":"03:36.845","Text":"I just need 11 times 84."},{"Start":"03:36.845 ","End":"03:43.895","Text":"And that is going to be 840 plus 84."},{"Start":"03:43.895 ","End":"03:49.265","Text":"That would be 924 hopefully."},{"Start":"03:49.265 ","End":"03:54.200","Text":"So back here, this we evaluated at 924."},{"Start":"03:54.200 ","End":"03:55.515","Text":"This is 6,"},{"Start":"03:55.515 ","End":"03:58.770","Text":"and this is also 6."},{"Start":"03:58.770 ","End":"04:05.570","Text":"By chance, this 6 equals this 6 so it\u0027s going to come out neater than I planned."},{"Start":"04:05.570 ","End":"04:07.430","Text":"So what we get is"},{"Start":"04:07.430 ","End":"04:13.160","Text":"924 times square root"},{"Start":"04:13.160 ","End":"04:18.680","Text":"of t to the 6th times 1 over square root of t to the 6th."},{"Start":"04:18.680 ","End":"04:24.110","Text":"I can take them both the power of 6."},{"Start":"04:24.110 ","End":"04:27.725","Text":"Now square root of t over square root of t is 1."},{"Start":"04:27.725 ","End":"04:30.455","Text":"This thing just comes out to be 1."},{"Start":"04:30.455 ","End":"04:34.010","Text":"So we actually get a numerical answer without t,"},{"Start":"04:34.010 ","End":"04:40.590","Text":"which is 924 and we are done."}],"ID":26922},{"Watched":false,"Name":"Exercise 13","Duration":"10m 45s","ChapterTopicVideoID":26102,"CourseChapterTopicPlaylistID":264993,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26102.jpeg","UploadDate":"2021-08-04T16:09:28.5700000","DurationForVideoObject":"PT10M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.610","Text":"This exercise is 1 of the practical applications of"},{"Start":"00:05.610 ","End":"00:13.026","Text":"the Newton\u0027s binomial expansion theorem as a practical,"},{"Start":"00:13.026 ","End":"00:15.495","Text":"well, in the age of calculators."},{"Start":"00:15.495 ","End":"00:17.760","Text":"You can imagine the old days,"},{"Start":"00:17.760 ","End":"00:21.080","Text":"or if you\u0027re stuck without a calculator for whatever reason."},{"Start":"00:21.080 ","End":"00:29.789","Text":"You want to approximate 0.99^18 to 3 decimal places."},{"Start":"00:29.789 ","End":"00:34.245","Text":"Now, obviously we\u0027re going to use Newton\u0027s binomial theorem,"},{"Start":"00:34.245 ","End":"00:36.360","Text":"and here it is,"},{"Start":"00:36.360 ","End":"00:38.950","Text":"but how is this?"},{"Start":"00:39.340 ","End":"00:47.810","Text":"This exercise is 1 of the applications of Newton\u0027s binomial expansion theorem,"},{"Start":"00:47.810 ","End":"00:52.400","Text":"and shows how we can do some approximations without calculator."},{"Start":"00:52.400 ","End":"00:56.090","Text":"This was more useful before the age of the calculator."},{"Start":"00:56.090 ","End":"00:59.225","Text":"We want to compute this to 3 decimal places."},{"Start":"00:59.225 ","End":"01:03.340","Text":"Let\u0027s bring the theorem in."},{"Start":"01:03.340 ","End":"01:07.745","Text":"There we go. Somehow we\u0027re going to choose a and b,"},{"Start":"01:07.745 ","End":"01:11.764","Text":"and n judiciously to help us with this task."},{"Start":"01:11.764 ","End":"01:16.708","Text":"Now, the thing is that we do know what is 1^18,"},{"Start":"01:16.708 ","End":"01:19.520","Text":"and 1 is very close to 0.99."