Basic Problems In Probability
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- Random trial
- Sample space
- Event and size of a sample space
- Complementary event
- A symmetric sample space
- Probability in a symmetric sample space
- Probability in an asymmetric sample space
- The probability of a complementary event
- Exercise 1
- Exercise 2 - Parts a-c
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6 - Parts a-d

Operations Between Events
0/48 completed

- Intersecting Events
- Union of Two Events
- Example - Part a
- Example - Part b
- Example - Part c
- The Magic Square
- Mutually Exclusive
- Exercise 1
- Exercise 2 - Part a
- Exercise 2 - Part b
- Exercise 2 - Part c
- Exercise 2 - Part d
- Exercise 2 - Part e
- Exercise 2 - Part f
- Exercise 3 - Part a
- Exercise 3 - Part b
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7 - Part a
- Exercise 7 - Part b
- Exercise 7 - Part c
- Exercise 8
- Exercise 9 - Part a
- Exercise 9 - Part b
- Exercise 9 - Part c
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 16 - Part a
- Exercise 16 - Part d
- Exercise 16 - Parts b-c
- Exercise 17 - Part a
- Exercise 17 - Part b
- Exercise 17 - Parts c-d
- Exercise 18 - Part a
- Exercise 18 - Part b
- Exercise 18 - Part c
- Exercise 18 - Part d
- Exercise 18 - Part e
- Exercise 18 - Parts f-g
- Exercise 19 - Part a
- Exercise 19 - Part b
- Exercise 19 - Part c
- Exercise 20
- Exercise 21

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[{"Name":"Basic Problems In Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Random trial","Duration":"36s","ChapterTopicVideoID":12246,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12246.jpeg","UploadDate":"2018-07-22T16:10:22.0330000","DurationForVideoObject":"PT36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.130","Text":"In this chapter, I\u0027ll introduce the concept of"},{"Start":"00:02.130 ","End":"00:05.325","Text":"probability theory by defining some basic terms."},{"Start":"00:05.325 ","End":"00:09.270","Text":"Let\u0027s get started. The first term is that of a random trial."},{"Start":"00:09.270 ","End":"00:15.210","Text":"Now, that\u0027s defined as a process that produces 1 result out of several possible results."},{"Start":"00:15.210 ","End":"00:19.920","Text":"Now this 1 result isn\u0027t known until after the process ends."},{"Start":"00:19.920 ","End":"00:23.910","Text":"For example, a random trial could be throwing a die."},{"Start":"00:23.910 ","End":"00:29.070","Text":"Now, I don\u0027t know what the result of throwing a die is until the die comes to rest."},{"Start":"00:29.070 ","End":"00:32.340","Text":"Or by guessing the weather 2 weeks from now."},{"Start":"00:32.340 ","End":"00:37.060","Text":"Now, I won\u0027t know what the weather would be until 2 weeks pass."}],"ID":12714},{"Watched":false,"Name":"Sample space","Duration":"1m 13s","ChapterTopicVideoID":12247,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12247.jpeg","UploadDate":"2018-07-22T16:10:25.1800000","DurationForVideoObject":"PT1M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.475","Text":"The next term is a sample space."},{"Start":"00:02.475 ","End":"00:07.140","Text":"Now that\u0027s defined as all the possible results of a random trial."},{"Start":"00:07.140 ","End":"00:10.530","Text":"Now that\u0027s denoted by the Greek letter Omega."},{"Start":"00:10.530 ","End":"00:15.375","Text":"In our case, a random trial would be throwing a die."},{"Start":"00:15.375 ","End":"00:19.320","Text":"Now what would be all the possible results of throwing a die?"},{"Start":"00:19.320 ","End":"00:22.815","Text":"Well, that\u0027s the set of numbers from 1 to 6."},{"Start":"00:22.815 ","End":"00:26.985","Text":"That\u0027s the possible results that I can get when I throw a die."},{"Start":"00:26.985 ","End":"00:30.765","Text":"I can get either a 1 or a 2 or 3, or 4 or 5,"},{"Start":"00:30.765 ","End":"00:35.451","Text":"or a 6, I can\u0027t get a 0 and I can\u0027t get an 8."},{"Start":"00:35.451 ","End":"00:40.590","Text":"These are the other possibilities that I can have when I throw the dice."},{"Start":"00:40.590 ","End":"00:45.185","Text":"That would be defined as my Omega or my sample space."},{"Start":"00:45.185 ","End":"00:47.300","Text":"Now, in the other example,"},{"Start":"00:47.300 ","End":"00:50.090","Text":"what would be my sample space?"},{"Start":"00:50.090 ","End":"00:56.360","Text":"What would be the weather or all the possible types of weathers in 2 weeks time?"},{"Start":"00:56.360 ","End":"00:59.450","Text":"That would be included in my sample space here."},{"Start":"00:59.450 ","End":"01:00.500","Text":"It could be pleasant,"},{"Start":"01:00.500 ","End":"01:02.480","Text":"could be a hot desert wind,"},{"Start":"01:02.480 ","End":"01:05.830","Text":"it could be snowy or rainy or whatever."},{"Start":"01:05.830 ","End":"01:10.460","Text":"All these possibilities will be included in my sample space,"},{"Start":"01:10.460 ","End":"01:13.800","Text":"which will be defined as omega."}],"ID":12715},{"Watched":false,"Name":"Event and size of a sample space","Duration":"2m 9s","ChapterTopicVideoID":12237,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12237.jpeg","UploadDate":"2018-07-22T16:08:30.0530000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.721","Text":"The next term is event."},{"Start":"00:01.721 ","End":"00:05.100","Text":"That\u0027s defined as a subset of a sample space."},{"Start":"00:05.100 ","End":"00:08.170","Text":"We\u0027ll denote that by capital letters."},{"Start":"00:08.170 ","End":"00:10.335","Text":"In our example, when throwing a die,"},{"Start":"00:10.335 ","End":"00:15.525","Text":"let\u0027s define the event A as throwing at least a 5."},{"Start":"00:15.525 ","End":"00:18.219","Text":"That means that A will have 2 numbers in it."},{"Start":"00:18.219 ","End":"00:20.985","Text":"It\u0027ll have either a 5 or a 6."},{"Start":"00:20.985 ","End":"00:27.075","Text":"Those are the possible values that the event A can have."},{"Start":"00:27.075 ","End":"00:32.690","Text":"Let\u0027s define B as getting an even number when throwing a die."},{"Start":"00:32.690 ","End":"00:34.880","Text":"How many even numbers do I have?"},{"Start":"00:34.880 ","End":"00:36.880","Text":"I have a 2, a 4, and a 6."},{"Start":"00:36.880 ","End":"00:40.730","Text":"So that\u0027ll be the set of even numbers which will make up"},{"Start":"00:40.730 ","End":"00:46.670","Text":"the event B. I can also define events that don\u0027t make much sense."},{"Start":"00:46.670 ","End":"00:48.150","Text":"For example, when I throw a die,"},{"Start":"00:48.150 ","End":"00:51.500","Text":"I want to get a number which is greater than 8."},{"Start":"00:51.500 ","End":"00:57.345","Text":"That\u0027ll be the definition of the event D. That\u0027ll be an empty set."},{"Start":"00:57.345 ","End":"01:01.005","Text":"We\u0027ll define an empty set like that."},{"Start":"01:01.005 ","End":"01:04.755","Text":"What about the size of a sample space?"},{"Start":"01:04.755 ","End":"01:07.180","Text":"The size of a sample space,"},{"Start":"01:07.180 ","End":"01:11.080","Text":"that\u0027s the number of all the possible results in the sample space."},{"Start":"01:11.080 ","End":"01:15.740","Text":"That will be denoted as the absolute value of Omega right here."},{"Start":"01:15.740 ","End":"01:17.975","Text":"In our case, when throwing a die,"},{"Start":"01:17.975 ","End":"01:20.300","Text":"what\u0027s the size of our sample space?"},{"Start":"01:20.300 ","End":"01:23.660","Text":"How many possible results can I get?"},{"Start":"01:23.660 ","End":"01:27.170","Text":"When throwing a die, I get the numbers from 1,"},{"Start":"01:27.170 ","End":"01:30.470","Text":"2, 3, 4, 5, and 6."},{"Start":"01:30.470 ","End":"01:32.914","Text":"These are all the possible results."},{"Start":"01:32.914 ","End":"01:36.470","Text":"So the size of the sample space when throwing"},{"Start":"01:36.470 ","End":"01:40.885","Text":"a die would be 6 because I have 6 numbers here."},{"Start":"01:40.885 ","End":"01:43.310","Text":"What about the size of an event?"},{"Start":"01:43.310 ","End":"01:47.450","Text":"That would be the number of possible results in the event itself."},{"Start":"01:47.450 ","End":"01:49.360","Text":"In event A,"},{"Start":"01:49.360 ","End":"01:52.050","Text":"we\u0027re throwing at least a 5."},{"Start":"01:52.050 ","End":"01:53.925","Text":"I have 2 numbers here,"},{"Start":"01:53.925 ","End":"01:56.600","Text":"so the absolute value of A would be 2."},{"Start":"01:56.600 ","End":"01:59.805","Text":"What\u0027s the size of event B?"},{"Start":"01:59.805 ","End":"02:02.270","Text":"That\u0027s getting an even number right here,"},{"Start":"02:02.270 ","End":"02:04.144","Text":"so that\u0027ll be either a 2, 4, or 6."},{"Start":"02:04.144 ","End":"02:05.960","Text":"So I have 3 numbers here."},{"Start":"02:05.960 ","End":"02:09.510","Text":"So the size of B would be 3."}],"ID":12716},{"Watched":false,"Name":"Complementary event","Duration":"2m 27s","ChapterTopicVideoID":12236,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12236.jpeg","UploadDate":"2018-07-22T16:08:23.7070000","DurationForVideoObject":"PT2M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"The next term is the complementary event."},{"Start":"00:02.940 ","End":"00:06.390","Text":"That\u0027s defined as an event that contains"},{"Start":"00:06.390 ","End":"00:09.750","Text":"all the possible results in the sample space that"},{"Start":"00:09.750 ","End":"00:13.275","Text":"are not included in the original event that it complements."},{"Start":"00:13.275 ","End":"00:17.040","Text":"The results of an event and the results of"},{"Start":"00:17.040 ","End":"00:21.510","Text":"the complementary event give the results of the whole sample space."},{"Start":"00:21.510 ","End":"00:24.750","Text":"For example, when throwing a die,"},{"Start":"00:24.750 ","End":"00:28.469","Text":"the complementary set of A would be this,"},{"Start":"00:28.469 ","End":"00:30.690","Text":"and the complementary set of B would be this."},{"Start":"00:30.690 ","End":"00:35.025","Text":"Let\u0027s recall Omega, the sample space,"},{"Start":"00:35.025 ","End":"00:36.755","Text":"was 1, 2,"},{"Start":"00:36.755 ","End":"00:38.615","Text":"3, 4,"},{"Start":"00:38.615 ","End":"00:40.745","Text":"5, and 6."},{"Start":"00:40.745 ","End":"00:46.325","Text":"A was defined as the event of getting at least a 5,"},{"Start":"00:46.325 ","End":"00:49.655","Text":"so that means it\u0027s a 5 and a 6,"},{"Start":"00:49.655 ","End":"00:53.810","Text":"and B is the event of getting even numbers,"},{"Start":"00:53.810 ","End":"00:57.945","Text":"so that\u0027s 2, 4, and 6."},{"Start":"00:57.945 ","End":"01:00.900","Text":"A complementary event,"},{"Start":"01:00.900 ","End":"01:06.440","Text":"that\u0027s denoted by the capital letter with a bar over it"},{"Start":"01:06.440 ","End":"01:12.700","Text":"or the capital letter with a small c above the letter."},{"Start":"01:12.700 ","End":"01:16.625","Text":"These are the 2 ways we can denote a complementary event."},{"Start":"01:16.625 ","End":"01:22.400","Text":"I\u0027ll be using this way of denoting a complementary event."},{"Start":"01:22.400 ","End":"01:25.570","Text":"Let\u0027s take a look at our example again."},{"Start":"01:25.570 ","End":"01:30.075","Text":"What\u0027s the complementary set of A?"},{"Start":"01:30.075 ","End":"01:31.656","Text":"A is 5 and 6,"},{"Start":"01:31.656 ","End":"01:37.370","Text":"and we\u0027ve defined a complementary set as the results in"},{"Start":"01:37.370 ","End":"01:43.550","Text":"the sample space which are not included in the original set."},{"Start":"01:43.550 ","End":"01:46.580","Text":"If it\u0027s not included in the original set,"},{"Start":"01:46.580 ","End":"01:49.775","Text":"so 5 and 6 go away."},{"Start":"01:49.775 ","End":"01:54.770","Text":"So that means that the complementary set of A would be 1, 2, 3, and 4,"},{"Start":"01:54.770 ","End":"01:58.145","Text":"and that\u0027s exactly what we have here,"},{"Start":"01:58.145 ","End":"01:59.975","Text":"or when looking at B,"},{"Start":"01:59.975 ","End":"02:02.930","Text":"B was all the even numbers."},{"Start":"02:02.930 ","End":"02:05.390","Text":"So B was 2, 4, and 6."},{"Start":"02:05.390 ","End":"02:10.325","Text":"So the complementary set of B would be all the values"},{"Start":"02:10.325 ","End":"02:15.920","Text":"in the sample space which are not included in the original event."},{"Start":"02:15.920 ","End":"02:19.280","Text":"That means we take away 2, 4, and 6,"},{"Start":"02:19.280 ","End":"02:23.360","Text":"and the complementary set of B would be 1,"},{"Start":"02:23.360 ","End":"02:27.390","Text":"3, and 5, and that\u0027s exactly what we have here."}],"ID":12717},{"Watched":false,"Name":"A symmetric sample space","Duration":"58s","ChapterTopicVideoID":12235,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12235.jpeg","UploadDate":"2018-07-22T16:08:17.5600000","DurationForVideoObject":"PT58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"The next term is a symmetric sample space."},{"Start":"00:02.880 ","End":"00:05.370","Text":"That\u0027s defined as a sample space in which"},{"Start":"00:05.370 ","End":"00:08.340","Text":"all the results have the same chance of occurring."},{"Start":"00:08.340 ","End":"00:13.095","Text":"An example of that would be the sample space for throwing a balanced die."},{"Start":"00:13.095 ","End":"00:20.085","Text":"Omega here would be all the numbers from 1-6."},{"Start":"00:20.085 ","End":"00:24.220","Text":"The chances of getting a 1 would be equal to the chances of getting a 2,"},{"Start":"00:24.220 ","End":"00:27.945","Text":"or a 3, or basically any other result in the sample space."},{"Start":"00:27.945 ","End":"00:32.195","Text":"What\u0027s not a symmetric sample space would then be the sample space,"},{"Start":"00:32.195 ","End":"00:36.680","Text":"let\u0027s say, for the weather a week from now. Why is that?"},{"Start":"00:36.680 ","End":"00:38.960","Text":"That would depend on the season."},{"Start":"00:38.960 ","End":"00:40.310","Text":"If we were in summer,"},{"Start":"00:40.310 ","End":"00:44.540","Text":"then the chances of it snowing would be close to 0,"},{"Start":"00:44.540 ","End":"00:48.965","Text":"whereas the chance of it being sunny would be great."},{"Start":"00:48.965 ","End":"00:53.619","Text":"So the 2 events will not have the same chance of occurring."},{"Start":"00:53.619 ","End":"00:58.500","Text":"So therefore, that would not be a symmetric sample space."}],"ID":12718},{"Watched":false,"Name":"Probability in a symmetric sample space","Duration":"2m 22s","ChapterTopicVideoID":12244,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12244.jpeg","UploadDate":"2018-07-22T16:10:10.6600000","DurationForVideoObject":"PT2M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Having defined the symmetric sample space,"},{"Start":"00:03.210 ","End":"00:07.920","Text":"we can now go ahead and calculate the probability in a symmetric sample space."},{"Start":"00:07.920 ","End":"00:09.825","Text":"Now, in a symmetric sample space,"},{"Start":"00:09.825 ","End":"00:14.460","Text":"the chance of an event would be this expression right here."},{"Start":"00:14.460 ","End":"00:17.385","Text":"Now, how do we read this expression?"},{"Start":"00:17.385 ","End":"00:19.125","Text":"This is P of A,"},{"Start":"00:19.125 ","End":"00:21.060","Text":"that\u0027s the probability of A."},{"Start":"00:21.060 ","End":"00:26.000","Text":"The probability of event A occurring will now would be equal to"},{"Start":"00:26.000 ","End":"00:32.780","Text":"the size of the event A divided by the size of the sample space."},{"Start":"00:32.780 ","End":"00:37.755","Text":"Now, let\u0027s take the example of event A."},{"Start":"00:37.755 ","End":"00:43.410","Text":"If we recall, A not equal to the event of getting at least a 5."},{"Start":"00:43.410 ","End":"00:49.005","Text":"That means that A was the event of getting a 5 or a 6."},{"Start":"00:49.005 ","End":"00:52.440","Text":"The size of A, well,"},{"Start":"00:52.440 ","End":"00:53.700","Text":"we calculated that,"},{"Start":"00:53.700 ","End":"00:55.755","Text":"that equal to 2."},{"Start":"00:55.755 ","End":"00:57.870","Text":"What about omega?"},{"Start":"00:57.870 ","End":"01:03.460","Text":"Omega was all the numbers from 1-6."},{"Start":"01:04.010 ","End":"01:07.275","Text":"1, 2, 3, 4, 5, 6."},{"Start":"01:07.275 ","End":"01:11.360","Text":"The size of Omega that equals to 6,"},{"Start":"01:11.360 ","End":"01:14.960","Text":"because we have a total of 6 possible numbers."},{"Start":"01:14.960 ","End":"01:19.410","Text":"How do we calculate the probability of A?"},{"Start":"01:19.410 ","End":"01:21.770","Text":"Well, that\u0027s equals, as we said,"},{"Start":"01:21.770 ","End":"01:27.650","Text":"to the size of event A divided by the size of Omega."},{"Start":"01:27.650 ","End":"01:29.810","Text":"The size of A that equals to 2,"},{"Start":"01:29.810 ","End":"01:31.235","Text":"we\u0027ve calculated that,"},{"Start":"01:31.235 ","End":"01:34.595","Text":"and the size of Omega that\u0027s equal to 6."},{"Start":"01:34.595 ","End":"01:37.265","Text":"So then B equal to 2 divided by 6."},{"Start":"01:37.265 ","End":"01:41.360","Text":"Now, what about the probability of B?"},{"Start":"01:41.360 ","End":"01:42.980","Text":"Well, if we recall,"},{"Start":"01:42.980 ","End":"01:45.620","Text":"B, was getting the even numbers."},{"Start":"01:45.620 ","End":"01:49.595","Text":"B was 2, 4, and 6."},{"Start":"01:49.595 ","End":"01:52.400","Text":"The size of B, well,"},{"Start":"01:52.400 ","End":"01:53.660","Text":"we have 3 numbers,"},{"Start":"01:53.660 ","End":"01:55.805","Text":"so the size of B is 3."},{"Start":"01:55.805 ","End":"02:00.180","Text":"The probability of event B occurring,"},{"Start":"02:00.180 ","End":"02:03.950","Text":"well that equals to the size of B, which is 3,"},{"Start":"02:03.950 ","End":"02:07.520","Text":"divided by the size of the sample space Omega,"},{"Start":"02:07.520 ","End":"02:08.975","Text":"which is equal to 6."},{"Start":"02:08.975 ","End":"02:16.760","Text":"This then are the probabilities for event A occurring and for event B occurring,"},{"Start":"02:16.760 ","End":"02:22.800","Text":"and that\u0027s how we calculate a probability in a symmetric sample space."}],"ID":12719},{"Watched":false,"Name":"Probability in an asymmetric sample space","Duration":"2m 53s","ChapterTopicVideoID":12245,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12245.jpeg","UploadDate":"2018-07-22T16:10:20.5170000","DurationForVideoObject":"PT2M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.770","Text":"Now that we know how to calculate the probability in a symmetric sample space,"},{"Start":"00:04.770 ","End":"00:09.375","Text":"let\u0027s see how we can calculate the probability and then asymmetric sample space."},{"Start":"00:09.375 ","End":"00:14.925","Text":"Now, the probability of an event is calculated according to its relative frequency."},{"Start":"00:14.925 ","End":"00:17.295","Text":"Now this is also called the proportion."},{"Start":"00:17.295 ","End":"00:19.320","Text":"Now, let\u0027s see what we mean by that."},{"Start":"00:19.320 ","End":"00:21.135","Text":"Well, we have an example here."},{"Start":"00:21.135 ","End":"00:25.925","Text":"The following table displays the distribution of grades in a certain class."},{"Start":"00:25.925 ","End":"00:27.470","Text":"Here we have the grades,"},{"Start":"00:27.470 ","End":"00:28.670","Text":"that\u0027ll be our X,"},{"Start":"00:28.670 ","End":"00:30.800","Text":"and the grades will have values of 5,"},{"Start":"00:30.800 ","End":"00:32.720","Text":"6, 7, 8, 9, and 10."},{"Start":"00:32.720 ","End":"00:37.915","Text":"Here we have the frequency of students who receive these grades, for example,"},{"Start":"00:37.915 ","End":"00:40.115","Text":"2 students who received a grade of 5,"},{"Start":"00:40.115 ","End":"00:41.780","Text":"4 students a grade of 6,"},{"Start":"00:41.780 ","End":"00:43.640","Text":"8 students who received a grade of 7,"},{"Start":"00:43.640 ","End":"00:44.945","Text":"and so on and so forth."},{"Start":"00:44.945 ","End":"00:52.970","Text":"We can see that the chance of getting either a 5 or 6 or 7 or an 8,"},{"Start":"00:52.970 ","End":"00:54.710","Text":"basically any number here,"},{"Start":"00:54.710 ","End":"00:57.065","Text":"the chances are not equal."},{"Start":"00:57.065 ","End":"01:02.660","Text":"The chances are dependent on the frequency of students receiving these grades."},{"Start":"01:02.660 ","End":"01:05.750","Text":"Now, let\u0027s just take a look at an example to see how we"},{"Start":"01:05.750 ","End":"01:09.385","Text":"can calculate these types of probabilities."},{"Start":"01:09.385 ","End":"01:15.350","Text":"What\u0027s the probability that a randomly chosen student class received a grade of 8?"},{"Start":"01:15.350 ","End":"01:19.520","Text":"Well, first of all, let\u0027s take a look at our X values right here,"},{"Start":"01:19.520 ","End":"01:22.805","Text":"and here we have a grade of 8 right here."},{"Start":"01:22.805 ","End":"01:27.845","Text":"Now, next we have to go to the frequency."},{"Start":"01:27.845 ","End":"01:33.600","Text":"We have 5 students who received a grade of 8."},{"Start":"01:33.600 ","End":"01:36.865","Text":"F here, that would be equal to 5."},{"Start":"01:36.865 ","End":"01:40.595","Text":"Now, how many students do I have here in total?"},{"Start":"01:40.595 ","End":"01:42.020","Text":"That would be my n. Well,"},{"Start":"01:42.020 ","End":"01:44.060","Text":"all I have to do some of these numbers,"},{"Start":"01:44.060 ","End":"01:45.680","Text":"sum up all the frequencies,"},{"Start":"01:45.680 ","End":"01:47.990","Text":"and that comes out to 25."},{"Start":"01:47.990 ","End":"01:53.450","Text":"The proportion or the relative frequency of students who"},{"Start":"01:53.450 ","End":"01:59.720","Text":"received a grade of 8 would be 5 divided by 25 and that equals to 0.2."},{"Start":"01:59.720 ","End":"02:02.405","Text":"Now let\u0027s take a look at another example."},{"Start":"02:02.405 ","End":"02:05.240","Text":"We\u0027re asked, what\u0027s the probability that"},{"Start":"02:05.240 ","End":"02:08.630","Text":"a randomly chosen student in the class will fail?"},{"Start":"02:08.630 ","End":"02:13.265","Text":"Well, let\u0027s define fail as getting a 5,"},{"Start":"02:13.265 ","End":"02:16.205","Text":"getting below 6, so that\u0027s a fail."},{"Start":"02:16.205 ","End":"02:20.690","Text":"Now, that means that I have to go to my X value,"},{"Start":"02:20.690 ","End":"02:22.180","Text":"which is 5,"},{"Start":"02:22.180 ","End":"02:27.365","Text":"and then I have to see what my frequency is here, and that\u0027s 2."},{"Start":"02:27.365 ","End":"02:32.900","Text":"What\u0027s the relative frequency of getting a 5?"},{"Start":"02:32.900 ","End":"02:37.200","Text":"Well, that\u0027ll be f divided by n. Now f that equals to 2,"},{"Start":"02:37.200 ","End":"02:40.120","Text":"only 2 students received a grade of 5,"},{"Start":"02:40.120 ","End":"02:43.730","Text":"divided by the total number of students that I have,"},{"Start":"02:43.730 ","End":"02:44.900","Text":"which is 25,"},{"Start":"02:44.900 ","End":"02:47.855","Text":"so the proportion here or the probability"},{"Start":"02:47.855 ","End":"02:52.770","Text":"would be 2 divided by 25 and that equals to 0.08."}],"ID":12720},{"Watched":false,"Name":"The probability of a complementary event","Duration":"1m 56s","ChapterTopicVideoID":12248,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12248.jpeg","UploadDate":"2018-07-22T16:10:29.6530000","DurationForVideoObject":"PT1M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"The next term is the probability of a complimentary event."},{"Start":"00:04.290 ","End":"00:08.625","Text":"We\u0027ve learned about events and complimentary events."},{"Start":"00:08.625 ","End":"00:14.265","Text":"We know that if we have an event and the complimentary event together,"},{"Start":"00:14.265 ","End":"00:17.985","Text":"and that will give us the total sample space."},{"Start":"00:17.985 ","End":"00:21.450","Text":"Well, when we translate that into probabilities."},{"Start":"00:21.450 ","End":"00:27.570","Text":"Well, the probability of an event plus the probability of a complimentary event,"},{"Start":"00:27.570 ","End":"00:29.880","Text":"well, that will always give us 1."},{"Start":"00:29.880 ","End":"00:32.460","Text":"Again, that\u0027s because the event and"},{"Start":"00:32.460 ","End":"00:35.935","Text":"the complimentary event will always give us the sample space."},{"Start":"00:35.935 ","End":"00:39.500","Text":"But this means that the probability of"},{"Start":"00:39.500 ","End":"00:45.140","Text":"a complimentary event that equals to 1 minus the probability of an event."},{"Start":"00:45.140 ","End":"00:47.150","Text":"This is exactly what we have here."},{"Start":"00:47.150 ","End":"00:49.085","Text":"Now, when can we use that?"},{"Start":"00:49.085 ","End":"00:54.635","Text":"Well, if calculating the probability of an event turns out to be very complicated,"},{"Start":"00:54.635 ","End":"00:59.540","Text":"and calculating the probability of a complimentary event turns out to be simple."},{"Start":"00:59.540 ","End":"01:01.940","Text":"Then we\u0027ll use this expression."},{"Start":"01:01.940 ","End":"01:04.670","Text":"Now, let\u0027s just take a look at an example."},{"Start":"01:04.670 ","End":"01:06.650","Text":"In the preceding example,"},{"Start":"01:06.650 ","End":"01:09.290","Text":"the chances of passing the examination can be"},{"Start":"01:09.290 ","End":"01:12.410","Text":"calculated according to the chances of failing."},{"Start":"01:12.410 ","End":"01:17.605","Text":"Now, we\u0027ve calculated the probability of failing."},{"Start":"01:17.605 ","End":"01:22.245","Text":"That was equal to 2 divided by 25."},{"Start":"01:22.245 ","End":"01:27.635","Text":"Only 2 people out of 25 received a grade of 5."},{"Start":"01:27.635 ","End":"01:31.895","Text":"What\u0027s the probability of passing?"},{"Start":"01:31.895 ","End":"01:37.355","Text":"Well, that equals to 1 minus the probability of failing,"},{"Start":"01:37.355 ","End":"01:42.200","Text":"and that equals to 23 divided by 25. There we go."},{"Start":"01:42.200 ","End":"01:46.900","Text":"That\u0027s how we use the probability of a complimentary event."},{"Start":"01:46.900 ","End":"01:52.145","Text":"This concludes the theoretical part of this chapter."},{"Start":"01:52.145 ","End":"01:56.490","Text":"Let\u0027s get started and do a lot of exercises."}],"ID":12721},{"Watched":false,"Name":"Exercise 1","Duration":"3m 50s","ChapterTopicVideoID":12238,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12238.jpeg","UploadDate":"2018-07-22T16:08:46.8470000","DurationForVideoObject":"PT3M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this question, a 2-letter word is formed from the letters E,"},{"Start":"00:03.540 ","End":"00:06.000","Text":"F, and G, not necessarily a real word."},{"Start":"00:06.000 ","End":"00:09.270","Text":"We\u0027re asked in Section 8 to list all the possible words."},{"Start":"00:09.270 ","End":"00:11.730","Text":"By asking us that,"},{"Start":"00:11.730 ","End":"00:18.210","Text":"we\u0027re basically asked to create a sample space that\u0027s denoted by the Greek letter Omega."},{"Start":"00:18.210 ","End":"00:22.170","Text":"So let\u0027s get started. We want a word that\u0027s"},{"Start":"00:22.170 ","End":"00:26.610","Text":"comprised of 2 letters out of these 3 letters."},{"Start":"00:26.610 ","End":"00:32.850","Text":"Let\u0027s begin with words that begin with the letter E. We can have EE,"},{"Start":"00:32.850 ","End":"00:37.480","Text":"EF, and EG."},{"Start":"00:37.480 ","End":"00:40.190","Text":"Words that begin with the letter F,"},{"Start":"00:40.190 ","End":"00:42.515","Text":"that\u0027ll be FE,"},{"Start":"00:42.515 ","End":"00:46.110","Text":"FF, and FG."},{"Start":"00:46.110 ","End":"00:48.370","Text":"Now words that begin with the letter G,"},{"Start":"00:48.370 ","End":"00:54.265","Text":"that\u0027ll be GE, GF, and GG."},{"Start":"00:54.265 ","End":"00:59.010","Text":"Excellent. So these then are all the possible words,"},{"Start":"00:59.010 ","End":"01:07.000","Text":"these are all the possible 2-letter combinations that we can form out of using 3 letters."},{"Start":"01:07.000 ","End":"01:10.245","Text":"So A is complete. What about B?"},{"Start":"01:10.245 ","End":"01:15.360","Text":"B asks us to list all the cases for events A and B."},{"Start":"01:15.360 ","End":"01:17.385","Text":"Let\u0027s begin with event A."},{"Start":"01:17.385 ","End":"01:21.489","Text":"Event A asks us to list all the words that contain"},{"Start":"01:21.489 ","End":"01:25.950","Text":"the letter E. Here are the words: EE,"},{"Start":"01:25.950 ","End":"01:27.465","Text":"EF, EG,"},{"Start":"01:27.465 ","End":"01:30.720","Text":"FE, and GE."},{"Start":"01:30.720 ","End":"01:33.435","Text":"Let\u0027s just write this out."},{"Start":"01:33.435 ","End":"01:37.080","Text":"That\u0027ll be EE, EF,"},{"Start":"01:37.080 ","End":"01:44.760","Text":"EG, FE, and GE."},{"Start":"01:44.760 ","End":"01:48.755","Text":"There aren\u0027t any other words here that contain the letter"},{"Start":"01:48.755 ","End":"01:53.650","Text":"E. So these are all the words that belong to event A."},{"Start":"01:53.650 ","End":"01:56.195","Text":"What about event B?"},{"Start":"01:56.195 ","End":"02:04.054","Text":"We\u0027re asked to include all the words that their letters are different."},{"Start":"02:04.054 ","End":"02:05.735","Text":"What would that be?"},{"Start":"02:05.735 ","End":"02:13.215","Text":"EE won\u0027t be in event B because the letters here are the same,"},{"Start":"02:13.215 ","End":"02:15.520","Text":"EF would be included,"},{"Start":"02:15.520 ","End":"02:17.300","Text":"EG would be included,"},{"Start":"02:17.300 ","End":"02:19.280","Text":"FE, FF, no,"},{"Start":"02:19.280 ","End":"02:20.870","Text":"FG, GE,"},{"Start":"02:20.870 ","End":"02:22.660","Text":"GF, and GG, no."},{"Start":"02:22.660 ","End":"02:24.028","Text":"So let\u0027s just write these out."},{"Start":"02:24.028 ","End":"02:27.350","Text":"So that will be EF,"},{"Start":"02:27.350 ","End":"02:32.760","Text":"EG, FE,"},{"Start":"02:32.760 ","End":"02:37.995","Text":"FG, GE, and GF."},{"Start":"02:37.995 ","End":"02:41.690","Text":"So B is complete. What about C?"},{"Start":"02:41.690 ","End":"02:46.820","Text":"C asks us to list all the cases for event A bar."},{"Start":"02:46.820 ","End":"02:50.150","Text":"A bar is the complimentary set of A,"},{"Start":"02:50.150 ","End":"02:54.260","Text":"and we know that if we have event A,"},{"Start":"02:54.260 ","End":"02:57.260","Text":"and we have the complimentary set A bar,"},{"Start":"02:57.260 ","End":"03:00.830","Text":"both of them make up the total sample space."},{"Start":"03:00.830 ","End":"03:04.040","Text":"So all we need to do is go to the sample space"},{"Start":"03:04.040 ","End":"03:08.114","Text":"and pick out all the words that don\u0027t belong to A."},{"Start":"03:08.114 ","End":"03:10.040","Text":"All these words,"},{"Start":"03:10.040 ","End":"03:14.390","Text":"the words that are not marked with"},{"Start":"03:14.390 ","End":"03:19.050","Text":"a red underscore because the red underscore are the words that belong in A,"},{"Start":"03:19.050 ","End":"03:20.605","Text":"so A bar,"},{"Start":"03:20.605 ","End":"03:23.580","Text":"that will be the following words."},{"Start":"03:23.580 ","End":"03:26.070","Text":"That will be FF,"},{"Start":"03:26.070 ","End":"03:29.055","Text":"FF doesn\u0027t have a red underscore,"},{"Start":"03:29.055 ","End":"03:34.200","Text":"FG, GE belongs to A,"},{"Start":"03:34.200 ","End":"03:36.105","Text":"so that will be GF,"},{"Start":"03:36.105 ","End":"03:39.405","Text":"that\u0027s the next word, and then GG."},{"Start":"03:39.405 ","End":"03:45.815","Text":"These are all the words that belong to the complimentary set of A or A bar."},{"Start":"03:45.815 ","End":"03:50.340","Text":"So there we go. We\u0027ve completed our first exercise."}],"ID":12722},{"Watched":false,"Name":"Exercise 2 - Parts a-c","Duration":"6m 34s","ChapterTopicVideoID":12239,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12239.jpeg","UploadDate":"2018-07-22T16:09:03.3700000","DurationForVideoObject":"PT6M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.015","Text":"In this question, a pair of dice is thrown,"},{"Start":"00:03.015 ","End":"00:05.710","Text":"and we\u0027re told to list the sample space of the trial,"},{"Start":"00:05.710 ","End":"00:08.925","Text":"and we are asked whether the sample space is symmetric or not."},{"Start":"00:08.925 ","End":"00:10.440","Text":"We have to pay attention."},{"Start":"00:10.440 ","End":"00:15.495","Text":"Here we\u0027re told that a pair of dice is thrown, not 1 die."},{"Start":"00:15.495 ","End":"00:20.340","Text":"So that means that whenever we throw the pair of dice,"},{"Start":"00:20.340 ","End":"00:21.945","Text":"then we get 2 numbers."},{"Start":"00:21.945 ","End":"00:24.870","Text":"We\u0027re asked to list the sample space of the trial."},{"Start":"00:24.870 ","End":"00:28.440","Text":"Sample space is denoted by the Greek letter Omega,"},{"Start":"00:28.440 ","End":"00:35.490","Text":"and it contains all the possible combination of numbers when we throw a pair of dice."},{"Start":"00:35.490 ","End":"00:41.180","Text":"So what would that be? When we throw a pair of dice,"},{"Start":"00:41.180 ","End":"00:44.960","Text":"the first die could give me a result of"},{"Start":"00:44.960 ","End":"00:49.520","Text":"1 and the second die would also give me a result of 1,"},{"Start":"00:49.520 ","End":"00:51.950","Text":"or we can have a 1 and a 2,"},{"Start":"00:51.950 ","End":"00:53.740","Text":"a 1 and a 3,"},{"Start":"00:53.740 ","End":"00:55.140","Text":"1 and 4,"},{"Start":"00:55.140 ","End":"00:56.580","Text":"1 and 5,"},{"Start":"00:56.580 ","End":"00:58.125","Text":"and 1 and 6."},{"Start":"00:58.125 ","End":"01:02.570","Text":"These would be the results of the first die,"},{"Start":"01:02.570 ","End":"01:09.490","Text":"and this would be the result of the second die in a throw."},{"Start":"01:09.490 ","End":"01:12.200","Text":"We can have other results,"},{"Start":"01:12.200 ","End":"01:16.130","Text":"the first die can give me a result of 2,"},{"Start":"01:16.130 ","End":"01:20.990","Text":"and the second die will give me also a result of either 1,"},{"Start":"01:20.990 ","End":"01:23.690","Text":"or 2, or 3,"},{"Start":"01:23.690 ","End":"01:28.255","Text":"or 4, or 5, or 6."},{"Start":"01:28.255 ","End":"01:31.840","Text":"Let\u0027s just complete the table here."},{"Start":"01:31.840 ","End":"01:35.120","Text":"Here we go. This is the complete table."},{"Start":"01:35.120 ","End":"01:38.030","Text":"These are all the possible results of"},{"Start":"01:38.030 ","End":"01:43.228","Text":"the numbers that we can get when we throw a pair of dice,"},{"Start":"01:43.228 ","End":"01:48.770","Text":"that would be the first number in each column would be the results of the first die and"},{"Start":"01:48.770 ","End":"01:56.570","Text":"the second column in the pair of numbers here would be the result of the second dice."},{"Start":"01:56.570 ","End":"02:00.080","Text":"Is the sample space symmetric?"},{"Start":"02:00.080 ","End":"02:02.255","Text":"Yes, it is. Why is that?"},{"Start":"02:02.255 ","End":"02:06.230","Text":"The chances of getting any pair of"},{"Start":"02:06.230 ","End":"02:10.820","Text":"numbers is equal to the chances of getting any other pair of numbers."},{"Start":"02:10.820 ","End":"02:16.520","Text":"For example, the chances of getting a 3 and a 2 is the same chance as getting,"},{"Start":"02:16.520 ","End":"02:19.345","Text":"let\u0027s say, a 6 and a 3."},{"Start":"02:19.345 ","End":"02:23.030","Text":"So the answer to this question is yes."},{"Start":"02:23.030 ","End":"02:25.430","Text":"The sample space is symmetric."},{"Start":"02:25.430 ","End":"02:29.585","Text":"In this section, we\u0027re asked to list all the possibilities for the following events,"},{"Start":"02:29.585 ","End":"02:31.190","Text":"event A and event B."},{"Start":"02:31.190 ","End":"02:36.275","Text":"Event A is defined as the sum of the results that will equal 7,"},{"Start":"02:36.275 ","End":"02:40.775","Text":"and B is that the product of the results is equal to 12."},{"Start":"02:40.775 ","End":"02:43.685","Text":"Let\u0027s begin with A."},{"Start":"02:43.685 ","End":"02:47.286","Text":"We\u0027re asked to find a pair of numbers in the sample space,"},{"Start":"02:47.286 ","End":"02:48.767","Text":"here\u0027s our sample space,"},{"Start":"02:48.767 ","End":"02:54.845","Text":"that\u0027s all the possible pair of numbers that I can get when I throw a pair of dice."},{"Start":"02:54.845 ","End":"03:01.565","Text":"We\u0027re asked to find a pair of numbers here that their sum equals to 7."},{"Start":"03:01.565 ","End":"03:07.740","Text":"That will be equal to the following: that will be 1 and 6,"},{"Start":"03:07.740 ","End":"03:10.155","Text":"that will be 2 and 5,"},{"Start":"03:10.155 ","End":"03:12.975","Text":"that will be 3 and 4,"},{"Start":"03:12.975 ","End":"03:15.600","Text":"that will be 4 and 3,"},{"Start":"03:15.600 ","End":"03:19.785","Text":"that will be 5 and 2,"},{"Start":"03:19.785 ","End":"03:23.220","Text":"and 6 and 1."},{"Start":"03:23.220 ","End":"03:27.214","Text":"So these are all the pairs of numbers,"},{"Start":"03:27.214 ","End":"03:29.060","Text":"and the only pairs of numbers,"},{"Start":"03:29.060 ","End":"03:33.590","Text":"that their sum would be equal to 7 in this sample space."},{"Start":"03:33.590 ","End":"03:41.330","Text":"What about B? B talks about the product of a pair of numbers being equal to 12."},{"Start":"03:41.330 ","End":"03:44.720","Text":"That will be equal to 2 and 6,"},{"Start":"03:44.720 ","End":"03:47.820","Text":"3 and 4,"},{"Start":"03:47.820 ","End":"03:50.880","Text":"4 and 3,"},{"Start":"03:50.880 ","End":"03:54.210","Text":"and then 6 and 2."},{"Start":"03:54.210 ","End":"03:59.045","Text":"So again these are all the possible pairs of numbers"},{"Start":"03:59.045 ","End":"04:03.860","Text":"in the sample space that their product equals to 12."},{"Start":"04:03.860 ","End":"04:05.150","Text":"So there we go."},{"Start":"04:05.150 ","End":"04:07.790","Text":"That\u0027s the answer to section B."},{"Start":"04:07.790 ","End":"04:10.595","Text":"In this section, we\u0027re asked to calculate the chances"},{"Start":"04:10.595 ","End":"04:13.600","Text":"of the events defined in section B above."},{"Start":"04:13.600 ","End":"04:17.525","Text":"Here we have our sample space, Omega,"},{"Start":"04:17.525 ","End":"04:21.380","Text":"and these are all the pairs of numbers"},{"Start":"04:21.380 ","End":"04:25.400","Text":"that we can get by throwing a die or a pair of dice."},{"Start":"04:25.400 ","End":"04:28.585","Text":"Here in event A,"},{"Start":"04:28.585 ","End":"04:35.540","Text":"these are all the pairs of numbers that add up to 7 that are in the sample space."},{"Start":"04:35.540 ","End":"04:40.415","Text":"In B, these are all the pairs of numbers in the sample space that,"},{"Start":"04:40.415 ","End":"04:45.370","Text":"when multiplied together, will give me a result of 12."},{"Start":"04:45.370 ","End":"04:48.169","Text":"We\u0027re asked to calculate the chances."},{"Start":"04:48.169 ","End":"04:52.310","Text":"The chance of an event, let\u0027s say A,"},{"Start":"04:52.310 ","End":"04:59.480","Text":"that equals to the size of the event divided by the size of the sample space."},{"Start":"04:59.480 ","End":"05:02.675","Text":"What\u0027s the size of our sample space?"},{"Start":"05:02.675 ","End":"05:06.890","Text":"How many pairs of numbers do I have in my sample space here?"},{"Start":"05:06.890 ","End":"05:12.540","Text":"I have 6 pairs of numbers in 1 column,"},{"Start":"05:12.540 ","End":"05:14.874","Text":"and I have 6 columns."},{"Start":"05:14.874 ","End":"05:17.885","Text":"So 6 times 6, that equals to 36."},{"Start":"05:17.885 ","End":"05:20.900","Text":"I have 36 pairs of numbers here."},{"Start":"05:20.900 ","End":"05:23.840","Text":"What about the size of A?"},{"Start":"05:23.840 ","End":"05:26.435","Text":"How many pairs of numbers do I have in A?"},{"Start":"05:26.435 ","End":"05:27.710","Text":"I have 1, 2, 3,"},{"Start":"05:27.710 ","End":"05:29.850","Text":"4, 5, and 6."},{"Start":"05:29.850 ","End":"05:32.450","Text":"What about the size of B?"},{"Start":"05:32.450 ","End":"05:34.790","Text":"How many pairs of numbers do I have in B?"},{"Start":"05:34.790 ","End":"05:37.310","Text":"That\u0027s 1, 2, 3, and 4."},{"Start":"05:37.310 ","End":"05:43.505","Text":"Let\u0027s calculate the chances of event A occurring."},{"Start":"05:43.505 ","End":"05:46.635","Text":"Again the probability of A,"},{"Start":"05:46.635 ","End":"05:53.015","Text":"that equals to the size of A divided by the size of Omega, the sample space."},{"Start":"05:53.015 ","End":"05:54.410","Text":"The size of A,"},{"Start":"05:54.410 ","End":"05:58.820","Text":"that equals to 6 divided by the size of the sample space,"},{"Start":"05:58.820 ","End":"06:02.584","Text":"that\u0027s 36, and that equals to 1 divided by 6."},{"Start":"06:02.584 ","End":"06:06.380","Text":"What about the chance of event B occurring?"},{"Start":"06:06.380 ","End":"06:12.245","Text":"That\u0027s the size of B divided by the size of my sample space."},{"Start":"06:12.245 ","End":"06:18.020","Text":"The size of B is equal to 4 divided by the size of my sample space,"},{"Start":"06:18.020 ","End":"06:23.180","Text":"that\u0027s 36, and that equals to 1 divided by 9."},{"Start":"06:23.180 ","End":"06:26.345","Text":"So there we go. These are the chances."