},{"Start":"01:19.520 ","End":"01:26.895","Text":"If we wrote 0.99 as 1 minus 0.01,"},{"Start":"01:26.895 ","End":"01:35.675","Text":"then it makes sense to take a equals 1,"},{"Start":"01:35.675 ","End":"01:40.700","Text":"to take b equals minus 0.01,"},{"Start":"01:40.700 ","End":"01:45.390","Text":"and to take n equals 18."},{"Start":"01:45.390 ","End":"01:48.555","Text":"Then we\u0027ll start with the expansion."},{"Start":"01:48.555 ","End":"01:50.550","Text":"Normally, there would be a lot of terms,"},{"Start":"01:50.550 ","End":"01:55.970","Text":"actually 19 terms but you\u0027ll see that when we start writing them out,"},{"Start":"01:55.970 ","End":"01:58.000","Text":"they get small very quickly."},{"Start":"01:58.000 ","End":"02:02.045","Text":"Only the first few terms will have any effect on the third decimal place."},{"Start":"02:02.045 ","End":"02:03.515","Text":"Let\u0027s start doing that."},{"Start":"02:03.515 ","End":"02:12.080","Text":"What we have is that 1 minus 0.01^18,"},{"Start":"02:12.080 ","End":"02:18.750","Text":"where this is 0.99 and b is minus 0.01."},{"Start":"02:18.920 ","End":"02:24.875","Text":"This will equal 18 choose 0,"},{"Start":"02:24.875 ","End":"02:28.865","Text":"it\u0027s always 1 but just for the formality,"},{"Start":"02:28.865 ","End":"02:40.100","Text":"a is 1^18 and minus 0.01^0."},{"Start":"02:40.100 ","End":"02:44.105","Text":"Next we take 18 on 1,"},{"Start":"02:44.105 ","End":"02:52.175","Text":"1 to the 17 minus 0.01^1."},{"Start":"02:52.175 ","End":"02:56.430","Text":"Let\u0027s add another couple, and if we need more, we\u0027ll add them."},{"Start":"02:56.530 ","End":"03:01.230","Text":"18, choose 2,"},{"Start":"03:01.230 ","End":"03:07.468","Text":"1^16 minus 0.01 squared,"},{"Start":"03:07.468 ","End":"03:09.920","Text":"and 1 more for good measure,"},{"Start":"03:09.920 ","End":"03:14.120","Text":"18 choose 3,"},{"Start":"03:14.120 ","End":"03:21.510","Text":"1^15 to be 18"},{"Start":"03:21.510 ","End":"03:27.685","Text":"minus 3, and minus 0.01^3."},{"Start":"03:27.685 ","End":"03:31.590","Text":"Now, this goes on and we\u0027ll see maybe this will do."},{"Start":"03:31.970 ","End":"03:34.830","Text":"This is 1,"},{"Start":"03:34.830 ","End":"03:35.918","Text":"this is 1,"},{"Start":"03:35.918 ","End":"03:37.455","Text":"and this is just 1,"},{"Start":"03:37.455 ","End":"03:39.720","Text":"the very first term."},{"Start":"03:39.720 ","End":"03:44.570","Text":"Now, the next 1, this bit,"},{"Start":"03:44.570 ","End":"03:52.220","Text":"18 on 1 is 18 and 18 times minus 0.01,"},{"Start":"03:52.220 ","End":"03:54.860","Text":"I\u0027ll just indicate that this computation is 18,"},{"Start":"03:54.860 ","End":"03:56.450","Text":"which we don\u0027t really compute,"},{"Start":"03:56.450 ","End":"04:00.545","Text":"and this is just minus 0.01."},{"Start":"04:00.545 ","End":"04:06.810","Text":"All together, we get minus 0.18."},{"Start":"04:06.810 ","End":"04:08.840","Text":"The next 1 obviously is going to be a plus,"},{"Start":"04:08.840 ","End":"04:15.280","Text":"it\u0027s a minus squared so I can safely write plus 18 choose 2."},{"Start":"04:15.280 ","End":"04:22.294","Text":"This bit is 18 times 17 over 2 times 1,"},{"Start":"04:22.294 ","End":"04:29.445","Text":"which is 9 times 17, and that\u0027s 153."},{"Start":"04:29.445 ","End":"04:33.270","Text":"How did I do that? 17 times 10 is 170 minus 17."