},{"Start":"06:26.345 ","End":"06:29.975","Text":"This is a chance of event B occurring,"},{"Start":"06:29.975 ","End":"06:34.589","Text":"and this is the chance of event A occurring."}],"ID":12723},{"Watched":false,"Name":"Exercise 3","Duration":"2m 59s","ChapterTopicVideoID":12240,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12240.jpeg","UploadDate":"2018-07-22T16:09:10.7330000","DurationForVideoObject":"PT2M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this question, we have a number from 0-9 that is randomly selected,"},{"Start":"00:04.170 ","End":"00:08.865","Text":"and we\u0027re asked, what\u0027s the probability that the number is larger than 5?"},{"Start":"00:08.865 ","End":"00:11.370","Text":"First of all, what\u0027s our sample space?"},{"Start":"00:11.370 ","End":"00:12.630","Text":"What\u0027s our Omega here?"},{"Start":"00:12.630 ","End":"00:15.480","Text":"Well, that\u0027s all the numbers from 0-9."},{"Start":"00:15.480 ","End":"00:17.340","Text":"That\u0027ll be 0, 1,"},{"Start":"00:17.340 ","End":"00:20.025","Text":"2, 3, 4,"},{"Start":"00:20.025 ","End":"00:22.005","Text":"5, 6,"},{"Start":"00:22.005 ","End":"00:25.425","Text":"7, 8, and 9."},{"Start":"00:25.425 ","End":"00:30.105","Text":"Now, what\u0027s the size of my sample space?"},{"Start":"00:30.105 ","End":"00:31.530","Text":"Well, do I have here?"},{"Start":"00:31.530 ","End":"00:34.140","Text":"How many numbers? I have 10 numbers here."},{"Start":"00:34.140 ","End":"00:37.035","Text":"10 would be the size of my sample space."},{"Start":"00:37.035 ","End":"00:38.610","Text":"Now in Section A, we\u0027re asked,"},{"Start":"00:38.610 ","End":"00:42.270","Text":"what\u0027s the probability that the number is larger than 5?"},{"Start":"00:42.270 ","End":"00:44.335","Text":"Let\u0027s define the event A,"},{"Start":"00:44.335 ","End":"00:47.270","Text":"which should be all the numbers that are larger than 5."},{"Start":"00:47.270 ","End":"00:50.290","Text":"That doesn\u0027t mean that it includes 5, it doesn\u0027t."},{"Start":"00:50.290 ","End":"00:53.010","Text":"A will include the number 6,"},{"Start":"00:53.010 ","End":"00:57.165","Text":"7, 8, and 9."},{"Start":"00:57.165 ","End":"00:59.455","Text":"Now, what\u0027s the size of A?"},{"Start":"00:59.455 ","End":"01:01.100","Text":"Well, how many numbers do I have here?"},{"Start":"01:01.100 ","End":"01:03.260","Text":"I have 1, 2, 3, and 4."},{"Start":"01:03.260 ","End":"01:07.670","Text":"I have 4 numbers and that\u0027ll be the size of A."},{"Start":"01:07.670 ","End":"01:13.540","Text":"Now we\u0027re asked about the probability that A will occur."},{"Start":"01:13.540 ","End":"01:17.480","Text":"Now, because the sample space is symmetric,"},{"Start":"01:17.480 ","End":"01:22.910","Text":"the chances of getting 1 number is equal to the chances of getting any other number."},{"Start":"01:22.910 ","End":"01:29.735","Text":"Well, that means that the probability that A will occur that will be 4 divided by 10."},{"Start":"01:29.735 ","End":"01:34.280","Text":"The size of the event divided by the size of the sample space."},{"Start":"01:34.280 ","End":"01:36.740","Text":"Now that equals to 0.4."},{"Start":"01:36.740 ","End":"01:42.275","Text":"Now b asks, what\u0027s the probability that the number is at most 3?"},{"Start":"01:42.275 ","End":"01:47.060","Text":"Let\u0027s define B as all the numbers that are at most 3."},{"Start":"01:47.060 ","End":"01:49.930","Text":"Well, that does include 3."},{"Start":"01:49.930 ","End":"01:53.160","Text":"The set of numbers in B would be 0,"},{"Start":"01:53.160 ","End":"01:56.050","Text":"1, 2, and 3."},{"Start":"01:56.050 ","End":"01:58.310","Text":"Now, the size of B,"},{"Start":"01:58.310 ","End":"02:00.050","Text":"well, how many numbers do I have here?"},{"Start":"02:00.050 ","End":"02:02.155","Text":"1, 2, 3, and 4."},{"Start":"02:02.155 ","End":"02:04.065","Text":"The size of B is 4."},{"Start":"02:04.065 ","End":"02:06.695","Text":"What\u0027s the chances of B occurring?"},{"Start":"02:06.695 ","End":"02:10.100","Text":"Well, again, the sample space is symmetrical."},{"Start":"02:10.100 ","End":"02:14.855","Text":"The chances of getting 1 number is the same as getting any other number."},{"Start":"02:14.855 ","End":"02:17.285","Text":"That\u0027ll be again 4 divided by 10,"},{"Start":"02:17.285 ","End":"02:19.775","Text":"that equals to 0.4."},{"Start":"02:19.775 ","End":"02:24.200","Text":"C asks, what\u0027s the probability that the number is odd?"},{"Start":"02:24.200 ","End":"02:28.595","Text":"Well, how many odd numbers do I have in my sample space?"},{"Start":"02:28.595 ","End":"02:30.915","Text":"Well, I have 1, 3,"},{"Start":"02:30.915 ","End":"02:34.440","Text":"5, 7, and 9."},{"Start":"02:34.440 ","End":"02:37.835","Text":"The size of C would be equal to what?"},{"Start":"02:37.835 ","End":"02:39.845","Text":"1, 2, 3, 4, and 5."},{"Start":"02:39.845 ","End":"02:43.070","Text":"The chances of C occurring, well,"},{"Start":"02:43.070 ","End":"02:46.340","Text":"again, the sample space is symmetrical."},{"Start":"02:46.340 ","End":"02:51.140","Text":"The chances of getting 1 number is the same as getting any other number."},{"Start":"02:51.140 ","End":"02:56.045","Text":"That\u0027ll be 5 divided by 10 and that equals to 0.5."},{"Start":"02:56.045 ","End":"03:00.270","Text":"These then are the probabilities."}],"ID":12724},{"Watched":false,"Name":"Exercise 4","Duration":"5m 14s","ChapterTopicVideoID":12241,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12241.jpeg","UploadDate":"2018-07-22T16:09:30.0370000","DurationForVideoObject":"PT5M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this question, we have a table that shows the distribution of"},{"Start":"00:02.910 ","End":"00:05.700","Text":"families according to the number of TV sets that they own."},{"Start":"00:05.700 ","End":"00:08.070","Text":"Here\u0027s the cable right here."},{"Start":"00:08.070 ","End":"00:09.495","Text":"In the first column,"},{"Start":"00:09.495 ","End":"00:11.490","Text":"we have the number of TV sets,"},{"Start":"00:11.490 ","End":"00:13.800","Text":"that\u0027s a variable, and we\u0027ll call that x."},{"Start":"00:13.800 ","End":"00:16.155","Text":"In the second column, we\u0027ll call that f,"},{"Start":"00:16.155 ","End":"00:20.820","Text":"that\u0027s the frequency of families having this number of TV set."},{"Start":"00:20.820 ","End":"00:22.350","Text":"In our example here,"},{"Start":"00:22.350 ","End":"00:24.975","Text":"22 families have no TV sets,"},{"Start":"00:24.975 ","End":"00:27.390","Text":"28 families have 1 TV set,"},{"Start":"00:27.390 ","End":"00:28.845","Text":"and so on and so forth."},{"Start":"00:28.845 ","End":"00:34.720","Text":"Now, we can see that this is not a symmetric sample space. Why is that?"},{"Start":"00:34.720 ","End":"00:39.500","Text":"Well, when a family is randomly selected,"},{"Start":"00:39.500 ","End":"00:43.220","Text":"the chances of getting a family with no TV set is not"},{"Start":"00:43.220 ","End":"00:48.065","Text":"equal to the chances of getting a family with 1 TV set,"},{"Start":"00:48.065 ","End":"00:49.895","Text":"or 2, or 3, or 4."},{"Start":"00:49.895 ","End":"00:52.730","Text":"The chances are different and as such,"},{"Start":"00:52.730 ","End":"00:55.460","Text":"this is not a symmetric sample space."},{"Start":"00:55.460 ","End":"01:00.950","Text":"Now, when calculating the probabilities of this type of sample space,"},{"Start":"01:00.950 ","End":"01:06.920","Text":"then the probability is calculated using the relative frequency or the proportion."},{"Start":"01:06.920 ","End":"01:10.385","Text":"That\u0027s defined as f divided by n,"},{"Start":"01:10.385 ","End":"01:15.425","Text":"where f is the frequency of the event that we\u0027ve defined and n"},{"Start":"01:15.425 ","End":"01:20.720","Text":"is the total number or the sum of all the frequencies."},{"Start":"01:20.720 ","End":"01:22.385","Text":"In our case,"},{"Start":"01:22.385 ","End":"01:23.930","Text":"n here, what would that be?"},{"Start":"01:23.930 ","End":"01:28.880","Text":"Well, that would be 22 plus 28 plus 18 plus 22 plus 10,"},{"Start":"01:28.880 ","End":"01:32.170","Text":"and that equals to 100."},{"Start":"01:32.170 ","End":"01:36.300","Text":"Let\u0027s start answering questions."},{"Start":"01:36.300 ","End":"01:38.735","Text":"Now, the first question is,"},{"Start":"01:38.735 ","End":"01:42.950","Text":"what\u0027s the probability that a family does not own a TV set?"},{"Start":"01:42.950 ","End":"01:46.235","Text":"We\u0027re looking for the probability."},{"Start":"01:46.235 ","End":"01:49.895","Text":"Let\u0027s say we\u0027ll define that as Event A."},{"Start":"01:49.895 ","End":"01:59.370","Text":"Now, A, the probability that a family does not own a TV set."},{"Start":"01:59.370 ","End":"02:01.585","Text":"Now, what does that mean?"},{"Start":"02:01.585 ","End":"02:03.624","Text":"That means that x,"},{"Start":"02:03.624 ","End":"02:06.055","Text":"that has to equal to 0."},{"Start":"02:06.055 ","End":"02:10.780","Text":"This is the number of TV sets in a family that a family has."},{"Start":"02:10.780 ","End":"02:16.060","Text":"If a family doesn\u0027t have any TV sets then that means that the number here is 0."},{"Start":"02:16.060 ","End":"02:18.400","Text":"Now, what\u0027s the frequency of 0?"},{"Start":"02:18.400 ","End":"02:20.880","Text":"Well, that\u0027s right here, that\u0027s 22."},{"Start":"02:20.880 ","End":"02:23.875","Text":"22 families have no TV sets."},{"Start":"02:23.875 ","End":"02:30.970","Text":"Now, we know that the probability of this event is the relative frequency,"},{"Start":"02:30.970 ","End":"02:34.450","Text":"or the frequency of getting 0 TV sets,"},{"Start":"02:34.450 ","End":"02:39.475","Text":"that\u0027s 22, divided by the total number of families, that\u0027s 100."},{"Start":"02:39.475 ","End":"02:42.285","Text":"That equals to 0.22."},{"Start":"02:42.285 ","End":"02:44.195","Text":"Now, what about B?"},{"Start":"02:44.195 ","End":"02:50.045","Text":"Well b asks, what\u0027s the probability that the family owns at least 1 TV set?"},{"Start":"02:50.045 ","End":"02:53.975","Text":"Well, if a family owns at least 1 TV set, well,"},{"Start":"02:53.975 ","End":"02:59.620","Text":"that\u0027s the complimentary set of this event right here."},{"Start":"02:59.620 ","End":"03:05.600","Text":"Why is that? Because A means that there are no TV sets."},{"Start":"03:05.600 ","End":"03:09.485","Text":"Now, the complimentary set of A is everything else."},{"Start":"03:09.485 ","End":"03:12.305","Text":"Apart from A, that means that it\u0027s here,"},{"Start":"03:12.305 ","End":"03:14.570","Text":"here, here, and here."},{"Start":"03:14.570 ","End":"03:20.365","Text":"Now, we know that the probability of A-bar,"},{"Start":"03:20.365 ","End":"03:27.670","Text":"that means owning at least 1 TV set,"},{"Start":"03:28.270 ","End":"03:34.220","Text":"that equals to 1 minus the probability of A."},{"Start":"03:34.220 ","End":"03:36.365","Text":"That\u0027s what we\u0027ve learned previously."},{"Start":"03:36.365 ","End":"03:43.220","Text":"Now, that equals to 1 minus 0.22 and that equals to 0.78."},{"Start":"03:43.220 ","End":"03:47.655","Text":"That\u0027s the probability of the complimentary set of A,"},{"Start":"03:47.655 ","End":"03:52.540","Text":"that a family owns at least 1 TV set."},{"Start":"03:52.540 ","End":"03:57.080","Text":"Now, I could have calculated this by using this formula,"},{"Start":"03:57.080 ","End":"04:05.925","Text":"by adding up all the frequencies of families that have 1 or 2 or 3 or 4 TV sets,"},{"Start":"04:05.925 ","End":"04:08.495","Text":"28 plus 18, plus 22 plus 10,"},{"Start":"04:08.495 ","End":"04:10.280","Text":"and then dividing that by 100,"},{"Start":"04:10.280 ","End":"04:12.200","Text":"but I\u0027d get the same value here."},{"Start":"04:12.200 ","End":"04:19.625","Text":"Now, c asks, what\u0027s the probability that the family owns at least 3 TV sets?"},{"Start":"04:19.625 ","End":"04:23.930","Text":"We\u0027re looking for the probability of"},{"Start":"04:23.930 ","End":"04:32.070","Text":"owning at least 3 TV sets."},{"Start":"04:32.070 ","End":"04:34.710","Text":"Now, that\u0027s equals to what?"},{"Start":"04:34.710 ","End":"04:40.065","Text":"We have to take that frequency of these events right here,"},{"Start":"04:40.065 ","End":"04:42.600","Text":"of family owning at least 3 TV sets."},{"Start":"04:42.600 ","End":"04:47.480","Text":"That means that they own either 3 TVs or 4 TVs."},{"Start":"04:47.480 ","End":"04:51.775","Text":"Now, what\u0027s the frequency of a family owning 3 TVs, that\u0027s 22."},{"Start":"04:51.775 ","End":"04:57.365","Text":"The frequency of a family owning 4 TVs, so that\u0027s 10."},{"Start":"04:57.365 ","End":"05:04.790","Text":"That\u0027ll be 22 plus 10 divided by 100 and that equals to 32 divided by 100,"},{"Start":"05:04.790 ","End":"05:07.070","Text":"and that equals to 0.32."},{"Start":"05:07.070 ","End":"05:15.120","Text":"That\u0027s the probability or the chance that a family owns at least 3 TV sets."}],"ID":12725},{"Watched":false,"Name":"Exercise 5","Duration":"4m 25s","ChapterTopicVideoID":12242,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12242.jpeg","UploadDate":"2018-07-22T16:09:44.9530000","DurationForVideoObject":"PT4M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"In this question, we have a table that shows"},{"Start":"00:02.190 ","End":"00:05.430","Text":"the distribution of families by the number of cars that they own."},{"Start":"00:05.430 ","End":"00:08.640","Text":"Now, here\u0027s the table."},{"Start":"00:08.640 ","End":"00:10.380","Text":"Here\u0027s the variable X,"},{"Start":"00:10.380 ","End":"00:11.400","Text":"the number of cars,"},{"Start":"00:11.400 ","End":"00:14.430","Text":"that\u0027s 0,1,2,3 and 4 and here we have f,"},{"Start":"00:14.430 ","End":"00:20.310","Text":"the frequency, the number of families that own this amount of cars."},{"Start":"00:20.310 ","End":"00:23.535","Text":"We have 20 families that own 0 cars,"},{"Start":"00:23.535 ","End":"00:25.680","Text":"40 families that own 1 car,"},{"Start":"00:25.680 ","End":"00:27.390","Text":"100 families that own 2 cars,"},{"Start":"00:27.390 ","End":"00:28.710","Text":"and so on and so forth."},{"Start":"00:28.710 ","End":"00:33.740","Text":"Now, since this is not a symmetric sample space,"},{"Start":"00:33.740 ","End":"00:36.230","Text":"then in order to calculate probabilities,"},{"Start":"00:36.230 ","End":"00:39.155","Text":"we\u0027re going to have to use the relative frequency."},{"Start":"00:39.155 ","End":"00:41.090","Text":"Now, the relative frequency, well,"},{"Start":"00:41.090 ","End":"00:44.840","Text":"that\u0027s defined as the frequency of the event"},{"Start":"00:44.840 ","End":"00:48.965","Text":"divided by the total number of families that I have."},{"Start":"00:48.965 ","End":"00:50.345","Text":"Now in our case,"},{"Start":"00:50.345 ","End":"00:51.861","Text":"n here, well,"},{"Start":"00:51.861 ","End":"00:54.290","Text":"that equals to the sum of these numbers right here."},{"Start":"00:54.290 ","End":"00:57.920","Text":"That will be 20 plus 40 plus 100 plus 30 plus 10,"},{"Start":"00:57.920 ","End":"01:00.870","Text":"that equals to 200."},{"Start":"01:01.910 ","End":"01:06.230","Text":"Let\u0027s start with section A, which asks,"},{"Start":"01:06.230 ","End":"01:10.400","Text":"\"What\u0027s the probability that a family does not own a car when"},{"Start":"01:10.400 ","End":"01:15.145","Text":"a family is randomly selected from this sample space?\""},{"Start":"01:15.145 ","End":"01:18.180","Text":"Well, the probability,"},{"Start":"01:18.180 ","End":"01:24.170","Text":"let\u0027s define A as the probability of a family does not own any car,"},{"Start":"01:24.170 ","End":"01:26.570","Text":"so there\u0027s no car."},{"Start":"01:26.570 ","End":"01:30.080","Text":"Now, that equals to this,"},{"Start":"01:30.080 ","End":"01:31.580","Text":"well, not owning a car,"},{"Start":"01:31.580 ","End":"01:33.440","Text":"that means that X equals to 0."},{"Start":"01:33.440 ","End":"01:35.225","Text":"That\u0027s this line right here."},{"Start":"01:35.225 ","End":"01:37.195","Text":"What\u0027s the frequency?"},{"Start":"01:37.195 ","End":"01:40.115","Text":"What\u0027s the frequency of this event right here?"},{"Start":"01:40.115 ","End":"01:41.855","Text":"That\u0027s 20 right here,"},{"Start":"01:41.855 ","End":"01:45.680","Text":"so that would be 20 divided by 200,"},{"Start":"01:45.680 ","End":"01:47.555","Text":"f divided by n,"},{"Start":"01:47.555 ","End":"01:51.755","Text":"and that equals to 0.1."},{"Start":"01:51.755 ","End":"01:53.915","Text":"Now, what about B?"},{"Start":"01:53.915 ","End":"01:58.445","Text":"What\u0027s the probability that the family owns at least 3 cars?"},{"Start":"01:58.445 ","End":"02:07.700","Text":"That\u0027s the probability of owning at least 3 cars."},{"Start":"02:07.700 ","End":"02:12.815","Text":"Well, that equals to the frequency of this event,"},{"Start":"02:12.815 ","End":"02:14.495","Text":"owning at least 3 cars."},{"Start":"02:14.495 ","End":"02:20.370","Text":"That means that I own either 3 cars or 4 cars right here."},{"Start":"02:20.370 ","End":"02:24.000","Text":"That\u0027ll be 30 plus 10."},{"Start":"02:24.000 ","End":"02:27.690","Text":"The frequency of owning 3 cars is 30 and the frequency"},{"Start":"02:27.690 ","End":"02:31.865","Text":"of owning 4 cars that\u0027s 10 divided by 200."},{"Start":"02:31.865 ","End":"02:35.240","Text":"Now that equals to 40 divided by 200,"},{"Start":"02:35.240 ","End":"02:38.870","Text":"and that equals to 0.2."},{"Start":"02:38.870 ","End":"02:41.030","Text":"Now, section C asks,"},{"Start":"02:41.030 ","End":"02:45.530","Text":"what\u0027s the probability that the family owns less than 3 cars?"},{"Start":"02:45.530 ","End":"02:53.815","Text":"We\u0027re looking for the probability of owning less than 3 cars."},{"Start":"02:53.815 ","End":"02:58.760","Text":"If we can see, we see that owning less than 3 cars,"},{"Start":"02:58.760 ","End":"03:01.865","Text":"this event is owning this,"},{"Start":"03:01.865 ","End":"03:05.230","Text":"this, and this number of cars."},{"Start":"03:05.230 ","End":"03:12.235","Text":"Now, we can see that this is a complimentary set of this thing right here."},{"Start":"03:12.235 ","End":"03:15.470","Text":"When dealing with a complimentary set,"},{"Start":"03:15.470 ","End":"03:21.845","Text":"then the probability of a complimentary set is 1 minus the probability of"},{"Start":"03:21.845 ","End":"03:31.955","Text":"owning at least 3 cars and that equals to 1 minus 0.2,"},{"Start":"03:31.955 ","End":"03:34.880","Text":"and that equals to 0.8."},{"Start":"03:34.880 ","End":"03:37.925","Text":"Now, I could have calculated this differently."},{"Start":"03:37.925 ","End":"03:43.637","Text":"I could\u0027ve said, the probability of owning less than 3 cars, well,"},{"Start":"03:43.637 ","End":"03:47.970","Text":"that\u0027s the frequency of owning 0 cars,"},{"Start":"03:47.970 ","End":"03:49.500","Text":"1 car, and 2 cars."},{"Start":"03:49.500 ","End":"03:54.230","Text":"That means that I\u0027ll add these frequencies that will be 20"},{"Start":"03:54.230 ","End":"03:59.990","Text":"plus 40 plus 100 divided by 200."},{"Start":"03:59.990 ","End":"04:03.980","Text":"Now that equals to 160 divided by 200,"},{"Start":"04:03.980 ","End":"04:06.245","Text":"and that equals to 0.8."},{"Start":"04:06.245 ","End":"04:10.700","Text":"We can see that these are the same numbers."},{"Start":"04:10.700 ","End":"04:17.660","Text":"I got the same answer here by using 2 types or 2 techniques to calculate the same event."},{"Start":"04:17.660 ","End":"04:25.350","Text":"1, by using the complementary event and the other by calculating the relative frequency."}],"ID":12726},{"Watched":false,"Name":"Exercise 6 - Parts a-d","Duration":"6m 45s","ChapterTopicVideoID":12243,"CourseChapterTopicPlaylistID":264984,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12243.jpeg","UploadDate":"2018-07-22T16:10:03.6870000","DurationForVideoObject":"PT6M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this question, I have a fair coin that has"},{"Start":"00:03.030 ","End":"00:06.420","Text":"heads on 1 side and tails on the other and is tossed 3 times."},{"Start":"00:06.420 ","End":"00:09.990","Text":"Now I\u0027m asked to list the sample space of the trial and"},{"Start":"00:09.990 ","End":"00:14.544","Text":"also I\u0027m asked whether the sample space is symmetrical or not."},{"Start":"00:14.544 ","End":"00:16.005","Text":"Let\u0027s get to work."},{"Start":"00:16.005 ","End":"00:18.300","Text":"Omega, the sample space,"},{"Start":"00:18.300 ","End":"00:21.270","Text":"that\u0027s all the combinations of heads and tails that"},{"Start":"00:21.270 ","End":"00:24.135","Text":"it can have when I toss the coin 3 times."},{"Start":"00:24.135 ","End":"00:26.100","Text":"Let\u0027s just get to work."