},{"Start":"04:33.270 ","End":"04:41.480","Text":"This thing squared is now 0.01,"},{"Start":"04:41.480 ","End":"04:45.210","Text":"then it\u0027s going to be 0001."},{"Start":"04:45.210 ","End":"04:49.650","Text":"What we get is 153 times this,"},{"Start":"04:49.650 ","End":"04:52.500","Text":"which is 0.0153,"},{"Start":"04:52.500 ","End":"05:03.260","Text":"but we have to keep going because we\u0027re even in the second decimal place."},{"Start":"05:03.590 ","End":"05:08.795","Text":"Next term is obviously a minus to an odd power."},{"Start":"05:08.795 ","End":"05:11.765","Text":"Now, 18 choose 3,"},{"Start":"05:11.765 ","End":"05:16.100","Text":"let\u0027s see, over 3 times 2 times 1."},{"Start":"05:16.100 ","End":"05:22.140","Text":"Here 18, 17, 16. What do we get?"},{"Start":"05:22.140 ","End":"05:23.610","Text":"3 times 2 is 6,"},{"Start":"05:23.610 ","End":"05:27.540","Text":"6 into 18 goes 3 times,"},{"Start":"05:27.540 ","End":"05:32.770","Text":"so it\u0027s 3 times 17 times 16."},{"Start":"05:32.870 ","End":"05:38.330","Text":"That would be 51 times 16."},{"Start":"05:38.330 ","End":"05:45.630","Text":"Well, 50 times 16 is 800, so that\u0027ll be 816."},{"Start":"05:46.000 ","End":"05:48.845","Text":"See, that\u0027s 51,"},{"Start":"05:48.845 ","End":"05:54.740","Text":"50 times 16 is 816."},{"Start":"05:54.740 ","End":"06:00.185","Text":"This is already 0.00,"},{"Start":"06:00.185 ","End":"06:03.680","Text":"another 00 and then a 01."},{"Start":"06:03.680 ","End":"06:08.711","Text":"This will come out 0 point."},{"Start":"06:08.711 ","End":"06:17.350","Text":"Now I\u0027ll need 3 zeros and 816."},{"Start":"06:17.350 ","End":"06:21.850","Text":"Then it\u0027ll be a plus, and it\u0027ll get small very quickly because this thing,"},{"Start":"06:21.850 ","End":"06:26.215","Text":"the 1 keeps getting further to the right and more and more zeros."},{"Start":"06:26.215 ","End":"06:31.450","Text":"The next term looks like this is in the fourth place."},{"Start":"06:31.450 ","End":"06:32.530","Text":"It could have an influence,"},{"Start":"06:32.530 ","End":"06:34.870","Text":"but the rest of them won\u0027t, so we\u0027ll stop here."},{"Start":"06:34.870 ","End":"06:40.135","Text":"What we need to do is evaluate this here."},{"Start":"06:40.135 ","End":"06:42.955","Text":"I\u0027m going to do it without a calculator."},{"Start":"06:42.955 ","End":"06:51.680","Text":"1 minus 0.18 will be 0.82, that\u0027s this minus this."},{"Start":"06:51.680 ","End":"06:56.982","Text":"Then if we add 0.0153,"},{"Start":"06:56.982 ","End":"07:00.158","Text":"we\u0027ll get 0.8,"},{"Start":"07:00.158 ","End":"07:04.695","Text":"and then the 1 with the 353."},{"Start":"07:04.695 ","End":"07:07.865","Text":"Then if we subtract this,"},{"Start":"07:07.865 ","End":"07:12.875","Text":"we\u0027re subtracting the 816."},{"Start":"07:12.875 ","End":"07:18.130","Text":"The 816 comes out 816 here."},{"Start":"07:18.130 ","End":"07:25.790","Text":"If we subtract, I give up."},{"Start":"07:25.790 ","End":"07:27.815","Text":"I used the calculator."},{"Start":"07:27.815 ","End":"07:30.900","Text":"Now, here I\u0027m in a bit of a dilemma,"},{"Start":"07:30.900 ","End":"07:34.670","Text":"because if I want to round it to 3 places,"},{"Start":"07:34.670 ","End":"07:39.375","Text":"really I round down the 4 because this is less than 5,"},{"Start":"07:39.375 ","End":"07:45.175","Text":"but it\u0027s a bit doubtful maybe if next 1 is a plus."},{"Start":"07:45.175 ","End":"07:49.380","Text":"It might just bring it over the 5 here."},{"Start":"07:49.380 ","End":"07:52.720","Text":"You know what, better do another term."},{"Start":"07:52.850 ","End":"07:56.640","Text":"We\u0027ll add the next term,"},{"Start":"07:56.640 ","End":"08:05.530","Text":"which would be 18 choose"},{"Start":"08:05.530 ","End":"08:14.310","Text":"4 times 1 to the 14th,"},{"Start":"08:14.310 ","End":"08:15.818","Text":"not going to count,"},{"Start":"08:15.818 ","End":"08:20.145","Text":"and it\u0027s going to be minus 0.01 to the fourth,"},{"Start":"08:20.145 ","End":"08:25.570","Text":"so it\u0027s going to be times 0.1,"},{"Start":"08:25.790 ","End":"08:30.000","Text":"2, 3, 4."},{"Start":"08:30.000 ","End":"08:32.835","Text":"That\u0027s going to be what it is to the fourth,"},{"Start":"08:32.835 ","End":"08:35.480","Text":"0.01 to the fourth,"},{"Start":"08:35.480 ","End":"08:38.555","Text":"with a minus, but it\u0027s plus."},{"Start":"08:38.555 ","End":"08:45.200","Text":"This we have to compute,18, 17."},{"Start":"08:45.200 ","End":"08:47.875","Text":"We\u0027ll start with the denominator 4, 3,"},{"Start":"08:47.875 ","End":"08:53.460","Text":"2, 1, 16, 15,"},{"Start":"08:54.980 ","End":"08:59.460","Text":"4 and 2,"},{"Start":"08:59.460 ","End":"09:03.255","Text":"8 into 16 goes twice,"},{"Start":"09:03.255 ","End":"09:11.430","Text":"and 3 into 18 goes 6 times."},{"Start":"09:11.430 ","End":"09:13.650","Text":"6 times 2 is 12,"},{"Start":"09:13.650 ","End":"09:17.490","Text":"12 times 15 is 180."},{"Start":"09:17.490 ","End":"09:21.940","Text":"I need a 180 times 17."},{"Start":"09:21.940 ","End":"09:27.500","Text":"I make it 3,060,"},{"Start":"09:27.500 ","End":"09:36.320","Text":"so that this bit here will become 0.00."},{"Start":"09:36.320 ","End":"09:41.985","Text":"Let\u0027s see, 3,060 means that this becomes 3,060,"},{"Start":"09:41.985 ","End":"09:47.190","Text":"so there\u0027s 4 zeros, and then 306."},{"Start":"09:47.190 ","End":"09:49.370","Text":"I don\u0027t have to put the 0 at the end."},{"Start":"09:49.370 ","End":"09:53.240","Text":"Now add these 2."},{"Start":"09:53.610 ","End":"10:07.040","Text":"What we get is 0.8345146."},{"Start":"10:07.040 ","End":"10:10.490","Text":"Now, I want it to 3 decimal places."},{"Start":"10:10.490 ","End":"10:13.460","Text":"I suppose you could just truncate,"},{"Start":"10:13.460 ","End":"10:16.285","Text":"but that\u0027s not the best thing you can do."},{"Start":"10:16.285 ","End":"10:22.580","Text":"You could say 0.834, but really,"},{"Start":"10:22.580 ","End":"10:24.440","Text":"because this is a 5,"},{"Start":"10:24.440 ","End":"10:26.345","Text":"anything that\u0027s 5 and above,"},{"Start":"10:26.345 ","End":"10:30.580","Text":"we should really round this 1 up and not down."},{"Start":"10:30.580 ","End":"10:37.220","Text":"I think a better answer would be 0.835,"},{"Start":"10:37.220 ","End":"10:39.890","Text":"although you might get away with this."},{"Start":"10:39.890 ","End":"10:42.980","Text":"I\u0027m going to say that this is our answer,"},{"Start":"10:42.980 ","End":"10:45.600","Text":"and we are done."}],"ID":26923}],"Thumbnail":null,"ID":264993}]

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