},{"Start":"00:26.100 ","End":"00:29.370","Text":"That\u0027ll be heads, heads, heads,"},{"Start":"00:29.370 ","End":"00:31.920","Text":"you can have heads on the first toss,"},{"Start":"00:31.920 ","End":"00:33.119","Text":"heads in the second toss,"},{"Start":"00:33.119 ","End":"00:34.860","Text":"and heads on the third toss."},{"Start":"00:34.860 ","End":"00:38.040","Text":"What other combinations can I have?"},{"Start":"00:38.040 ","End":"00:40.495","Text":"Heads, heads, tails."},{"Start":"00:40.495 ","End":"00:43.160","Text":"Heads, tails, heads."},{"Start":"00:43.160 ","End":"00:45.505","Text":"Heads, tails, and tails."},{"Start":"00:45.505 ","End":"00:48.305","Text":"Then I have tails, heads, heads."},{"Start":"00:48.305 ","End":"00:50.885","Text":"Tails, heads, tails."},{"Start":"00:50.885 ","End":"00:53.360","Text":"Tails, tails, heads."},{"Start":"00:53.360 ","End":"00:56.055","Text":"Then tails, tails, and tails."},{"Start":"00:56.055 ","End":"00:59.120","Text":"These are all the possible combinations of heads and"},{"Start":"00:59.120 ","End":"01:02.570","Text":"tails that I can have when I toss a coin 3 times."},{"Start":"01:02.570 ","End":"01:04.910","Text":"Now, is this sample space symmetrical?"},{"Start":"01:04.910 ","End":"01:08.480","Text":"Well yes, it is because any combination of"},{"Start":"01:08.480 ","End":"01:14.150","Text":"3 results of heads and tails is equal to any other combination."},{"Start":"01:14.150 ","End":"01:15.230","Text":"The chances of getting,"},{"Start":"01:15.230 ","End":"01:18.005","Text":"let\u0027s say, this combination right here."},{"Start":"01:18.005 ","End":"01:21.950","Text":"Well, that\u0027s the same as getting this combination or any other combination."},{"Start":"01:21.950 ","End":"01:26.870","Text":"The sample space is symmetrical."},{"Start":"01:26.870 ","End":"01:28.585","Text":"Now, what about B?"},{"Start":"01:28.585 ","End":"01:34.670","Text":"B asks us to list the possibilities for the following events A and D,"},{"Start":"01:34.670 ","End":"01:40.940","Text":"where A is heads was thrown exactly once and D tails was thrown at least once."},{"Start":"01:40.940 ","End":"01:43.355","Text":"Let\u0027s get to work."},{"Start":"01:43.355 ","End":"01:48.800","Text":"A, that means heads was thrown exactly once."},{"Start":"01:48.800 ","End":"01:51.980","Text":"Well, that means it\u0027s not this 1, not this 1,"},{"Start":"01:51.980 ","End":"01:53.935","Text":"not this 1 here,"},{"Start":"01:53.935 ","End":"01:56.730","Text":"here, and here."},{"Start":"01:56.730 ","End":"02:03.515","Text":"Okay, so A would include heads, tails, tails."},{"Start":"02:03.515 ","End":"02:05.840","Text":"Tails, heads, tails."},{"Start":"02:05.840 ","End":"02:08.425","Text":"And tails, tails, heads."},{"Start":"02:08.425 ","End":"02:13.530","Text":"We see that we have only 3 results,"},{"Start":"02:13.530 ","End":"02:17.870","Text":"3 combinations where heads was thrown only once."},{"Start":"02:17.870 ","End":"02:24.050","Text":"Now what about D? D says the tails was thrown at least once."},{"Start":"02:24.050 ","End":"02:25.700","Text":"Well, what does that mean?"},{"Start":"02:25.700 ","End":"02:27.620","Text":"Well, let\u0027s look at our sample space."},{"Start":"02:27.620 ","End":"02:32.330","Text":"At least once means that it\u0027s either once or twice or 3 times."},{"Start":"02:32.330 ","End":"02:34.460","Text":"Here tails was thrown once,"},{"Start":"02:34.460 ","End":"02:37.445","Text":"once here tails was thrown 2 times,"},{"Start":"02:37.445 ","End":"02:39.090","Text":"once, 2 times,"},{"Start":"02:39.090 ","End":"02:40.838","Text":"2 times, and 3 times."},{"Start":"02:40.838 ","End":"02:45.965","Text":"That\u0027s the definition of tails being thrown at least once."},{"Start":"02:45.965 ","End":"02:51.500","Text":"Let\u0027s just write out these combination of 3\u0027s that\u0027ll be heads,"},{"Start":"02:51.500 ","End":"02:54.170","Text":"heads, and tails."},{"Start":"02:54.170 ","End":"02:56.630","Text":"Heads, tails, heads."},{"Start":"02:56.630 ","End":"02:59.570","Text":"Heads, tails, tails."},{"Start":"02:59.570 ","End":"03:02.195","Text":"Tails, heads, heads."},{"Start":"03:02.195 ","End":"03:04.895","Text":"Tails, heads, tails."},{"Start":"03:04.895 ","End":"03:07.460","Text":"Tails, tails, and heads."},{"Start":"03:07.460 ","End":"03:10.265","Text":"Tails, tails, and tails."},{"Start":"03:10.265 ","End":"03:16.100","Text":"Again, we have 7 possible combinations here."},{"Start":"03:16.100 ","End":"03:20.690","Text":"Now, C asks what\u0027s the complimentary event of D?"},{"Start":"03:20.690 ","End":"03:23.195","Text":"That means we\u0027re looking at D bar,"},{"Start":"03:23.195 ","End":"03:27.530","Text":"the complimentary set of D. That means that we\u0027re looking at all the results"},{"Start":"03:27.530 ","End":"03:35.090","Text":"all the combinations in the sample space that are not included in the event D. Well,"},{"Start":"03:35.090 ","End":"03:38.435","Text":"there\u0027s only 1 such combination,"},{"Start":"03:38.435 ","End":"03:42.690","Text":"and that\u0027s this combination right here."},{"Start":"03:43.100 ","End":"03:47.685","Text":"The complimentary set of D would be heads,"},{"Start":"03:47.685 ","End":"03:50.860","Text":"heads, and heads."},{"Start":"03:51.940 ","End":"03:55.310","Text":"In this section, we\u0027re asked to calculate the chances of"},{"Start":"03:55.310 ","End":"03:58.745","Text":"the events defined in section B and C. Okay?"},{"Start":"03:58.745 ","End":"04:01.280","Text":"If we recall, this was our omega,"},{"Start":"04:01.280 ","End":"04:03.350","Text":"this is our sample space right here."},{"Start":"04:03.350 ","End":"04:04.630","Text":"This is event A,"},{"Start":"04:04.630 ","End":"04:06.230","Text":"this is event D,"},{"Start":"04:06.230 ","End":"04:10.580","Text":"and this is the complimentary event D bar."},{"Start":"04:10.580 ","End":"04:13.430","Text":"Now we\u0027re asked to calculate the chances of A,"},{"Start":"04:13.430 ","End":"04:15.095","Text":"D and D bar."},{"Start":"04:15.095 ","End":"04:16.745","Text":"Let\u0027s get to work here."},{"Start":"04:16.745 ","End":"04:18.035","Text":"How do we do that?"},{"Start":"04:18.035 ","End":"04:23.000","Text":"Well, because the sample space is symmetrical,"},{"Start":"04:23.000 ","End":"04:26.900","Text":"then the probability or the chances of an event,"},{"Start":"04:26.900 ","End":"04:33.235","Text":"well that equals to the size of the event divided by the size of the sample space."},{"Start":"04:33.235 ","End":"04:37.340","Text":"First of all, what\u0027s the size of my sample space?"},{"Start":"04:37.340 ","End":"04:40.610","Text":"Well, how many combinations of 3\u0027s do I have?"},{"Start":"04:40.610 ","End":"04:43.835","Text":"Well, I have 1,2,3,4,5,6,7,8,"},{"Start":"04:43.835 ","End":"04:48.770","Text":"so the size of my sample space that equals to 8."},{"Start":"04:48.770 ","End":"04:51.965","Text":"Now, what about the size of event A?"},{"Start":"04:51.965 ","End":"04:55.010","Text":"Now that equals to 1, 2, and 3."},{"Start":"04:55.010 ","End":"04:57.264","Text":"I have 3 combination of 3\u0027s."},{"Start":"04:57.264 ","End":"04:59.290","Text":"That\u0027s the size of A."},{"Start":"04:59.290 ","End":"05:02.255","Text":"What about the size of D here?"},{"Start":"05:02.255 ","End":"05:03.860","Text":"Well, that\u0027s 1, 2,"},{"Start":"05:03.860 ","End":"05:05.540","Text":"3, 4, 5, 6,"},{"Start":"05:05.540 ","End":"05:12.395","Text":"7 and the size of D bar well that equals to 1."},{"Start":"05:12.395 ","End":"05:18.480","Text":"Okay, now we have to just note that if we add up the size of D and D bar,"},{"Start":"05:18.480 ","End":"05:20.800","Text":"that will give us the total space."},{"Start":"05:20.800 ","End":"05:26.020","Text":"The size of an event plus the size of the complimentary event,"},{"Start":"05:26.020 ","End":"05:29.060","Text":"will give us the size of the sample space."},{"Start":"05:29.060 ","End":"05:32.405","Text":"But, okay let\u0027s now calculate the chances."},{"Start":"05:32.405 ","End":"05:37.560","Text":"Well, the chances of A that equals to"},{"Start":"05:37.560 ","End":"05:42.650","Text":"the size of A divided by the size of the sample space."},{"Start":"05:42.650 ","End":"05:49.610","Text":"Now the size of A that equals to 3 and the size of the sample space that equals to 8."},{"Start":"05:49.610 ","End":"05:51.935","Text":"That\u0027s the chances right here."},{"Start":"05:51.935 ","End":"05:55.085","Text":"What\u0027s the chances of D?"},{"Start":"05:55.085 ","End":"06:02.060","Text":"Well, that equals to the size of D divided by the size of the sample space."},{"Start":"06:02.060 ","End":"06:10.790","Text":"Now that equals to 7 divided by 8 and the probability or the chances of"},{"Start":"06:10.790 ","End":"06:14.630","Text":"the complimentary event D. Well that equals to"},{"Start":"06:14.630 ","End":"06:21.170","Text":"the size of the complimentary set D divided by the size of the sample space,"},{"Start":"06:21.170 ","End":"06:23.195","Text":"and that equals to 1 divided by 8."},{"Start":"06:23.195 ","End":"06:26.840","Text":"Now, again, note that the probability of D"},{"Start":"06:26.840 ","End":"06:31.400","Text":"and plus the probability of the complimentary set of"},{"Start":"06:31.400 ","End":"06:35.120","Text":"D. That would 7/8 plus 1/8 and that equals to"},{"Start":"06:35.120 ","End":"06:40.255","Text":"1 and that\u0027s the probability of the sample space here."},{"Start":"06:40.255 ","End":"06:45.870","Text":"There we go. These are the probabilities that we\u0027re asked to calculate."}],"ID":12727}],"Thumbnail":null,"ID":264984},{"Name":"Operations Between Events","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intersecting Events","Duration":"4m 52s","ChapterTopicVideoID":12294,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12294.jpeg","UploadDate":"2019-01-31T23:13:28.6670000","DurationForVideoObject":"PT4M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.765","Text":"In this chapter I\u0027ll be talking about operations between events."},{"Start":"00:03.765 ","End":"00:07.830","Text":"That means I\u0027ll be talking about the intersection and union of events."},{"Start":"00:07.830 ","End":"00:12.600","Text":"Also, I\u0027ll be talking about mutually exclusive events and subset events."},{"Start":"00:12.600 ","End":"00:15.450","Text":"Let\u0027s get started with intersecting events."},{"Start":"00:15.450 ","End":"00:20.880","Text":"The intersection between 2 events gives what\u0027s common to both of them."},{"Start":"00:20.880 ","End":"00:27.720","Text":"The intersection of Event A and Event B will be denoted as A intersect B."},{"Start":"00:27.720 ","End":"00:30.420","Text":"This will be the sign for intersect."},{"Start":"00:30.420 ","End":"00:36.695","Text":"Now, this consists of the results that are in both A and in B."},{"Start":"00:36.695 ","End":"00:41.915","Text":"Let\u0027s take a look at an example here and we\u0027ll be using the Venn diagram."},{"Start":"00:41.915 ","End":"00:44.090","Text":"This is this diagram right here."},{"Start":"00:44.090 ","End":"00:48.560","Text":"Now, Venn diagram is a diagram that\u0027s a square,"},{"Start":"00:48.560 ","End":"00:51.020","Text":"and within that square we have circles."},{"Start":"00:51.020 ","End":"00:54.665","Text":"Now, the square denotes the whole sample space,"},{"Start":"00:54.665 ","End":"00:59.740","Text":"Omega, and each circle denotes an event."},{"Start":"00:59.740 ","End":"01:02.780","Text":"Now, when we\u0027re talking about intersecting events,"},{"Start":"01:02.780 ","End":"01:03.950","Text":"what are we talking about?"},{"Start":"01:03.950 ","End":"01:07.295","Text":"Well, let\u0027s take a look at Event A."},{"Start":"01:07.295 ","End":"01:10.610","Text":"This right here is Event A,"},{"Start":"01:10.610 ","End":"01:13.200","Text":"that\u0027s this circle right here."},{"Start":"01:13.760 ","End":"01:16.140","Text":"What about Event B?"},{"Start":"01:16.140 ","End":"01:21.570","Text":"well, Event B is this circle right here,"},{"Start":"01:21.570 ","End":"01:26.355","Text":"that\u0027s denoted in red."},{"Start":"01:26.355 ","End":"01:30.975","Text":"Now, what\u0027s the intersection between A and B?"},{"Start":"01:30.975 ","End":"01:33.365","Text":"Well, that\u0027s this thing right here,"},{"Start":"01:33.365 ","End":"01:40.260","Text":"this area right here where the blue lines intersect the red lines,"},{"Start":"01:40.260 ","End":"01:42.465","Text":"what\u0027s colored in yellow."},{"Start":"01:42.465 ","End":"01:46.624","Text":"Here, this is how we denote the intersect,"},{"Start":"01:46.624 ","End":"01:49.130","Text":"that\u0027s A intersect B."},{"Start":"01:49.130 ","End":"01:53.980","Text":"Again, let\u0027s take a look at an example when rolling a die."},{"Start":"01:53.980 ","End":"01:58.770","Text":"Let\u0027s define the Event A of getting at least a 5."},{"Start":"01:58.770 ","End":"02:05.703","Text":"That means that A has a 5 and a 6 and B is getting an even results."},{"Start":"02:05.703 ","End":"02:07.920","Text":"B has the values 2, 4, and 6,"},{"Start":"02:07.920 ","End":"02:13.065","Text":"so the intersection of Event A and Event B is 6."},{"Start":"02:13.065 ","End":"02:15.705","Text":"We\u0027re talking about A intersect B."},{"Start":"02:15.705 ","End":"02:18.135","Text":"Well that equals to 6. Why is that?"},{"Start":"02:18.135 ","End":"02:23.509","Text":"Well, we said that the intersection is what\u0027s common to both events."},{"Start":"02:23.509 ","End":"02:27.065","Text":"What\u0027s common to A which is 5 and 6,"},{"Start":"02:27.065 ","End":"02:29.780","Text":"and B, which is 2,4, and 6?"},{"Start":"02:29.780 ","End":"02:33.855","Text":"Well, that\u0027s the result, 6."},{"Start":"02:33.855 ","End":"02:37.065","Text":"That\u0027s what\u0027s common to both events right here."},{"Start":"02:37.065 ","End":"02:41.460","Text":"A intersect B, that equals to 6."},{"Start":"02:41.460 ","End":"02:46.135","Text":"Let\u0027s see how this relates to our Venn diagram here."},{"Start":"02:46.135 ","End":"02:49.350","Text":"First of all, what\u0027s your sample space?"},{"Start":"02:49.350 ","End":"02:50.655","Text":"When rolling a die,"},{"Start":"02:50.655 ","End":"02:53.850","Text":"the sample space are the results 1, 2,"},{"Start":"02:53.850 ","End":"02:55.455","Text":"3, 4,"},{"Start":"02:55.455 ","End":"02:57.120","Text":"5, and 6."},{"Start":"02:57.120 ","End":"02:59.190","Text":"Now, what\u0027s Event A?"},{"Start":"02:59.190 ","End":"03:02.340","Text":"Well, Event A is getting at least a 5."},{"Start":"03:02.340 ","End":"03:09.915","Text":"That means that Event A has within this circle the results 5 and 6."},{"Start":"03:09.915 ","End":"03:12.260","Text":"Let\u0027s just erase it from here,"},{"Start":"03:12.260 ","End":"03:15.350","Text":"from the general list and put them in Event A."},{"Start":"03:15.350 ","End":"03:17.270","Text":"Now, Event A is in the sample space,"},{"Start":"03:17.270 ","End":"03:20.210","Text":"so we didn\u0027t erase the results 5 and 6,"},{"Start":"03:20.210 ","End":"03:27.050","Text":"we just moved them from the general sample space to the Event A."},{"Start":"03:27.050 ","End":"03:29.180","Text":"Now, what about event B?"},{"Start":"03:29.180 ","End":"03:33.290","Text":"Well, Event B is getting the result and even result, 2, 4,"},{"Start":"03:33.290 ","End":"03:36.705","Text":"and 6, so event B has 2,"},{"Start":"03:36.705 ","End":"03:38.400","Text":"4, and 6."},{"Start":"03:38.400 ","End":"03:42.020","Text":"Now, let\u0027s erase what\u0027s left in the general sample space,"},{"Start":"03:42.020 ","End":"03:45.080","Text":"what we\u0027ve placed in Event B,"},{"Start":"03:45.080 ","End":"03:46.850","Text":"well that\u0027s 2, 4, and 6."},{"Start":"03:46.850 ","End":"03:49.110","Text":"Well, 6 is right here as well,"},{"Start":"03:49.110 ","End":"03:52.110","Text":"and 6 is right here, so we can erase 2 and 4."},{"Start":"03:52.110 ","End":"03:58.360","Text":"Let\u0027s just erase 2 and 4 from the general list in the sample space."},{"Start":"03:58.360 ","End":"04:02.220","Text":"Now, let\u0027s take a look at Event A and Event B."},{"Start":"04:02.220 ","End":"04:04.680","Text":"What\u0027s common to Event A and Event B?"},{"Start":"04:04.680 ","End":"04:07.880","Text":"Well, that\u0027s 6, so we\u0027ll put 6 in"},{"Start":"04:07.880 ","End":"04:12.310","Text":"the common space right here and we\u0027ll erase it from here."},{"Start":"04:12.310 ","End":"04:19.095","Text":"What happened? We see here that Event A has the numbers 5 and 6."},{"Start":"04:19.095 ","End":"04:22.645","Text":"6 is still in A."},{"Start":"04:22.645 ","End":"04:25.010","Text":"Event B has 2, 4, and 6."},{"Start":"04:25.010 ","End":"04:26.090","Text":"2, 4 here,"},{"Start":"04:26.090 ","End":"04:28.895","Text":"and 6 in the common space right here."},{"Start":"04:28.895 ","End":"04:33.238","Text":"Again, 2, 4, and 6 are in Event B."},{"Start":"04:33.238 ","End":"04:36.620","Text":"Again, the common area right here,"},{"Start":"04:36.620 ","End":"04:39.960","Text":"that\u0027s A intersect B,"},{"Start":"04:39.960 ","End":"04:42.090","Text":"that\u0027s 6 right here."},{"Start":"04:42.090 ","End":"04:44.360","Text":"This is how we use the Venn diagram."},{"Start":"04:44.360 ","End":"04:48.430","Text":"It\u0027s a very important technique and we\u0027ll use that a lot."},{"Start":"04:48.430 ","End":"04:52.549","Text":"Let\u0027s go to the next definition."}],"ID":12728},{"Watched":false,"Name":"Union of Two Events","Duration":"2m 9s","ChapterTopicVideoID":12297,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12297.jpeg","UploadDate":"2019-01-31T23:13:35.1370000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:04.110","Text":"Let\u0027s take a look now at the union of 2 events."},{"Start":"00:04.110 ","End":"00:06.345","Text":"Now, the union of 2 events consists of"},{"Start":"00:06.345 ","End":"00:09.855","Text":"all the outcomes that there are in at least 1 of the events."},{"Start":"00:09.855 ","End":"00:12.735","Text":"The union of event A and event B,"},{"Start":"00:12.735 ","End":"00:16.320","Text":"that will be denoted as A union B."},{"Start":"00:16.320 ","End":"00:18.465","Text":"That\u0027s the sign for union right here."},{"Start":"00:18.465 ","End":"00:23.745","Text":"Now, this consists of results that are A or in B, or in both."},{"Start":"00:23.745 ","End":"00:28.035","Text":"Basically what we\u0027re doing is we\u0027re combining all the results in A and"},{"Start":"00:28.035 ","End":"00:33.360","Text":"all the results in B into 1 general list of results."},{"Start":"00:33.360 ","End":"00:37.110","Text":"Now, let\u0027s just take a look at this in the Venn diagram."},{"Start":"00:37.110 ","End":"00:38.880","Text":"In the Venn diagram,"},{"Start":"00:38.880 ","End":"00:43.740","Text":"this is event A right here,"},{"Start":"00:43.740 ","End":"00:48.495","Text":"and this would be event B right here."},{"Start":"00:48.495 ","End":"00:51.500","Text":"Now, what\u0027s A union B?"},{"Start":"00:51.500 ","End":"00:55.230","Text":"Well, that\u0027s this area right here."},{"Start":"00:55.760 ","End":"01:01.685","Text":"We\u0027ve combined A and B together to form 1 big event."},{"Start":"01:01.685 ","End":"01:07.165","Text":"Now let\u0027s take a look at our example here when rolling a die."},{"Start":"01:07.165 ","End":"01:11.160","Text":"Here, we\u0027ve defined A as 5 and 6,"},{"Start":"01:11.160 ","End":"01:14.400","Text":"and B is getting an even result 2, 4, and 6."},{"Start":"01:14.400 ","End":"01:18.440","Text":"Now the union of event A and event B is 2,"},{"Start":"01:18.440 ","End":"01:20.525","Text":"4, 5, and 6. Now why is that?"},{"Start":"01:20.525 ","End":"01:22.459","Text":"We took 5 and 6,"},{"Start":"01:22.459 ","End":"01:24.905","Text":"and we\u0027ve added to it,"},{"Start":"01:24.905 ","End":"01:27.350","Text":"we\u0027ve unified it with 2, 4,"},{"Start":"01:27.350 ","End":"01:30.764","Text":"and 6 to get this oneness to 2, 4,"},{"Start":"01:30.764 ","End":"01:32.630","Text":"5 and 6, 2,4 from B,"},{"Start":"01:32.630 ","End":"01:35.465","Text":"5 from A and 6 from both of them."},{"Start":"01:35.465 ","End":"01:38.510","Text":"Now, let\u0027s take a look at this in the Venn diagram."},{"Start":"01:38.510 ","End":"01:41.075","Text":"We had the results 1 and 3,"},{"Start":"01:41.075 ","End":"01:44.640","Text":"which were outside of A and B."},{"Start":"01:44.640 ","End":"01:46.170","Text":"We had 5 here,"},{"Start":"01:46.170 ","End":"01:50.175","Text":"6 in the intersects and 2 and 4 in B."},{"Start":"01:50.175 ","End":"01:55.200","Text":"Again, the union of A and B,"},{"Start":"01:55.200 ","End":"01:57.315","Text":"that\u0027s this guy right here,"},{"Start":"01:57.315 ","End":"01:59.730","Text":"and this has the values 2, 4,"},{"Start":"01:59.730 ","End":"02:03.380","Text":"5 and 6 which is exactly this list right here."},{"Start":"02:03.380 ","End":"02:09.180","Text":"This then is the definition of the union of 2 events."}],"ID":12729},{"Watched":false,"Name":"Example - Part a","Duration":"5m 13s","ChapterTopicVideoID":12249,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12249.jpeg","UploadDate":"2018-07-22T16:13:35.6130000","DurationForVideoObject":"PT5M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Let\u0027s take a look now at how we can use"},{"Start":"00:03.150 ","End":"00:06.690","Text":"the intersection in the union of events to solve problems."},{"Start":"00:06.690 ","End":"00:08.070","Text":"Now in our example,"},{"Start":"00:08.070 ","End":"00:10.530","Text":"a student takes 2 semester exams,"},{"Start":"00:10.530 ","End":"00:12.810","Text":"1 in statistics and 1 in economics."},{"Start":"00:12.810 ","End":"00:16.995","Text":"The probability of passing the statistics exam is 0.9,"},{"Start":"00:16.995 ","End":"00:20.990","Text":"and the probability of passing the economics exam is 0.8,"},{"Start":"00:20.990 ","End":"00:25.620","Text":"and the probability of passing both exams is 0.75."},{"Start":"00:25.620 ","End":"00:32.160","Text":"We\u0027re asked, what\u0027s the probability that the student will pass only the statistics exam?"},{"Start":"00:32.160 ","End":"00:37.350","Text":"The first thing that I want to do is write down what we\u0027re given."},{"Start":"00:37.350 ","End":"00:45.560","Text":"Let\u0027s define the event A as passing stats,"},{"Start":"00:45.560 ","End":"00:54.605","Text":"and B, that\u0027ll be passing economics."},{"Start":"00:54.605 ","End":"00:57.560","Text":"The probability of A, well,"},{"Start":"00:57.560 ","End":"00:58.700","Text":"that\u0027s given to us,"},{"Start":"00:58.700 ","End":"01:03.170","Text":"the probability of passing the statistics exam, that\u0027s 0.9."},{"Start":"01:03.170 ","End":"01:05.960","Text":"So we\u0027ll write that down, 0.9."},{"Start":"01:05.960 ","End":"01:08.540","Text":"What\u0027s the probability of B?"},{"Start":"01:08.540 ","End":"01:13.910","Text":"That\u0027s the probability of passing the economics exam, that\u0027s 0.8."},{"Start":"01:14.260 ","End":"01:18.920","Text":"What about the probability of passing both exams?"},{"Start":"01:18.920 ","End":"01:20.600","Text":"Now passing both exams,"},{"Start":"01:20.600 ","End":"01:27.270","Text":"that means that we\u0027re taking A and we have A intersect B."},{"Start":"01:27.270 ","End":"01:30.345","Text":"The probability of A intersect B,"},{"Start":"01:30.345 ","End":"01:34.685","Text":"that means that that\u0027s the probability of A and B,"},{"Start":"01:34.685 ","End":"01:41.720","Text":"and A and B means passing the statistics exam and passing the economics exam."},{"Start":"01:41.720 ","End":"01:45.530","Text":"That equals to 0.75."},{"Start":"01:45.530 ","End":"01:51.275","Text":"We\u0027re asked what\u0027s the probability that the student will pass only the statistics exam?"},{"Start":"01:51.275 ","End":"01:52.880","Text":"In order to answer that,"},{"Start":"01:52.880 ","End":"01:56.980","Text":"let\u0027s take a look at our Venn diagram here."},{"Start":"01:56.980 ","End":"01:59.264","Text":"Here\u0027s our Venn diagram."},{"Start":"01:59.264 ","End":"02:02.860","Text":"Here\u0027s event A and here\u0027s event B."},{"Start":"02:02.860 ","End":"02:07.010","Text":"Now we know that the probability of event A, that\u0027s 90 percent,"},{"Start":"02:07.010 ","End":"02:13.081","Text":"that means that all the probability of this circle right here, that\u0027s 90 percent."},{"Start":"02:13.081 ","End":"02:14.690","Text":"The probability of B,"},{"Start":"02:14.690 ","End":"02:19.340","Text":"that\u0027s the probability of this circle right here, that\u0027s 0.8."},{"Start":"02:19.340 ","End":"02:24.080","Text":"The probability, the intersect that\u0027s right here, this area,"},{"Start":"02:24.080 ","End":"02:26.465","Text":"which is common to both A and B,"},{"Start":"02:26.465 ","End":"02:29.725","Text":"that equals to 0.75."},{"Start":"02:29.725 ","End":"02:32.880","Text":"Let\u0027s just write this out right here."},{"Start":"02:32.880 ","End":"02:35.865","Text":"We have 0.9 for A,"},{"Start":"02:35.865 ","End":"02:41.645","Text":"0.8 for B, and 0.75 for the intersect."},{"Start":"02:41.645 ","End":"02:43.205","Text":"What do we ask?"},{"Start":"02:43.205 ","End":"02:48.890","Text":"We\u0027re asked, what\u0027s the probability that the student will pass only the statistics exam?"},{"Start":"02:48.890 ","End":"02:54.470","Text":"Now we have to play close attention to the wordings of these questions. They\u0027re tricky."},{"Start":"02:54.470 ","End":"02:59.240","Text":"We know what the probability is to pass the statistics exam,"},{"Start":"02:59.240 ","End":"03:00.500","Text":"but we\u0027re not asked that."},{"Start":"03:00.500 ","End":"03:05.180","Text":"We\u0027re asked what\u0027s the probability that it will pass only the statistics exam?"},{"Start":"03:05.180 ","End":"03:06.919","Text":"Now in our Venn diagram,"},{"Start":"03:06.919 ","End":"03:09.905","Text":"when we say only the statistics exam,"},{"Start":"03:09.905 ","End":"03:14.610","Text":"that means we\u0027re looking at this area right here."},{"Start":"03:14.780 ","End":"03:19.760","Text":"It can\u0027t be here on the common area right here, on the intersect,"},{"Start":"03:19.760 ","End":"03:22.340","Text":"because in this intersect right here, well,"},{"Start":"03:22.340 ","End":"03:26.900","Text":"the student passed both the statistics and economics exam,"},{"Start":"03:26.900 ","End":"03:28.322","Text":"and we weren\u0027t asked that."},{"Start":"03:28.322 ","End":"03:33.490","Text":"We\u0027re asked about the probability of passing only the statistics exam."},{"Start":"03:33.490 ","End":"03:37.645","Text":"That means that we\u0027re looking at this area right here."},{"Start":"03:37.645 ","End":"03:39.410","Text":"How do we figure that out?"},{"Start":"03:39.410 ","End":"03:41.285","Text":"How do we calculate that?"},{"Start":"03:41.285 ","End":"03:50.300","Text":"Well, we know that we\u0027re looking at the event A,"},{"Start":"03:50.300 ","End":"03:57.590","Text":"but we have to take away from that, the intersect."},{"Start":"03:57.590 ","End":"04:03.360","Text":"That means that it\u0027s A and not B."},{"Start":"04:03.360 ","End":"04:07.095","Text":"We\u0027re looking at A but not at B."},{"Start":"04:07.095 ","End":"04:10.430","Text":"That means that we\u0027re looking at this area right here."},{"Start":"04:10.430 ","End":"04:12.875","Text":"That\u0027s how we define this area."},{"Start":"04:12.875 ","End":"04:15.755","Text":"It\u0027s A but not B."},{"Start":"04:15.755 ","End":"04:21.690","Text":"What\u0027s the probability of A and not B,"},{"Start":"04:21.690 ","End":"04:23.845","Text":"or A intersect not B?"},{"Start":"04:23.845 ","End":"04:26.825","Text":"Well, if we take a look at this area,"},{"Start":"04:26.825 ","End":"04:32.185","Text":"that equals to the probability of A"},{"Start":"04:32.185 ","End":"04:40.385","Text":"minus the probability of the intersect, A intersect B."},{"Start":"04:40.385 ","End":"04:43.535","Text":"We know what that is."},{"Start":"04:43.535 ","End":"04:46.820","Text":"That is equal to the following."},{"Start":"04:46.820 ","End":"04:48.080","Text":"The probability of A,"},{"Start":"04:48.080 ","End":"04:50.960","Text":"that\u0027s given to us, that\u0027s 0.9,"},{"Start":"04:50.960 ","End":"04:56.938","Text":"and the probability of the intersect that\u0027s 0.75."},{"Start":"04:56.938 ","End":"05:01.195","Text":"Now, that equals to 0.15."},{"Start":"05:01.195 ","End":"05:05.075","Text":"The probability of passing"},{"Start":"05:05.075 ","End":"05:14.350","Text":"only the statistics exam is 0.15 or 15 percent."}],"ID":12730},{"Watched":false,"Name":"Example - Part b","Duration":"4m 25s","ChapterTopicVideoID":12250,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12250.jpeg","UploadDate":"2018-07-22T16:13:47.2100000","DurationForVideoObject":"PT4M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"00:02.790 ","End":"00:05.040","Text":"that the student will pass at least 1 exam?"},{"Start":"00:05.040 ","End":"00:09.975","Text":"Well, what do we mean when we say that the student will pass at least 1 exam,"},{"Start":"00:09.975 ","End":"00:18.975","Text":"that he\u0027ll pass either the statistics exam only or he\u0027ll pass the economics exam only,"},{"Start":"00:18.975 ","End":"00:22.750","Text":"or he\u0027ll pass both exams."},{"Start":"00:22.750 ","End":"00:24.705","Text":"Now, what does that mean?"},{"Start":"00:24.705 ","End":"00:27.585","Text":"Let\u0027s take a look at our Venn diagram."},{"Start":"00:27.585 ","End":"00:31.080","Text":"If he passes only the statistics exam."},{"Start":"00:31.080 ","End":"00:34.410","Text":"Well, that\u0027s what we\u0027ve calculated in section A."},{"Start":"00:34.410 ","End":"00:37.605","Text":"That\u0027s this area right here in blue."},{"Start":"00:37.605 ","End":"00:46.535","Text":"The mirror image of that is the event of the student passing only the economics exam."},{"Start":"00:46.535 ","End":"00:49.475","Text":"That\u0027s this event right here."},{"Start":"00:49.475 ","End":"00:55.295","Text":"Now, the event of the student passing both exams."},{"Start":"00:55.295 ","End":"00:57.740","Text":"Well, that\u0027s the intersect that\u0027s right here."},{"Start":"00:57.740 ","End":"01:00.170","Text":"When we\u0027re talking about the union,"},{"Start":"01:00.170 ","End":"01:03.800","Text":"we\u0027re talking about these 3 sections right here."},{"Start":"01:03.800 ","End":"01:05.750","Text":"All these 3 sections,"},{"Start":"01:05.750 ","End":"01:11.830","Text":"that\u0027s what comprise the union of A and B."},{"Start":"01:11.830 ","End":"01:18.905","Text":"Let\u0027s now calculate the probability of A union B."},{"Start":"01:18.905 ","End":"01:24.440","Text":"Well, what\u0027s the probability of the blue section right here?"},{"Start":"01:24.440 ","End":"01:30.945","Text":"Now remember, this blue section is A intersect not B."},{"Start":"01:30.945 ","End":"01:32.670","Text":"Now the probability then,"},{"Start":"01:32.670 ","End":"01:40.115","Text":"that would be equal to the probability of A minus the probability of A intersect B."},{"Start":"01:40.115 ","End":"01:44.015","Text":"That\u0027s what we calculated in section 8."},{"Start":"01:44.015 ","End":"01:47.360","Text":"Let\u0027s now take a look at the mirror image of that."},{"Start":"01:47.360 ","End":"01:53.015","Text":"We\u0027re looking at the probability of B and not A."},{"Start":"01:53.015 ","End":"01:56.035","Text":"It\u0027s not A and B."},{"Start":"01:56.035 ","End":"01:57.530","Text":"Well, that\u0027s the mirror image."},{"Start":"01:57.530 ","End":"02:04.195","Text":"That means the probability of B minus the probability of A intersect B."},{"Start":"02:04.195 ","End":"02:07.220","Text":"The 3rd section here is the intersect."},{"Start":"02:07.220 ","End":"02:11.495","Text":"We have the probability of A intersect B."},{"Start":"02:11.495 ","End":"02:14.690","Text":"Now, when we add all the probabilities,"},{"Start":"02:14.690 ","End":"02:17.990","Text":"the probabilities of each of the 3 sections,"},{"Start":"02:17.990 ","End":"02:21.980","Text":"then we\u0027ll have the probability of the union."},{"Start":"02:21.980 ","End":"02:24.805","Text":"Let\u0027s just add these guys up right here."},{"Start":"02:24.805 ","End":"02:32.890","Text":"That\u0027ll be the probability of A minus the probability of A intersect B."},{"Start":"02:32.890 ","End":"02:35.055","Text":"That\u0027s this guy right here."},{"Start":"02:35.055 ","End":"02:37.580","Text":"Plus now this guy,"},{"Start":"02:37.580 ","End":"02:41.930","Text":"that\u0027ll be the probability of B minus the probability"},{"Start":"02:41.930 ","End":"02:46.580","Text":"of A intersect B plus this guy right here,"},{"Start":"02:46.580 ","End":"02:50.270","Text":"the probability of A intersect B."},{"Start":"02:50.270 ","End":"02:54.260","Text":"Now let\u0027s just calculate that."},{"Start":"02:54.260 ","End":"02:58.535","Text":"Well here we have the probability of A intersect B,"},{"Start":"02:58.535 ","End":"03:02.435","Text":"and here we have minus the probability of A intersect B."},{"Start":"03:02.435 ","End":"03:04.820","Text":"These 2 guys cancel each other out,"},{"Start":"03:04.820 ","End":"03:10.800","Text":"and we\u0027re left with the probability of A plus"},{"Start":"03:10.800 ","End":"03:19.895","Text":"the probability of B minus the probability of A intersect B."},{"Start":"03:19.895 ","End":"03:22.430","Text":"That equals, as we said,"},{"Start":"03:22.430 ","End":"03:25.850","Text":"to the probability of A union B."},{"Start":"03:25.850 ","End":"03:28.670","Text":"Now, this equation right here is"},{"Start":"03:28.670 ","End":"03:35.255","Text":"a very important equation and it\u0027s very worthwhile for us to learn this by heart."},{"Start":"03:35.255 ","End":"03:38.870","Text":"Once we\u0027ve calculated this,"},{"Start":"03:38.870 ","End":"03:42.655","Text":"let\u0027s just do the actual math here."},{"Start":"03:42.655 ","End":"03:48.275","Text":"The probability of A union B,"},{"Start":"03:48.275 ","End":"03:50.900","Text":"well that would be equal to the probability of A,"},{"Start":"03:50.900 ","End":"03:55.880","Text":"well that equals to 0.9 plus the probability of B."},{"Start":"03:55.880 ","End":"04:04.040","Text":"Well that\u0027s 0.8 minus the probability of A intersect B. Well that\u0027s 0.75."},{"Start":"04:04.040 ","End":"04:07.580","Text":"Now that equals to 0.95."},{"Start":"04:07.580 ","End":"04:14.375","Text":"This then, would be the probability of passing at least 1 exam,"},{"Start":"04:14.375 ","End":"04:24.810","Text":"either the statistics exam only or the economics exam only, or both exams."}],"ID":12731},{"Watched":false,"Name":"Example - Part c","Duration":"4m 9s","ChapterTopicVideoID":12251,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12251.jpeg","UploadDate":"2018-07-22T16:14:00.0700000","DurationForVideoObject":"PT4M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"00:02.040 ","End":"00:05.745","Text":"the probability that the student will fail both exams?"},{"Start":"00:05.745 ","End":"00:09.555","Text":"Let\u0027s take a look at this event, failing both exams."},{"Start":"00:09.555 ","End":"00:13.590","Text":"A is the event of passing the statistics exam."},{"Start":"00:13.590 ","End":"00:16.995","Text":"That means that not A or A bar,"},{"Start":"00:16.995 ","End":"00:18.810","Text":"that\u0027s failing stats,"},{"Start":"00:18.810 ","End":"00:21.240","Text":"and B is passing the economics exam,"},{"Start":"00:21.240 ","End":"00:23.400","Text":"so B bar or not B,"},{"Start":"00:23.400 ","End":"00:26.175","Text":"well, that\u0027s failing the economics exam."},{"Start":"00:26.175 ","End":"00:34.020","Text":"Now, failing both exams means that I have the event not A, and not B."},{"Start":"00:34.020 ","End":"00:37.275","Text":"Now, how does that look under Venn diagram?"},{"Start":"00:37.275 ","End":"00:38.895","Text":"Well, let\u0027s take a look."},{"Start":"00:38.895 ","End":"00:47.640","Text":"Not A and not B means that I am outside of event A and event B,"},{"Start":"00:47.640 ","End":"00:49.980","Text":"not A and not B."},{"Start":"00:49.980 ","End":"00:52.635","Text":"That\u0027s the area in green."},{"Start":"00:52.635 ","End":"00:57.540","Text":"Let\u0027s take a look at this area now right here."},{"Start":"00:57.540 ","End":"01:06.125","Text":"This is the total area that outside of this area we have the green lines,"},{"Start":"01:06.125 ","End":"01:08.495","Text":"or not A and not B."},{"Start":"01:08.495 ","End":"01:12.170","Text":"Well, what\u0027s the area within the black lines?"},{"Start":"01:12.170 ","End":"01:15.965","Text":"Well, this is exactly A union B."},{"Start":"01:15.965 ","End":"01:18.460","Text":"That\u0027s this area right here."},{"Start":"01:18.460 ","End":"01:21.970","Text":"This is A union B."},{"Start":"01:23.030 ","End":"01:27.755","Text":"We now know that not A and not B,"},{"Start":"01:27.755 ","End":"01:32.750","Text":"that has to equal to not a union B."},{"Start":"01:32.750 ","End":"01:37.070","Text":"Now, let\u0027s take a look at the probabilities here."},{"Start":"01:37.070 ","End":"01:42.060","Text":"The probability of not A and not B,"},{"Start":"01:42.060 ","End":"01:47.870","Text":"well that equals to the probability of not A union B."},{"Start":"01:47.870 ","End":"01:51.005","Text":"Now, what\u0027s the probability of not A union B?"},{"Start":"01:51.005 ","End":"01:57.385","Text":"Well, that equals to 1 minus the probability of A union B."},{"Start":"01:57.385 ","End":"02:01.880","Text":"Now, this is called De Morgan\u0027s law."},{"Start":"02:01.880 ","End":"02:04.415","Text":"Now let\u0027s take a look at De Morgan\u0027s law."},{"Start":"02:04.415 ","End":"02:08.675","Text":"Here we have 2 ways of looking at De Morgan\u0027s law."},{"Start":"02:08.675 ","End":"02:14.325","Text":"We have here not A and not B,"},{"Start":"02:14.325 ","End":"02:18.390","Text":"which equals to not A union B,"},{"Start":"02:18.390 ","End":"02:20.840","Text":"and this is exactly what we have here."},{"Start":"02:20.840 ","End":"02:26.288","Text":"Now, this expression right here for the events and these probabilities,"},{"Start":"02:26.288 ","End":"02:30.799","Text":"well, I\u0027ll refer to them when we do the exercise."},{"Start":"02:30.799 ","End":"02:35.100","Text":"I want to concentrate on this and this."},{"Start":"02:35.100 ","End":"02:36.825","Text":"This is the event."},{"Start":"02:36.825 ","End":"02:38.490","Text":"Now, what\u0027s the probability?"},{"Start":"02:38.490 ","End":"02:41.810","Text":"Well the probability is we\u0027ve calculated that right here."},{"Start":"02:41.810 ","End":"02:44.570","Text":"The probability of not A and not B,"},{"Start":"02:44.570 ","End":"02:48.890","Text":"that equals to 1 minus the probability of A union B."},{"Start":"02:48.890 ","End":"02:51.020","Text":"Well, let\u0027s just switch things around here."},{"Start":"02:51.020 ","End":"02:53.895","Text":"The probability of A union B,"},{"Start":"02:53.895 ","End":"02:56.990","Text":"that equals, we\u0027ll just move that to the other side,"},{"Start":"02:56.990 ","End":"02:59.405","Text":"and this will move to this side of the equation,"},{"Start":"02:59.405 ","End":"03:06.050","Text":"and that would be 1 minus the probability of not A and not B."},{"Start":"03:06.050 ","End":"03:09.125","Text":"Well, this is exactly what we have here."},{"Start":"03:09.125 ","End":"03:11.030","Text":"This is called, as I said,"},{"Start":"03:11.030 ","End":"03:12.560","Text":"De Morgan\u0027s law,"},{"Start":"03:12.560 ","End":"03:14.945","Text":"and it\u0027s a very important law,"},{"Start":"03:14.945 ","End":"03:17.770","Text":"and we need to learn this by heart."},{"Start":"03:17.770 ","End":"03:21.500","Text":"Let\u0027s now do the math here and calculate"},{"Start":"03:21.500 ","End":"03:25.984","Text":"the probability that the student will fail both exams."},{"Start":"03:25.984 ","End":"03:30.120","Text":"Well, that\u0027s the probability, as we said,"},{"Start":"03:30.120 ","End":"03:35.025","Text":"of not A and not B,"},{"Start":"03:35.025 ","End":"03:40.325","Text":"which equals to 1 minus the probability of A union B."},{"Start":"03:40.325 ","End":"03:42.290","Text":"Now, in the last section,"},{"Start":"03:42.290 ","End":"03:47.820","Text":"we\u0027ve calculated the probability of A union B, that was 0.95."},{"Start":"03:47.820 ","End":"03:56.420","Text":"1 minus 0.95 would be equal to the probability that the student will fail both exams,"},{"Start":"03:56.420 ","End":"04:01.415","Text":"and that equals to 0.05 or 5 percent."},{"Start":"04:01.415 ","End":"04:10.079","Text":"This is it. This is the answer to the question of the probability of failing both exams."}],"ID":12732},{"Watched":false,"Name":"The Magic Square","Duration":"6m 7s","ChapterTopicVideoID":12296,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12296.jpeg","UploadDate":"2019-01-31T23:28:55.8000000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.230 ","End":"00:03.600","Text":"Here, I\u0027d like to show you a technique called the"},{"Start":"00:03.600 ","End":"00:06.600","Text":"magic square or the probability matrix method."},{"Start":"00:06.600 ","End":"00:12.120","Text":"This method is useful when dealing with exactly 2 simultaneous events,"},{"Start":"00:12.120 ","End":"00:14.010","Text":"as in the preceding exercise."},{"Start":"00:14.010 ","End":"00:15.494","Text":"In the preceding exercise,"},{"Start":"00:15.494 ","End":"00:18.240","Text":"we had 2 events, A and B."},{"Start":"00:18.240 ","End":"00:20.730","Text":"A would be passing the stats exam,"},{"Start":"00:20.730 ","End":"00:22.850","Text":"and B would be passing the economics exam,"},{"Start":"00:22.850 ","End":"00:25.170","Text":"and we were given the probabilities."},{"Start":"00:25.170 ","End":"00:28.559","Text":"How do we use the probability matrix?"},{"Start":"00:28.559 ","End":"00:31.005","Text":"Here we have, as we said,"},{"Start":"00:31.005 ","End":"00:33.090","Text":"2 events, A and B."},{"Start":"00:33.090 ","End":"00:34.860","Text":"In the columns and rows,"},{"Start":"00:34.860 ","End":"00:36.000","Text":"let\u0027s say in the columns,"},{"Start":"00:36.000 ","End":"00:37.875","Text":"we have the event A,"},{"Start":"00:37.875 ","End":"00:40.000","Text":"and then the event not A."},{"Start":"00:40.000 ","End":"00:42.950","Text":"In the row, we have the event B,"},{"Start":"00:42.950 ","End":"00:46.415","Text":"and then in the next row we have not B."},{"Start":"00:46.415 ","End":"00:51.770","Text":"In the cells, that\u0027s the intersection of these events."},{"Start":"00:51.770 ","End":"00:52.970","Text":"For example here,"},{"Start":"00:52.970 ","End":"00:56.375","Text":"that\u0027ll be the intersection of A and B."},{"Start":"00:56.375 ","End":"00:59.980","Text":"That means I\u0027m looking at A intersect B."},{"Start":"00:59.980 ","End":"01:03.825","Text":"In this cell, that\u0027ll be not A intersect B."},{"Start":"01:03.825 ","End":"01:07.125","Text":"Here, that\u0027ll be A intersect not B,"},{"Start":"01:07.125 ","End":"01:10.735","Text":"and here that\u0027ll be not A intersect not B."},{"Start":"01:10.735 ","End":"01:12.890","Text":"What do I have in the margins?"},{"Start":"01:12.890 ","End":"01:18.050","Text":"In the margins, I have the sum of the probabilities in the cells,"},{"Start":"01:18.050 ","End":"01:22.795","Text":"or I have the probability of the event itself."},{"Start":"01:22.795 ","End":"01:26.720","Text":"Here for example, I have the probability of A."},{"Start":"01:26.720 ","End":"01:29.170","Text":"Here, the probability of not A."},{"Start":"01:29.170 ","End":"01:31.760","Text":"Here, that\u0027ll be the probability of B,"},{"Start":"01:31.760 ","End":"01:34.460","Text":"and here that\u0027ll be the probability of not B."},{"Start":"01:34.460 ","End":"01:43.720","Text":"Let\u0027s just fill in this table by using the data that we were given in the example."},{"Start":"01:43.720 ","End":"01:47.100","Text":"The probability of A would be 0.9."},{"Start":"01:47.100 ","End":"01:49.855","Text":"Let\u0027s just write here 0.9,"},{"Start":"01:49.855 ","End":"01:56.135","Text":"and here the probability of B, that\u0027ll be 0.8."},{"Start":"01:56.135 ","End":"01:59.555","Text":"The probability of A intersect B,"},{"Start":"01:59.555 ","End":"02:01.340","Text":"that\u0027s not in the margins."},{"Start":"02:01.340 ","End":"02:03.775","Text":"That\u0027s in the cells itself."},{"Start":"02:03.775 ","End":"02:08.750","Text":"A intersect B, that\u0027s right here, that\u0027ll be 0.75."},{"Start":"02:08.750 ","End":"02:13.730","Text":"We know that the sum of all the probabilities have to add up to 1."},{"Start":"02:13.730 ","End":"02:17.480","Text":"That means that the sum of these margins have to add up to 1."},{"Start":"02:17.480 ","End":"02:19.370","Text":"The sum of these margins,"},{"Start":"02:19.370 ","End":"02:21.020","Text":"they have to add up to 1."},{"Start":"02:21.020 ","End":"02:25.700","Text":"Then the sums of each of all the cells within the table,"},{"Start":"02:25.700 ","End":"02:27.880","Text":"they also have to add up to 1."},{"Start":"02:27.880 ","End":"02:31.775","Text":"We can calculate the rest of the cells here."},{"Start":"02:31.775 ","End":"02:36.800","Text":"Let\u0027s just use another color because we want to"},{"Start":"02:36.800 ","End":"02:38.930","Text":"differentiate between the data that was"},{"Start":"02:38.930 ","End":"02:42.095","Text":"given to us and the data that we have to calculate."},{"Start":"02:42.095 ","End":"02:47.810","Text":"Here, we were given the probability of B, that\u0027s 0.80."},{"Start":"02:47.810 ","End":"02:51.300","Text":"0.8 plus what equals to 1?"},{"Start":"02:51.300 ","End":"02:52.995","Text":"That\u0027s 0.2."},{"Start":"02:52.995 ","End":"02:55.860","Text":"0.2 plus 0.8, that equals to 1."},{"Start":"02:55.860 ","End":"02:57.889","Text":"Here, in the margins,"},{"Start":"02:57.889 ","End":"03:01.265","Text":"0.9 plus what equals to 1?"},{"Start":"03:01.265 ","End":"03:03.655","Text":"That\u0027s 0.1."},{"Start":"03:03.655 ","End":"03:05.750","Text":"What do we have here?"},{"Start":"03:05.750 ","End":"03:07.160","Text":"Let\u0027s calculate this."},{"Start":"03:07.160 ","End":"03:13.655","Text":"0.75 plus what would give me 0.9? That\u0027s 0.15."},{"Start":"03:13.655 ","End":"03:18.270","Text":"0.75 plus what will give me 0.8?"},{"Start":"03:18.270 ","End":"03:24.750","Text":"That\u0027s 0.05, and 0.05 plus what gives me 0.1?"},{"Start":"03:24.750 ","End":"03:26.475","Text":"That\u0027s 0.05."},{"Start":"03:26.475 ","End":"03:34.700","Text":"We can check ourselves by adding this row right here, 0.15 plus 0.05."},{"Start":"03:34.700 ","End":"03:37.685","Text":"That will give me 0.2."},{"Start":"03:37.685 ","End":"03:43.430","Text":"If we recall in section A, in the example,"},{"Start":"03:43.430 ","End":"03:52.595","Text":"I wanted to know what was the probability of A and not B."},{"Start":"03:52.595 ","End":"03:54.770","Text":"In the previous section,"},{"Start":"03:54.770 ","End":"03:56.980","Text":"I had to go through the Venn diagram,"},{"Start":"03:56.980 ","End":"04:01.400","Text":"I had to color in the areas and then calculate the probability."},{"Start":"04:01.400 ","End":"04:03.470","Text":"But here, I don\u0027t have to do that."},{"Start":"04:03.470 ","End":"04:08.850","Text":"All I have to do is go to the cell with the intersection of A and not B."},{"Start":"04:08.850 ","End":"04:11.900","Text":"Here\u0027s not B in this row right here,"},{"Start":"04:11.900 ","End":"04:13.945","Text":"and this is A right here."},{"Start":"04:13.945 ","End":"04:16.740","Text":"The probability of A and not B,"},{"Start":"04:16.740 ","End":"04:18.720","Text":"that\u0027s this cell right here,"},{"Start":"04:18.720 ","End":"04:21.090","Text":"and that equals to 0.15."},{"Start":"04:21.090 ","End":"04:23.925","Text":"In section B above,"},{"Start":"04:23.925 ","End":"04:28.600","Text":"we were asked about the probability of A union B."},{"Start":"04:28.600 ","End":"04:32.430","Text":"A union B, the operator union,"},{"Start":"04:32.430 ","End":"04:35.330","Text":"that\u0027s not mentioned here because we\u0027re looking at"},{"Start":"04:35.330 ","End":"04:38.150","Text":"a table and the table deals with intersections."},{"Start":"04:38.150 ","End":"04:42.780","Text":"But we know that the probability of the union,"},{"Start":"04:42.780 ","End":"04:47.180","Text":"that equals to the probability of A plus the probability of"},{"Start":"04:47.180 ","End":"04:51.920","Text":"B minus the probability of A intersect B."},{"Start":"04:51.920 ","End":"04:53.900","Text":"That equals to the following."},{"Start":"04:53.900 ","End":"04:55.250","Text":"The probability of A,"},{"Start":"04:55.250 ","End":"04:59.105","Text":"that\u0027s 0.9, that\u0027s right here,"},{"Start":"04:59.105 ","End":"05:01.370","Text":"plus the probability of B,"},{"Start":"05:01.370 ","End":"05:03.995","Text":"that\u0027s 0.8, that\u0027s right here,"},{"Start":"05:03.995 ","End":"05:06.950","Text":"minus the probability of the intersect,"},{"Start":"05:06.950 ","End":"05:10.725","Text":"A and B, that\u0027s right here, 0.75."},{"Start":"05:10.725 ","End":"05:13.280","Text":"That equals to 0.95."},{"Start":"05:13.280 ","End":"05:18.215","Text":"I could also use DeMorgan\u0027s rule."},{"Start":"05:18.215 ","End":"05:20.540","Text":"The probability of A union B,"},{"Start":"05:20.540 ","End":"05:28.845","Text":"that equals to 1 minus the probability of not A intersect not B."},{"Start":"05:28.845 ","End":"05:30.990","Text":"That equals to 1 minus,"},{"Start":"05:30.990 ","End":"05:34.415","Text":"what\u0027s the probability of not A intersect not B?"},{"Start":"05:34.415 ","End":"05:36.670","Text":"That\u0027s right here."},{"Start":"05:36.670 ","End":"05:39.480","Text":"Here\u0027s not A and here\u0027s not B,"},{"Start":"05:39.480 ","End":"05:43.970","Text":"the intersect would be here and that equals to 0.05,"},{"Start":"05:43.970 ","End":"05:49.190","Text":"and that equals to 0.95."},{"Start":"05:49.190 ","End":"05:56.615","Text":"I\u0027ve calculated the probability of the union of A and B in 2 different ways."},{"Start":"05:56.615 ","End":"06:01.910","Text":"There we go. This is a very useful method of calculating"},{"Start":"06:01.910 ","End":"06:07.570","Text":"the probabilities when we have 2 simultaneous events."}],"ID":12733},{"Watched":false,"Name":"Mutually Exclusive","Duration":"3m 56s","ChapterTopicVideoID":12295,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12295.jpeg","UploadDate":"2019-01-31T23:28:37.5130000","DurationForVideoObject":"PT3M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"The next thing that I want to talk about are mutually exclusive events."},{"Start":"00:04.290 ","End":"00:08.520","Text":"These events have nothing in common whatsoever;"},{"Start":"00:08.520 ","End":"00:10.725","Text":"they can\u0027t occur simultaneously."},{"Start":"00:10.725 ","End":"00:12.465","Text":"Let\u0027s look at the Venn diagrams."},{"Start":"00:12.465 ","End":"00:16.480","Text":"You see here that we have 2 events in the sample space,"},{"Start":"00:16.480 ","End":"00:19.825","Text":"and there is no intersection between them."},{"Start":"00:19.825 ","End":"00:22.985","Text":"Let\u0027s just take a look at an example here."},{"Start":"00:22.985 ","End":"00:24.440","Text":"When rolling a die,"},{"Start":"00:24.440 ","End":"00:27.950","Text":"let\u0027s define event A as getting at least a 5."},{"Start":"00:27.950 ","End":"00:32.025","Text":"That means that A will have the results 5 and 6,"},{"Start":"00:32.025 ","End":"00:34.810","Text":"and B would be defined as getting a result of 3."},{"Start":"00:34.810 ","End":"00:37.910","Text":"B, the set of results,"},{"Start":"00:37.910 ","End":"00:39.830","Text":"it would be just 1 number, 3."},{"Start":"00:39.830 ","End":"00:42.335","Text":"What\u0027s common between A and B?"},{"Start":"00:42.335 ","End":"00:43.985","Text":"Well, there\u0027s nothing in common."},{"Start":"00:43.985 ","End":"00:47.060","Text":"That means that A intersect B,"},{"Start":"00:47.060 ","End":"00:48.410","Text":"well, that\u0027s an empty set."},{"Start":"00:48.410 ","End":"00:52.820","Text":"We can also write an empty set as the Greek letter Phi."},{"Start":"00:52.820 ","End":"00:56.750","Text":"As we said, A intersect B, that\u0027s an empty set."},{"Start":"00:56.750 ","End":"01:01.070","Text":"The probability of A intersect B, that equals to 0."},{"Start":"01:01.070 ","End":"01:05.165","Text":"What about the probability of the union?"},{"Start":"01:05.165 ","End":"01:09.110","Text":"Well, we know that the probability of A union B, well,"},{"Start":"01:09.110 ","End":"01:12.690","Text":"that equals to the probability of A plus"},{"Start":"01:12.690 ","End":"01:17.945","Text":"the probability of B minus the probability of A intersect B."},{"Start":"01:17.945 ","End":"01:22.730","Text":"Well, we know now that there\u0027s no intersect between A and B,"},{"Start":"01:22.730 ","End":"01:25.625","Text":"so the probability would be equal to 0."},{"Start":"01:25.625 ","End":"01:31.730","Text":"That means that that equals to the probability of A plus the probability of B,"},{"Start":"01:31.730 ","End":"01:34.025","Text":"only that\u0027s the probability of the union."},{"Start":"01:34.025 ","End":"01:36.829","Text":"This is exactly what\u0027s written right here."},{"Start":"01:36.829 ","End":"01:40.680","Text":"What about the next term?"},{"Start":"01:40.680 ","End":"01:44.350","Text":"The next term, our subset events."},{"Start":"01:44.350 ","End":"01:48.160","Text":"Let\u0027s take a look at the Venn diagram."},{"Start":"01:48.160 ","End":"01:49.855","Text":"If we have an event A,"},{"Start":"01:49.855 ","End":"01:54.715","Text":"then B is totally encased in event A,"},{"Start":"01:54.715 ","End":"01:57.655","Text":"that means that all the results of B,"},{"Start":"01:57.655 ","End":"02:00.614","Text":"we can see them in A as well."},{"Start":"02:00.614 ","End":"02:02.955","Text":"If B is a subset of A,"},{"Start":"02:02.955 ","End":"02:06.028","Text":"then event A contains event B,"},{"Start":"02:06.028 ","End":"02:10.240","Text":"that means that all the results in event B are also contained in event A."},{"Start":"02:10.240 ","End":"02:13.580","Text":"This connection is denoted like this;"},{"Start":"02:13.580 ","End":"02:17.340","Text":"if B is contained in A, this isn\u0027t the C,"},{"Start":"02:17.340 ","End":"02:22.074","Text":"this is the mathematical symbol for contained,"},{"Start":"02:22.074 ","End":"02:24.900","Text":"then we have this,"},{"Start":"02:24.900 ","End":"02:26.550","Text":"A intersect B,"},{"Start":"02:26.550 ","End":"02:28.275","Text":"well, that equals to B."},{"Start":"02:28.275 ","End":"02:31.155","Text":"What\u0027s common between A and B?"},{"Start":"02:31.155 ","End":"02:33.755","Text":"Well, B is common to both events,"},{"Start":"02:33.755 ","End":"02:36.260","Text":"so A intersect B would be B."},{"Start":"02:36.260 ","End":"02:41.780","Text":"The probability of A intersect B would be the probability of B itself."},{"Start":"02:41.780 ","End":"02:44.315","Text":"What about A union B?"},{"Start":"02:44.315 ","End":"02:46.505","Text":"Well, if we take"},{"Start":"02:46.505 ","End":"02:51.710","Text":"all the results in A and all the results of B and we put them in the same list,"},{"Start":"02:51.710 ","End":"02:54.380","Text":"then we\u0027ll get the results of A."},{"Start":"02:54.380 ","End":"02:56.840","Text":"That means that A union B,"},{"Start":"02:56.840 ","End":"02:58.010","Text":"well, that equals to A,"},{"Start":"02:58.010 ","End":"03:00.355","Text":"so the probability of A union B,"},{"Start":"03:00.355 ","End":"03:02.690","Text":"well, that would be equal to the probability of A."},{"Start":"03:02.690 ","End":"03:04.670","Text":"Let\u0027s take a look at an example."},{"Start":"03:04.670 ","End":"03:10.800","Text":"If, again, we\u0027re rolling a die and A will have the results 2,"},{"Start":"03:10.800 ","End":"03:12.560","Text":"4, 6, that\u0027s the even results,"},{"Start":"03:12.560 ","End":"03:16.605","Text":"and B would be 2 and 4 only,"},{"Start":"03:16.605 ","End":"03:21.015","Text":"then we see that the results 2,"},{"Start":"03:21.015 ","End":"03:24.285","Text":"4 are contained within 2,"},{"Start":"03:24.285 ","End":"03:27.765","Text":"4, 6; B is contained within A."},{"Start":"03:27.765 ","End":"03:29.580","Text":"There we go."},{"Start":"03:29.580 ","End":"03:34.430","Text":"These are the last 2 terms that I\u0027ll be talking about."},{"Start":"03:34.430 ","End":"03:42.245","Text":"The secret to succeeding in this long chapter is to do all the exercises,"},{"Start":"03:42.245 ","End":"03:43.820","Text":"and to do them by yourself."},{"Start":"03:43.820 ","End":"03:48.500","Text":"If you get stuck, then you can come back and see how I solve them."},{"Start":"03:48.500 ","End":"03:51.230","Text":"But, first do it by yourself,"},{"Start":"03:51.230 ","End":"03:52.925","Text":"see where you\u0027re at."},{"Start":"03:52.925 ","End":"03:55.440","Text":"Good luck."}],"ID":12734},{"Watched":false,"Name":"Exercise 1","Duration":"4m 32s","ChapterTopicVideoID":12252,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12252.jpeg","UploadDate":"2018-07-22T16:14:13.6430000","DurationForVideoObject":"PT4M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.380 ","End":"00:04.350","Text":"A 2 letter word is formed from the letters E, F,"},{"Start":"00:04.350 ","End":"00:09.060","Text":"and G. The word doesn\u0027t necessarily have to have any meaning."},{"Start":"00:09.060 ","End":"00:11.160","Text":"Now, we define the following events,"},{"Start":"00:11.160 ","End":"00:14.145","Text":"A is the letter E is in the word,"},{"Start":"00:14.145 ","End":"00:16.680","Text":"and B, the 2 letters in the word are different."},{"Start":"00:16.680 ","End":"00:20.070","Text":"We\u0027re asked in Section a to list"},{"Start":"00:20.070 ","End":"00:23.895","Text":"all the possible results in the intersection of A with B,"},{"Start":"00:23.895 ","End":"00:25.550","Text":"and in section b,"},{"Start":"00:25.550 ","End":"00:30.435","Text":"to list all the possible results in the union of A with B."},{"Start":"00:30.435 ","End":"00:37.120","Text":"The first thing that I want to do is to write my sample space, that will be Omega."},{"Start":"00:37.120 ","End":"00:45.635","Text":"Now, that\u0027ll be defined as all the possible words that I can create using these letters,"},{"Start":"00:45.635 ","End":"00:48.770","Text":"where the words have only 2 letters in them."},{"Start":"00:48.770 ","End":"00:53.060","Text":"Let\u0027s begin. That\u0027ll be EE, EF,"},{"Start":"00:53.060 ","End":"00:57.810","Text":"EG, then FE,"},{"Start":"00:57.810 ","End":"01:02.788","Text":"FF, FG,"},{"Start":"01:02.788 ","End":"01:11.760","Text":"then GE, GF, and GG."},{"Start":"01:11.760 ","End":"01:17.920","Text":"Again, these are all the 2 letter words that I can create using these 3 letters."},{"Start":"01:17.920 ","End":"01:20.075","Text":"Now, what about A?"},{"Start":"01:20.075 ","End":"01:27.910","Text":"Well, A is the set of all the words that have the letter E in them."},{"Start":"01:27.910 ","End":"01:30.940","Text":"Let\u0027s go to the sample space and see where we can"},{"Start":"01:30.940 ","End":"01:34.880","Text":"find words with the letter E. That\u0027ll be EE,"},{"Start":"01:34.880 ","End":"01:39.630","Text":"right here, EF, and EG."},{"Start":"01:39.630 ","End":"01:42.480","Text":"Now, FE,"},{"Start":"01:42.480 ","End":"01:45.075","Text":"FF, no, FG, no."},{"Start":"01:45.075 ","End":"01:49.560","Text":"Then GE, GF and GG, no."},{"Start":"01:49.560 ","End":"01:53.495","Text":"There we go. These are all the words with the letter E in them."},{"Start":"01:53.495 ","End":"01:55.510","Text":"Then what about event B?"},{"Start":"01:55.510 ","End":"02:01.335","Text":"Event B are words where the 2 letters in the word are different."},{"Start":"02:01.335 ","End":"02:04.980","Text":"Again, let\u0027s go to the sample space, EE, well,"},{"Start":"02:04.980 ","End":"02:10.335","Text":"that\u0027s no, EF yes, EG yes,"},{"Start":"02:10.335 ","End":"02:14.040","Text":"FE yes, FF no,"},{"Start":"02:14.040 ","End":"02:18.585","Text":"FG yes, GE yes,"},{"Start":"02:18.585 ","End":"02:22.200","Text":"GF yes and GG no."},{"Start":"02:22.200 ","End":"02:27.215","Text":"These are all the words where the 2 letters in the word are different."},{"Start":"02:27.215 ","End":"02:34.670","Text":"Excellent. Now we have a definition of our sample space of event A and event B."},{"Start":"02:34.670 ","End":"02:38.330","Text":"Now we can go ahead and answer a and b."},{"Start":"02:38.330 ","End":"02:45.620","Text":"In a, we\u0027re asked to list all the possible results in the intersection of A with B."},{"Start":"02:45.620 ","End":"02:47.389","Text":"We\u0027re looking for the intersection,"},{"Start":"02:47.389 ","End":"02:50.920","Text":"we\u0027re looking for A intersect B."},{"Start":"02:50.920 ","End":"02:52.895","Text":"Now, what would that be?"},{"Start":"02:52.895 ","End":"02:57.815","Text":"Well, what\u0027s common between A and B?"},{"Start":"02:57.815 ","End":"03:00.230","Text":"Well, EE?"},{"Start":"03:00.230 ","End":"03:01.710","Text":"No. What about EF?"},{"Start":"03:01.710 ","End":"03:05.055","Text":"Yes. EF is right here."},{"Start":"03:05.055 ","End":"03:09.010","Text":"EG, yes."},{"Start":"03:09.890 ","End":"03:13.500","Text":"FE, yes."},{"Start":"03:13.500 ","End":"03:15.480","Text":"FG, GE, no."},{"Start":"03:15.480 ","End":"03:17.650","Text":"GE, yes."},{"Start":"03:18.500 ","End":"03:20.904","Text":"GF, no."},{"Start":"03:20.904 ","End":"03:25.540","Text":"Here we go. These are all the words that are in common,"},{"Start":"03:25.540 ","End":"03:28.115","Text":"in event A and event B."},{"Start":"03:28.115 ","End":"03:30.710","Text":"Now, what about Section b?"},{"Start":"03:30.710 ","End":"03:35.960","Text":"It asks us to list all the possible results of the union of A with B."},{"Start":"03:35.960 ","End":"03:38.890","Text":"We\u0027re looking at A union B."},{"Start":"03:38.890 ","End":"03:42.485","Text":"Well, that\u0027ll be equal to the following, but first of all,"},{"Start":"03:42.485 ","End":"03:48.815","Text":"let\u0027s write out all the other values or other words in event A."},{"Start":"03:48.815 ","End":"03:51.210","Text":"That\u0027ll be EE,"},{"Start":"03:52.310 ","End":"03:56.970","Text":"EF, EG,"},{"Start":"03:56.970 ","End":"03:59.850","Text":"FE, and GE."},{"Start":"03:59.850 ","End":"04:02.155","Text":"All we did is we just copied A."},{"Start":"04:02.155 ","End":"04:04.855","Text":"Now let\u0027s take a look at the words in B."},{"Start":"04:04.855 ","End":"04:06.345","Text":"We have here, EF."},{"Start":"04:06.345 ","End":"04:07.520","Text":"Well, we have EF here,"},{"Start":"04:07.520 ","End":"04:09.680","Text":"so we don\u0027t need to write it again."},{"Start":"04:09.680 ","End":"04:12.380","Text":"EG, well, we have EG here again,"},{"Start":"04:12.380 ","End":"04:14.280","Text":"so we don\u0027t need to write it out."},{"Start":"04:14.280 ","End":"04:15.744","Text":"FE, we have that."},{"Start":"04:15.744 ","End":"04:17.990","Text":"FG, well, we don\u0027t have that,"},{"Start":"04:17.990 ","End":"04:20.240","Text":"so let\u0027s just add that on."},{"Start":"04:20.240 ","End":"04:23.346","Text":"GE, we have that and GF,"},{"Start":"04:23.346 ","End":"04:24.480","Text":"well, we don\u0027t have that,"},{"Start":"04:24.480 ","End":"04:26.400","Text":"so GF that\u0027ll be here."},{"Start":"04:26.400 ","End":"04:31.450","Text":"This then would be the union between A and B."}],"ID":12735},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"2m 19s","ChapterTopicVideoID":12253,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12253.jpeg","UploadDate":"2018-07-22T16:14:19.8330000","DurationForVideoObject":"PT2M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"A student takes 2 semester exams;"},{"Start":"00:03.075 ","End":"00:05.610","Text":"1 in economics and 1 in statistics."},{"Start":"00:05.610 ","End":"00:08.760","Text":"We define the following events: A is passing"},{"Start":"00:08.760 ","End":"00:12.450","Text":"the statistics exam and B is passing the economics exam."},{"Start":"00:12.450 ","End":"00:16.275","Text":"Using only the intersection of events,"},{"Start":"00:16.275 ","End":"00:17.790","Text":"union of events,"},{"Start":"00:17.790 ","End":"00:19.590","Text":"and complimentary events,"},{"Start":"00:19.590 ","End":"00:22.170","Text":"we\u0027re asked to define the following events and"},{"Start":"00:22.170 ","End":"00:25.214","Text":"mark the appropriate area on the Venn diagram."},{"Start":"00:25.214 ","End":"00:28.635","Text":"Before we get to that let\u0027s first recall."},{"Start":"00:28.635 ","End":"00:34.449","Text":"This sign right here, that\u0027s the intersection."},{"Start":"00:34.450 ","End":"00:36.980","Text":"What does that mean?"},{"Start":"00:36.980 ","End":"00:41.555","Text":"That means that both A and B have to occur at the same time."},{"Start":"00:41.555 ","End":"00:46.850","Text":"This is the union, and what does that mean?"},{"Start":"00:46.850 ","End":"00:51.755","Text":"This means that at least 1 of the events have to occur."},{"Start":"00:51.755 ","End":"00:53.270","Text":"If we have an event,"},{"Start":"00:53.270 ","End":"00:57.475","Text":"let\u0027s say S,"},{"Start":"00:57.475 ","End":"01:00.330","Text":"this is the complimentary set of S,"},{"Start":"01:00.330 ","End":"01:07.100","Text":"that means that we take everything that\u0027s not S. What are we asked to do?"},{"Start":"01:07.100 ","End":"01:14.900","Text":"We\u0027re asked to write the events that the student passed only the economics exam."},{"Start":"01:14.900 ","End":"01:18.310","Text":"That means that he passed the economics exam B,"},{"Start":"01:18.310 ","End":"01:23.385","Text":"but he only passed the economics exam he did not pass the statistics exam,"},{"Start":"01:23.385 ","End":"01:27.645","Text":"so it has to be B and not A."},{"Start":"01:27.645 ","End":"01:32.375","Text":"This sign, the intersection sign,"},{"Start":"01:32.375 ","End":"01:35.030","Text":"the union sign, and also the complimentary sign,"},{"Start":"01:35.030 ","End":"01:37.340","Text":"they have commutative property."},{"Start":"01:37.340 ","End":"01:40.670","Text":"That means that B and not A,"},{"Start":"01:40.670 ","End":"01:43.620","Text":"that equals to not A and B,"},{"Start":"01:43.620 ","End":"01:45.035","Text":"we can switch them around."},{"Start":"01:45.035 ","End":"01:49.175","Text":"How does this look on the Venn diagram?"},{"Start":"01:49.175 ","End":"01:52.190","Text":"We have to take B,"},{"Start":"01:52.190 ","End":"01:57.200","Text":"that\u0027s this circle right here, and not A."},{"Start":"01:57.200 ","End":"02:01.455","Text":"Let\u0027s just mark this like this."},{"Start":"02:01.455 ","End":"02:07.070","Text":"We have B but we don\u0027t take any portion of A right here."},{"Start":"02:07.070 ","End":"02:11.880","Text":"This is the portion of"},{"Start":"02:11.880 ","End":"02:18.480","Text":"B that says that the student passed only the economics exam."}],"ID":12736},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"40s","ChapterTopicVideoID":12254,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12254.jpeg","UploadDate":"2018-07-22T16:14:21.1770000","DurationForVideoObject":"PT40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this section, we\u0027re asked to write the event"},{"Start":"00:02.640 ","End":"00:05.730","Text":"the student passed only the Statistics exam."},{"Start":"00:05.730 ","End":"00:09.105","Text":"Well, this is a mirror image of what we had in section A."},{"Start":"00:09.105 ","End":"00:11.940","Text":"A is passing the Statistics exam,"},{"Start":"00:11.940 ","End":"00:16.110","Text":"so it\u0027s A and not B."},{"Start":"00:16.110 ","End":"00:20.340","Text":"If the student passed only the Statistics exam, that\u0027s A,"},{"Start":"00:20.340 ","End":"00:25.260","Text":"and he did not pass the Economics exam, that\u0027s not B."},{"Start":"00:25.260 ","End":"00:31.230","Text":"This will look like this on the Venn diagram."},{"Start":"00:31.230 ","End":"00:35.205","Text":"We\u0027re interested in this area right here,"},{"Start":"00:35.205 ","End":"00:39.700","Text":"all the area in A which is not B."}],"ID":12737},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"37s","ChapterTopicVideoID":12255,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12255.jpeg","UploadDate":"2018-07-22T16:14:22.5230000","DurationForVideoObject":"PT37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"This event is the student passed both exams,"},{"Start":"00:04.140 ","End":"00:09.220","Text":"so A is passing the Statistics exam,"},{"Start":"00:09.220 ","End":"00:12.720","Text":"and we want the student to pass also"},{"Start":"00:12.720 ","End":"00:17.790","Text":"the Economics exam so it\u0027ll be A and B or A intersect B."},{"Start":"00:17.790 ","End":"00:22.890","Text":"That\u0027s the definition of the intersect when both events happen simultaneously."},{"Start":"00:22.890 ","End":"00:25.845","Text":"Now how would that look under Venn diagram?"},{"Start":"00:25.845 ","End":"00:28.425","Text":"Well, it\u0027s this thing right here."},{"Start":"00:28.425 ","End":"00:31.890","Text":"It\u0027s this area that intersect between A and B,"},{"Start":"00:31.890 ","End":"00:36.490","Text":"so it\u0027ll be this area right here."}],"ID":12738},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"47s","ChapterTopicVideoID":12256,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12256.jpeg","UploadDate":"2018-07-22T16:14:26.0230000","DurationForVideoObject":"PT47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"In this event, we have the student passed at least 1 exam."},{"Start":"00:05.700 ","End":"00:10.350","Text":"He could pass either A or B,"},{"Start":"00:10.350 ","End":"00:12.600","Text":"or he can pass both of them."},{"Start":"00:12.600 ","End":"00:14.895","Text":"That\u0027s union."},{"Start":"00:14.895 ","End":"00:16.440","Text":"A union B."},{"Start":"00:16.440 ","End":"00:21.900","Text":"That\u0027s the definition of union where at least 1 of the events occur."},{"Start":"00:21.900 ","End":"00:24.585","Text":"Now, how would that look on the Venn diagram?"},{"Start":"00:24.585 ","End":"00:34.390","Text":"Well, he\u0027ll pass either B or A,"},{"Start":"00:34.390 ","End":"00:35.705","Text":"or the union,"},{"Start":"00:35.705 ","End":"00:37.745","Text":"or both A and B."},{"Start":"00:37.745 ","End":"00:46.230","Text":"This then would be the area on the Venn diagram that we would mark."}],"ID":12739},{"Watched":false,"Name":"Exercise 2 - Part e","Duration":"1m 29s","ChapterTopicVideoID":12257,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12257.jpeg","UploadDate":"2018-07-22T16:14:29.2570000","DurationForVideoObject":"PT1M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"Here we have the event that the student failed both exams."},{"Start":"00:05.085 ","End":"00:08.150","Text":"A means that he passed statistics,"},{"Start":"00:08.150 ","End":"00:10.530","Text":"so failing statistics is not A."},{"Start":"00:10.530 ","End":"00:13.084","Text":"B is passing economics,"},{"Start":"00:13.084 ","End":"00:15.430","Text":"so not B is failing economics."},{"Start":"00:15.430 ","End":"00:17.870","Text":"Now, the student has to fail both exams,"},{"Start":"00:17.870 ","End":"00:21.995","Text":"so it has to be not A and not B."},{"Start":"00:21.995 ","End":"00:25.685","Text":"Both these things have to happen simultaneously."},{"Start":"00:25.685 ","End":"00:28.280","Text":"Now, how does that look on the Venn diagram?"},{"Start":"00:28.280 ","End":"00:33.225","Text":"Well, that\u0027s the area outside both of the circles."},{"Start":"00:33.225 ","End":"00:37.925","Text":"The area inside the circles mean that the student passed,"},{"Start":"00:37.925 ","End":"00:41.960","Text":"either 1 or the other, or both exams."},{"Start":"00:41.960 ","End":"00:45.050","Text":"But here we\u0027re failing both exams."},{"Start":"00:45.050 ","End":"00:49.800","Text":"The area outside both circles,"},{"Start":"00:49.800 ","End":"00:52.435","Text":"that\u0027s not A and not B."},{"Start":"00:52.435 ","End":"00:57.010","Text":"Now again, let\u0027s take a look at this area right here."},{"Start":"00:57.010 ","End":"01:01.985","Text":"We\u0027re taking a look at the area outside both of the circles."},{"Start":"01:01.985 ","End":"01:05.785","Text":"But what\u0027s the area inside?"},{"Start":"01:05.785 ","End":"01:09.455","Text":"What\u0027s the area of both of the circles right here?"},{"Start":"01:09.455 ","End":"01:12.200","Text":"Well, that\u0027s A union B."},{"Start":"01:12.200 ","End":"01:15.380","Text":"But again, we\u0027re not looking at A union B,"},{"Start":"01:15.380 ","End":"01:21.050","Text":"we\u0027re looking at the complimentary set of A union B or not A union B."},{"Start":"01:21.050 ","End":"01:24.650","Text":"Here I\u0027ve shown that not A and not B,"},{"Start":"01:24.650 ","End":"01:29.280","Text":"well, that equals to not A union B."}],"ID":12740},{"Watched":false,"Name":"Exercise 2 - Part f","Duration":"51s","ChapterTopicVideoID":12258,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12258.jpeg","UploadDate":"2018-07-22T16:14:31.0070000","DurationForVideoObject":"PT51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"In this event, the student failed the economics exam."},{"Start":"00:03.855 ","End":"00:08.280","Text":"B is defined as passing the economics exam."},{"Start":"00:08.280 ","End":"00:11.250","Text":"Failing the economics exam is not B."},{"Start":"00:11.250 ","End":"00:13.870","Text":"That\u0027s the complimentary set of B."},{"Start":"00:13.870 ","End":"00:16.790","Text":"Now, how would that look on the Venn diagram?"},{"Start":"00:16.790 ","End":"00:20.640","Text":"Well, if this represents the event B,"},{"Start":"00:20.640 ","End":"00:28.070","Text":"well then we want not B everything outside of the circle right here."},{"Start":"00:28.070 ","End":"00:30.890","Text":"Now, it really doesn\u0027t matter what happens to A,"},{"Start":"00:30.890 ","End":"00:33.080","Text":"we didn\u0027t see anything about A."},{"Start":"00:33.080 ","End":"00:35.345","Text":"The student could pass or fail."},{"Start":"00:35.345 ","End":"00:37.790","Text":"The statistics examined really doesn\u0027t matter."},{"Start":"00:37.790 ","End":"00:42.079","Text":"The definition of this event is that the student failed economics."},{"Start":"00:42.079 ","End":"00:46.760","Text":"This then would be the area on the Venn diagram."},{"Start":"00:46.760 ","End":"00:50.790","Text":"That\u0027s all the area outside of the circle B."}],"ID":12741},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"2m 21s","ChapterTopicVideoID":12259,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12259.jpeg","UploadDate":"2018-07-22T16:14:37.2630000","DurationForVideoObject":"PT2M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this question, you\u0027re asked to randomly select a digit from 0-9."},{"Start":"00:04.800 ","End":"00:09.390","Text":"Now, we define A as the selection of an even number and B,"},{"Start":"00:09.390 ","End":"00:11.455","Text":"the selection of a number less than 5,"},{"Start":"00:11.455 ","End":"00:14.610","Text":"and we\u0027re asked to list the results of the following events."},{"Start":"00:14.610 ","End":"00:16.005","Text":"Now before we do that,"},{"Start":"00:16.005 ","End":"00:17.730","Text":"what\u0027s my sample space?"},{"Start":"00:17.730 ","End":"00:22.480","Text":"What\u0027s my Omega? What are all the possible results that I can have?"},{"Start":"00:22.480 ","End":"00:25.190","Text":"Well, that\u0027s all the digits from 0-9,"},{"Start":"00:25.190 ","End":"00:26.210","Text":"so let\u0027s write that out."},{"Start":"00:26.210 ","End":"00:30.773","Text":"That\u0027ll be 0, 1, 2,"},{"Start":"00:30.773 ","End":"00:33.018","Text":"3, 4, 5, 6,"},{"Start":"00:33.018 ","End":"00:36.255","Text":"7, 8 and 9."},{"Start":"00:36.255 ","End":"00:37.890","Text":"What\u0027s A?"},{"Start":"00:37.890 ","End":"00:43.268","Text":"A are the results of getting an even number."},{"Start":"00:43.268 ","End":"00:46.190","Text":"The number that\u0027s divisible by 2,"},{"Start":"00:46.190 ","End":"00:49.426","Text":"will then be 0,"},{"Start":"00:49.426 ","End":"00:52.275","Text":"2, 4, 6 and 8. What about B?"},{"Start":"00:52.275 ","End":"00:54.705","Text":"Selection of a number less than 5,"},{"Start":"00:54.705 ","End":"00:56.199","Text":"that will be 0,"},{"Start":"00:56.199 ","End":"01:00.200","Text":"1, 2, 3 and 4, not 5."},{"Start":"01:00.200 ","End":"01:03.290","Text":"We didn\u0027t say less than or equal to 5,"},{"Start":"01:03.290 ","End":"01:04.720","Text":"we said less than 5."},{"Start":"01:04.720 ","End":"01:06.315","Text":"What about not B?"},{"Start":"01:06.315 ","End":"01:11.840","Text":"Well, not B are other results in the sample space that are not in B."},{"Start":"01:11.840 ","End":"01:14.266","Text":"Well, that\u0027s 5, 6,"},{"Start":"01:14.266 ","End":"01:17.400","Text":"7, 8 and 9."},{"Start":"01:17.400 ","End":"01:19.725","Text":"What about A intersect B?"},{"Start":"01:19.725 ","End":"01:26.085","Text":"Well, that\u0027s the results that are common to both A and B."},{"Start":"01:26.085 ","End":"01:29.130","Text":"Well, 0 is common, so that\u0027s in,"},{"Start":"01:29.130 ","End":"01:31.410","Text":"1 is not common,"},{"Start":"01:31.410 ","End":"01:33.600","Text":"so that\u0027s out, 2 is common,"},{"Start":"01:33.600 ","End":"01:37.500","Text":"3 is not and 4, 6,"},{"Start":"01:37.500 ","End":"01:40.775","Text":"and 8 aren\u0027t in B, so A intersect B,"},{"Start":"01:40.775 ","End":"01:42.440","Text":"that\u0027ll be 0, 2, and 4."},{"Start":"01:42.440 ","End":"01:44.240","Text":"What about A union B?"},{"Start":"01:44.240 ","End":"01:49.620","Text":"Well, that\u0027s all the results that are in both A and B."},{"Start":"01:49.620 ","End":"01:54.080","Text":"Let\u0027s first of all write all the results in A that\u0027s 0,"},{"Start":"01:54.080 ","End":"01:55.750","Text":"2, 4, 6,"},{"Start":"01:55.750 ","End":"02:01.857","Text":"and 8 and let\u0027s now write all the results in B that weren\u0027t already in A,"},{"Start":"02:01.857 ","End":"02:03.900","Text":"so 0 is already here,"},{"Start":"02:03.900 ","End":"02:05.310","Text":"I have that, 1 I don\u0027t,"},{"Start":"02:05.310 ","End":"02:07.785","Text":"so let\u0027s add 1 here, 2 I have,"},{"Start":"02:07.785 ","End":"02:09.600","Text":"3 I don\u0027t have,"},{"Start":"02:09.600 ","End":"02:11.265","Text":"4 I have,"},{"Start":"02:11.265 ","End":"02:19.080","Text":"all and then 6 and 8 so A union B would be this result right here."}],"ID":12742},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"2m 45s","ChapterTopicVideoID":12260,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12260.jpeg","UploadDate":"2018-07-22T16:14:44.1070000","DurationForVideoObject":"PT2M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this section, we\u0027re asked to calculate"},{"Start":"00:02.040 ","End":"00:05.310","Text":"the probabilities of all the events in the preceding section."},{"Start":"00:05.310 ","End":"00:08.055","Text":"Here, the events from the preceding section."},{"Start":"00:08.055 ","End":"00:12.060","Text":"Now, if we recall calculating the probability of,"},{"Start":"00:12.060 ","End":"00:14.100","Text":"let\u0027s say, an event S, well,"},{"Start":"00:14.100 ","End":"00:20.190","Text":"that equals to the size of S divided by the size of the sample space."},{"Start":"00:20.190 ","End":"00:23.610","Text":"The first thing that we need to do is to calculate the size"},{"Start":"00:23.610 ","End":"00:27.240","Text":"of each 1 of the events. Let\u0027s do that."},{"Start":"00:27.240 ","End":"00:29.710","Text":"The size of A,"},{"Start":"00:30.260 ","End":"00:34.574","Text":"that equals to 1, 2, 3, 4 and 5."},{"Start":"00:34.574 ","End":"00:36.840","Text":"The size of B,"},{"Start":"00:36.840 ","End":"00:38.580","Text":"that equals to 1, 2,"},{"Start":"00:38.580 ","End":"00:40.515","Text":"3, 4, and 5 again."},{"Start":"00:40.515 ","End":"00:42.555","Text":"The size of B,"},{"Start":"00:42.555 ","End":"00:44.385","Text":"the complimentary set of B,"},{"Start":"00:44.385 ","End":"00:45.780","Text":"that\u0027s 1, 2, 3,"},{"Start":"00:45.780 ","End":"00:47.535","Text":"4, and 5 as well."},{"Start":"00:47.535 ","End":"00:51.150","Text":"The size of A intersect B,"},{"Start":"00:51.150 ","End":"00:53.790","Text":"that equals to 1, 2, and 3."},{"Start":"00:53.790 ","End":"00:57.690","Text":"The size of A union B,"},{"Start":"00:57.690 ","End":"00:59.460","Text":"that equals to 1, 2,"},{"Start":"00:59.460 ","End":"01:02.115","Text":"3, 4, 5, 6, 7."},{"Start":"01:02.115 ","End":"01:05.235","Text":"The size of Omega,"},{"Start":"01:05.235 ","End":"01:07.710","Text":"that equals to 1, 2, 3, 4,"},{"Start":"01:07.710 ","End":"01:10.365","Text":"5, 6, 7, 8, 9, 10."},{"Start":"01:10.365 ","End":"01:14.525","Text":"Now that we\u0027ve calculated the sizes of all the events here,"},{"Start":"01:14.525 ","End":"01:16.910","Text":"let\u0027s calculate the probabilities."},{"Start":"01:16.910 ","End":"01:19.415","Text":"The probability of A,"},{"Start":"01:19.415 ","End":"01:24.425","Text":"that equals to the size of A divided by the size of Omega."},{"Start":"01:24.425 ","End":"01:31.970","Text":"Now, that equals to 5 divided by 10, or 0.5."},{"Start":"01:31.970 ","End":"01:35.140","Text":"What about the probability of B?"},{"Start":"01:35.140 ","End":"01:40.573","Text":"That equals to the size of B divided by the size of Omega."},{"Start":"01:40.573 ","End":"01:44.090","Text":"The size of B is 5 divided by 10."},{"Start":"01:44.090 ","End":"01:47.045","Text":"The size of Omega, that\u0027s also 0.5."},{"Start":"01:47.045 ","End":"01:52.500","Text":"What about the probability of not B?"},{"Start":"01:52.500 ","End":"01:58.370","Text":"That\u0027s the size of not B divided by the size of Omega."},{"Start":"01:58.370 ","End":"02:02.195","Text":"Now, that equals to 5 divided by 10,"},{"Start":"02:02.195 ","End":"02:04.805","Text":"and that equals to 0.5."},{"Start":"02:04.805 ","End":"02:11.670","Text":"What about the probability of A intersect B?"},{"Start":"02:11.670 ","End":"02:18.255","Text":"That\u0027s the size of A intersect B divided by the size of Omega,"},{"Start":"02:18.255 ","End":"02:21.200","Text":"and the size of A intersect B,"},{"Start":"02:21.200 ","End":"02:25.610","Text":"that equals to 3 divided by 10, that\u0027s 0.3."},{"Start":"02:25.610 ","End":"02:30.535","Text":"The probability of A union B,"},{"Start":"02:30.535 ","End":"02:34.310","Text":"that\u0027s the size of A union B,"},{"Start":"02:34.310 ","End":"02:37.070","Text":"divided by the size of Omega."},{"Start":"02:37.070 ","End":"02:39.700","Text":"The size of A union B,"},{"Start":"02:39.700 ","End":"02:41.630","Text":"that\u0027s 7 divided by 10,"},{"Start":"02:41.630 ","End":"02:45.000","Text":"and that equals to 0.7."}],"ID":12743},{"Watched":false,"Name":"Exercise 4","Duration":"2m 34s","ChapterTopicVideoID":12261,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12261.jpeg","UploadDate":"2018-07-22T16:14:50.5000000","DurationForVideoObject":"PT2M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"Let Omega denote the sample space and Phi denote the empty set."},{"Start":"00:04.635 ","End":"00:07.755","Text":"Let A be an event in the sample space."},{"Start":"00:07.755 ","End":"00:11.715","Text":"Now, we\u0027re asked to find the solution for each event right here."},{"Start":"00:11.715 ","End":"00:14.100","Text":"I\u0027m also giving you the answers."},{"Start":"00:14.100 ","End":"00:19.365","Text":"The answer to these events are either Omega, Phi or A."},{"Start":"00:19.365 ","End":"00:22.380","Text":"How do we calculate these events?"},{"Start":"00:22.380 ","End":"00:24.795","Text":"Well, we\u0027ll use the Venn diagram."},{"Start":"00:24.795 ","End":"00:27.690","Text":"Here we have the sample space,"},{"Start":"00:27.690 ","End":"00:31.500","Text":"that\u0027s the rectangle and the circle would be event A."},{"Start":"00:31.500 ","End":"00:38.135","Text":"In section a, we\u0027re asked to calculate the complimentary set of A."},{"Start":"00:38.135 ","End":"00:40.115","Text":"That\u0027s A double bar."},{"Start":"00:40.115 ","End":"00:42.085","Text":"Let\u0027s take a look at this."},{"Start":"00:42.085 ","End":"00:44.630","Text":"We have A. The complimentary set of A,"},{"Start":"00:44.630 ","End":"00:46.430","Text":"that\u0027s everything outside of A."},{"Start":"00:46.430 ","End":"00:53.270","Text":"The complimentary set of that would be everything inside of the circle again."},{"Start":"00:53.270 ","End":"00:56.520","Text":"A double bar would be A."},{"Start":"00:56.520 ","End":"00:59.775","Text":"Here we have A intersect Phi."},{"Start":"00:59.775 ","End":"01:06.630","Text":"A intersect with nothing but what\u0027s common to A and nothing?"},{"Start":"01:06.630 ","End":"01:08.220","Text":"Nothing is common."},{"Start":"01:08.220 ","End":"01:12.555","Text":"What about A union nothing? The empty set."},{"Start":"01:12.555 ","End":"01:18.320","Text":"Well, we\u0027re unifying or combining all the results of A and other results in nothing."},{"Start":"01:18.320 ","End":"01:22.145","Text":"That\u0027s A. What about A intersect Omega?"},{"Start":"01:22.145 ","End":"01:27.220","Text":"What\u0027s common to all the sample space and to A?"},{"Start":"01:27.220 ","End":"01:30.615","Text":"A is common to both these events."},{"Start":"01:30.615 ","End":"01:32.660","Text":"What about A union omega?"},{"Start":"01:32.660 ","End":"01:35.870","Text":"We\u0027re taking all the other results in Omega and"},{"Start":"01:35.870 ","End":"01:40.115","Text":"the sample space and we unifying it with the results in A."},{"Start":"01:40.115 ","End":"01:41.935","Text":"We get Omega."},{"Start":"01:41.935 ","End":"01:46.215","Text":"Now, what about A intersect not A."},{"Start":"01:46.215 ","End":"01:48.565","Text":"We\u0027re taking all the results in A,"},{"Start":"01:48.565 ","End":"01:52.010","Text":"and we\u0027re seeing what\u0027s common between those results and"},{"Start":"01:52.010 ","End":"01:56.105","Text":"the results that are not in the complimentary set of A."},{"Start":"01:56.105 ","End":"02:00.340","Text":"Nothing is common to A and not A."},{"Start":"02:00.340 ","End":"02:03.980","Text":"What about the complimentary set of Phi?"},{"Start":"02:03.980 ","End":"02:05.900","Text":"If Phi is nothing,"},{"Start":"02:05.900 ","End":"02:11.250","Text":"it\u0027s an empty set with the complimentary set of nothing would be everything."},{"Start":"02:11.250 ","End":"02:13.890","Text":"Everything is Omega."},{"Start":"02:13.890 ","End":"02:18.140","Text":"The last event, A union not A."},{"Start":"02:18.140 ","End":"02:23.810","Text":"A is the circle and not A is everything outside the circle."},{"Start":"02:23.810 ","End":"02:25.655","Text":"A union not A,"},{"Start":"02:25.655 ","End":"02:30.475","Text":"that\u0027s the unified results of A and not A."},{"Start":"02:30.475 ","End":"02:33.310","Text":"That\u0027s Omega, that\u0027s everything."}],"ID":12744},{"Watched":false,"Name":"Exercise 5","Duration":"3m 36s","ChapterTopicVideoID":12262,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12262.jpeg","UploadDate":"2018-07-22T16:14:59.9070000","DurationForVideoObject":"PT3M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"In this question, we have the following events;"},{"Start":"00:03.645 ","End":"00:07.170","Text":"A is a person\u0027s height which is greater than 1.7 meters,"},{"Start":"00:07.170 ","End":"00:09.750","Text":"and B is a person\u0027s height which is less than"},{"Start":"00:09.750 ","End":"00:13.155","Text":"1.8 meters and we\u0027re asked to calculate the following events."},{"Start":"00:13.155 ","End":"00:15.090","Text":"But before we can do that,"},{"Start":"00:15.090 ","End":"00:20.240","Text":"let\u0027s just draw event A and B on an axis."},{"Start":"00:20.240 ","End":"00:23.210","Text":"Here we have the height axis right here,"},{"Start":"00:23.210 ","End":"00:25.190","Text":"here we have 1.7 meters,"},{"Start":"00:25.190 ","End":"00:28.730","Text":"and here we have 1.8 meters and A is defined as"},{"Start":"00:28.730 ","End":"00:32.300","Text":"a person\u0027s height which is greater than 1.7 meters,"},{"Start":"00:32.300 ","End":"00:34.975","Text":"so that\u0027s this line right here."},{"Start":"00:34.975 ","End":"00:37.790","Text":"1.7 meters and greater,"},{"Start":"00:37.790 ","End":"00:39.920","Text":"now, it doesn\u0027t include 1.7 meters,"},{"Start":"00:39.920 ","End":"00:45.690","Text":"so that\u0027s why we have an open dot here and this line would be B."},{"Start":"00:45.690 ","End":"00:48.920","Text":"This is a person\u0027s height which is less than 1.8 meters."},{"Start":"00:48.920 ","End":"00:51.065","Text":"It doesn\u0027t include 1.8 meters,"},{"Start":"00:51.065 ","End":"00:53.000","Text":"that\u0027s why we have an open dot here."},{"Start":"00:53.000 ","End":"00:56.170","Text":"Now, what\u0027s A intersect B?"},{"Start":"00:56.170 ","End":"01:01.805","Text":"A intersect B, are all the heights which are common to both A and B."},{"Start":"01:01.805 ","End":"01:05.420","Text":"Well, that\u0027s this section right here."},{"Start":"01:05.420 ","End":"01:08.740","Text":"These are the heights which are common to both A and B."},{"Start":"01:08.740 ","End":"01:16.169","Text":"That means that the height is between 1.7 and 1.8 meters."},{"Start":"01:16.169 ","End":"01:18.365","Text":"Now what about A union B?"},{"Start":"01:18.365 ","End":"01:23.085","Text":"A union B says that I take all the possible heights of A"},{"Start":"01:23.085 ","End":"01:28.400","Text":"and I unify that with all the possible heights in B and when I do that,"},{"Start":"01:28.400 ","End":"01:33.840","Text":"I have all the possible heights."},{"Start":"01:35.750 ","End":"01:40.530","Text":"What about not A intersect B?"},{"Start":"01:40.530 ","End":"01:43.860","Text":"Well, here is not A."},{"Start":"01:43.860 ","End":"01:45.900","Text":"Now, what\u0027s A?"},{"Start":"01:45.900 ","End":"01:49.590","Text":"A person\u0027s height which is greater than 1.7 meters,"},{"Start":"01:49.590 ","End":"01:55.955","Text":"so not A is a person\u0027s height which is less than or equal to 1.7 meters."},{"Start":"01:55.955 ","End":"01:58.770","Text":"That\u0027s why this is a full circle right here,"},{"Start":"01:58.770 ","End":"02:02.505","Text":"1.7 is included in not A."},{"Start":"02:02.505 ","End":"02:09.835","Text":"So what\u0027s common between not A and B?"},{"Start":"02:09.835 ","End":"02:17.600","Text":"Well, that\u0027s all the heights which are less than or equal to 1.7 meters."},{"Start":"02:17.600 ","End":"02:20.300","Text":"Well, that\u0027s exactly not A."},{"Start":"02:20.300 ","End":"02:24.575","Text":"The height here would be less than or equal to 1.7 meters."},{"Start":"02:24.575 ","End":"02:28.595","Text":"Now what about not A union not B?"},{"Start":"02:28.595 ","End":"02:31.255","Text":"Well, let\u0027s draw not B."},{"Start":"02:31.255 ","End":"02:34.530","Text":"So here we have not B."},{"Start":"02:34.530 ","End":"02:39.570","Text":"Again, B is a person\u0027s height which is less than 1.8 meters."},{"Start":"02:39.570 ","End":"02:45.740","Text":"So not B is a person\u0027s height which is greater or equal to 1.8 meters."},{"Start":"02:45.740 ","End":"02:47.840","Text":"That\u0027s why we have a full circle here."},{"Start":"02:47.840 ","End":"02:50.930","Text":"So not A union not B."},{"Start":"02:50.930 ","End":"02:58.280","Text":"Well, that\u0027s all the heights which are less than or equal to 1.7 meters,"},{"Start":"02:58.280 ","End":"03:03.230","Text":"and all the heights which are greater than 1.8 meters."},{"Start":"03:03.230 ","End":"03:06.950","Text":"So H would be less than or equal to 1.7"},{"Start":"03:06.950 ","End":"03:13.295","Text":"meters and H would be greater or equal to 1.8 meters."},{"Start":"03:13.295 ","End":"03:15.875","Text":"Now, what about not not A?"},{"Start":"03:15.875 ","End":"03:19.560","Text":"Well, not not A that equals to A."},{"Start":"03:19.560 ","End":"03:27.210","Text":"That means that the height is greater than 1.7 meters."},{"Start":"03:27.210 ","End":"03:32.580","Text":"That\u0027s the definition of A right here."},{"Start":"03:32.580 ","End":"03:36.070","Text":"These are the answers."}],"ID":12745},{"Watched":false,"Name":"Exercise 6","Duration":"3m 38s","ChapterTopicVideoID":12263,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12263.jpeg","UploadDate":"2018-07-22T16:15:10.8230000","DurationForVideoObject":"PT3M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this question, we define the following events: A,"},{"Start":"00:03.750 ","End":"00:05.790","Text":"a person speaking Chinese, B,"},{"Start":"00:05.790 ","End":"00:07.350","Text":"a person speaking French and C,"},{"Start":"00:07.350 ","End":"00:09.525","Text":"a person speaking English."},{"Start":"00:09.525 ","End":"00:13.110","Text":"We\u0027re asked to write the following events using intersection, union,"},{"Start":"00:13.110 ","End":"00:15.015","Text":"and complimentary operators."},{"Start":"00:15.015 ","End":"00:16.920","Text":"Let\u0027s get started."},{"Start":"00:16.920 ","End":"00:20.580","Text":"The first section, a person speaks all 3 languages."},{"Start":"00:20.580 ","End":"00:27.420","Text":"Well, all 3 languages means that he speaks A and a person speaks B and he"},{"Start":"00:27.420 ","End":"00:30.860","Text":"speaks C. That means that we\u0027re looking at"},{"Start":"00:30.860 ","End":"00:34.490","Text":"the intersection of all 3 languages right here,"},{"Start":"00:34.490 ","End":"00:38.170","Text":"all 3 events, and that\u0027s this area right here."},{"Start":"00:38.170 ","End":"00:42.995","Text":"The next event is that a person speaks only Chinese."},{"Start":"00:42.995 ","End":"00:46.070","Text":"That means that a person speaks only event A,"},{"Start":"00:46.070 ","End":"00:47.810","Text":"that only event A occurs."},{"Start":"00:47.810 ","End":"00:55.035","Text":"That means we\u0027re looking at A and not B and not C. Now,"},{"Start":"00:55.035 ","End":"00:58.590","Text":"how would that look on the Venn diagram?"},{"Start":"00:58.590 ","End":"01:06.430","Text":"Well, it\u0027ll be A and not B and not C. It\u0027s all the area in A,"},{"Start":"01:10.040 ","End":"01:16.790","Text":"which is not intersected with B or C. That\u0027s this area right here."},{"Start":"01:16.790 ","End":"01:21.620","Text":"The next event is that a person speaks at least 1 language."},{"Start":"01:21.620 ","End":"01:28.730","Text":"That means that a person can speak 1 language or 2 languages, or 3 languages."},{"Start":"01:28.730 ","End":"01:32.270","Text":"That means that we\u0027re looking at the union of the events."},{"Start":"01:32.270 ","End":"01:37.385","Text":"That means that it\u0027s A or B or C. Now,"},{"Start":"01:37.385 ","End":"01:40.805","Text":"that\u0027ll be this area right here."},{"Start":"01:40.805 ","End":"01:47.190","Text":"All the area of the 3 circles combined."},{"Start":"01:47.320 ","End":"01:51.830","Text":"The next event is that a person does not speak English."},{"Start":"01:51.830 ","End":"01:53.675","Text":"Well, speaking English is C,"},{"Start":"01:53.675 ","End":"01:59.300","Text":"so not speaking English is just not C. That means that we\u0027re looking"},{"Start":"01:59.300 ","End":"02:05.390","Text":"at all the area which is outside of the circle,"},{"Start":"02:05.390 ","End":"02:13.320","Text":"which is C. This would be all this area in the sample space right here."},{"Start":"02:13.690 ","End":"02:19.415","Text":"The last event is that the person speaks exactly 2 languages."},{"Start":"02:19.415 ","End":"02:22.639","Text":"Now, how would that look on the Venn diagram?"},{"Start":"02:22.639 ","End":"02:26.525","Text":"Well, that means that we\u0027re looking at this area right here,"},{"Start":"02:26.525 ","End":"02:29.615","Text":"a person speaks Chinese and French."},{"Start":"02:29.615 ","End":"02:31.475","Text":"That\u0027s event A and B,"},{"Start":"02:31.475 ","End":"02:37.490","Text":"or that a person speaks Chinese and English,"},{"Start":"02:37.490 ","End":"02:40.880","Text":"or that a person speaks French and English."},{"Start":"02:40.880 ","End":"02:44.585","Text":"Now, how can we write this event right here?"},{"Start":"02:44.585 ","End":"02:46.715","Text":"Well, there are many ways to do this,"},{"Start":"02:46.715 ","End":"02:48.530","Text":"but let\u0027s get started."},{"Start":"02:48.530 ","End":"02:54.290","Text":"This area right here would be A"},{"Start":"02:54.290 ","End":"03:02.600","Text":"intersect B intersect not C, union."},{"Start":"03:02.600 ","End":"03:05.330","Text":"Now let\u0027s take a look at this area right here."},{"Start":"03:05.330 ","End":"03:08.930","Text":"That\u0027ll be A intersect C,"},{"Start":"03:08.930 ","End":"03:13.145","Text":"intersect not B union."},{"Start":"03:13.145 ","End":"03:15.185","Text":"Well, let\u0027s take a look at this."},{"Start":"03:15.185 ","End":"03:21.860","Text":"That\u0027s B intersect C, intersect not A."},{"Start":"03:21.860 ","End":"03:29.480","Text":"This is the event where a person speaks exactly 2 language."},{"Start":"03:29.480 ","End":"03:31.700","Text":"We didn\u0027t say which 2, therefore,"},{"Start":"03:31.700 ","End":"03:37.230","Text":"we have a union of 3 events here."}],"ID":12746},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"2m 42s","ChapterTopicVideoID":12264,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12264.jpeg","UploadDate":"2018-07-22T16:15:18.5000000","DurationForVideoObject":"PT2M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this question, we have 2 political parties that are running for office,"},{"Start":"00:03.840 ","End":"00:05.490","Text":"the Blues and the Greys."},{"Start":"00:05.490 ","End":"00:10.170","Text":"Now, the probability of the Blue Party passing the voting threshold is 0.08 and"},{"Start":"00:10.170 ","End":"00:15.735","Text":"the probability of the Grey Party passing the voting threshold is 0.2."},{"Start":"00:15.735 ","End":"00:20.120","Text":"The chances that neither will pass the threshold is 76 percent."},{"Start":"00:20.120 ","End":"00:25.965","Text":"We\u0027re asked, what\u0027s the probability of at least 1 of the parties passing the threshold?"},{"Start":"00:25.965 ","End":"00:28.020","Text":"Before we can answer that,"},{"Start":"00:28.020 ","End":"00:30.090","Text":"let\u0027s just define our events."},{"Start":"00:30.090 ","End":"00:36.210","Text":"Let\u0027s define event A as the Blue Party passing"},{"Start":"00:36.210 ","End":"00:45.379","Text":"the voting threshold and B will be the Grey Party passing the threshold."},{"Start":"00:45.379 ","End":"00:49.093","Text":"The probability of A, well,"},{"Start":"00:49.093 ","End":"00:51.350","Text":"that equals to 0.08,"},{"Start":"00:51.350 ","End":"00:53.255","Text":"that\u0027s given to us right here."},{"Start":"00:53.255 ","End":"00:55.540","Text":"The probability of B,"},{"Start":"00:55.540 ","End":"00:57.680","Text":"that equals to 0.2,"},{"Start":"00:57.680 ","End":"00:59.420","Text":"that\u0027s given to us right here."},{"Start":"00:59.420 ","End":"01:04.250","Text":"Now we\u0027re also given that the chances that neither will pass the threshold is 76 percent."},{"Start":"01:04.250 ","End":"01:08.090","Text":"Well, the event that neither will pass the threshold, well,"},{"Start":"01:08.090 ","End":"01:11.790","Text":"that\u0027s not A intersect not B,"},{"Start":"01:11.790 ","End":"01:17.780","Text":"so the probability of this event will equal to 0.76."},{"Start":"01:17.780 ","End":"01:20.870","Text":"Now that we\u0027ve written all the information down,"},{"Start":"01:20.870 ","End":"01:22.250","Text":"let\u0027s answer this question."},{"Start":"01:22.250 ","End":"01:28.580","Text":"What\u0027s the probability of at least 1 of the parties passing the threshold?"},{"Start":"01:28.580 ","End":"01:31.550","Text":"Whenever we see the term, at least,"},{"Start":"01:31.550 ","End":"01:36.020","Text":"then we automatically have to think of union because at least 1"},{"Start":"01:36.020 ","End":"01:40.820","Text":"of the parties means that either A or B or both."},{"Start":"01:40.820 ","End":"01:44.420","Text":"The probability of a union B,"},{"Start":"01:44.420 ","End":"01:46.415","Text":"that\u0027s what we\u0027re looking for."},{"Start":"01:46.415 ","End":"01:49.295","Text":"What does that equal to?"},{"Start":"01:49.295 ","End":"01:51.380","Text":"Well, according to De Morgan\u0027s,"},{"Start":"01:51.380 ","End":"01:57.575","Text":"that equals to 1 minus the probability of not A and not B."},{"Start":"01:57.575 ","End":"02:00.380","Text":"Well, what does that equal to?"},{"Start":"02:00.380 ","End":"02:02.915","Text":"Well, we know what this probability is."},{"Start":"02:02.915 ","End":"02:05.780","Text":"This probability is 0.76,"},{"Start":"02:05.780 ","End":"02:12.545","Text":"so that equals to 1 minus 0.76 and that equals to 0.24."},{"Start":"02:12.545 ","End":"02:16.160","Text":"Now, this didn\u0027t makes sense. Why is that?"},{"Start":"02:16.160 ","End":"02:24.105","Text":"Well, this event means that neither party will pass the threshold."},{"Start":"02:24.105 ","End":"02:27.065","Text":"What\u0027s the complimentary set of that?"},{"Start":"02:27.065 ","End":"02:30.110","Text":"Well, the complimentary set of neither of"},{"Start":"02:30.110 ","End":"02:33.965","Text":"the parties passing means that at least 1 of the parties will pass,"},{"Start":"02:33.965 ","End":"02:36.575","Text":"and that\u0027s this event right here."},{"Start":"02:36.575 ","End":"02:41.280","Text":"This is De Morgan\u0027s rule."}],"ID":12747},{"Watched":false,"Name":"Exercise 7 - Part b","Duration":"1m 44s","ChapterTopicVideoID":12265,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12265.jpeg","UploadDate":"2018-07-22T16:15:22.7030000","DurationForVideoObject":"PT1M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this section we\u0027re asked,"},{"Start":"00:02.040 ","End":"00:05.340","Text":"what\u0027s the probability of both parties passing the threshold?"},{"Start":"00:05.340 ","End":"00:09.975","Text":"We\u0027re looking for the event that both parties pass the threshold."},{"Start":"00:09.975 ","End":"00:12.930","Text":"This is the information that we got from the question."},{"Start":"00:12.930 ","End":"00:16.125","Text":"This is what was calculated in section A."},{"Start":"00:16.125 ","End":"00:19.995","Text":"Now, the event both parties passing the threshold means that"},{"Start":"00:19.995 ","End":"00:24.240","Text":"we\u0027re looking for A intersect B."},{"Start":"00:24.240 ","End":"00:26.940","Text":"We\u0027re looking for this probability right here,"},{"Start":"00:26.940 ","End":"00:29.175","Text":"the probability of this event."},{"Start":"00:29.175 ","End":"00:35.840","Text":"Now, let\u0027s just recall that the probability of A union B,"},{"Start":"00:35.840 ","End":"00:40.610","Text":"that equals to the probability of A plus the probability of"},{"Start":"00:40.610 ","End":"00:45.700","Text":"B minus the probability of A intersect B."},{"Start":"00:45.700 ","End":"00:50.379","Text":"This is the expression that we want to calculate,"},{"Start":"00:50.379 ","End":"00:53.390","Text":"and it\u0027s in this equation right here."},{"Start":"00:53.390 ","End":"00:56.680","Text":"But we have all of the other components here,"},{"Start":"00:56.680 ","End":"00:58.115","Text":"we know what they are."},{"Start":"00:58.115 ","End":"01:00.920","Text":"The probability of A union B,"},{"Start":"01:00.920 ","End":"01:06.050","Text":"we\u0027ve calculated that in the previous section, that\u0027s 0.24."},{"Start":"01:06.050 ","End":"01:09.260","Text":"That equals to the probability of A, that\u0027s right here,"},{"Start":"01:09.260 ","End":"01:12.920","Text":"that\u0027s 0.08 plus the probability of B,"},{"Start":"01:12.920 ","End":"01:15.530","Text":"that\u0027s 0.2, it\u0027s right here,"},{"Start":"01:15.530 ","End":"01:18.710","Text":"minus the probability of A intersect B,"},{"Start":"01:18.710 ","End":"01:20.180","Text":"what we\u0027re trying to calculate."},{"Start":"01:20.180 ","End":"01:22.580","Text":"Now, let\u0027s just move things around."},{"Start":"01:22.580 ","End":"01:31.570","Text":"That\u0027ll be the probability of A intersect B and that equals to 0.08 plus 0.2 minus 0.24,"},{"Start":"01:31.570 ","End":"01:37.550","Text":"that equals to 0.04."},{"Start":"01:37.550 ","End":"01:44.430","Text":"This then would be the probability of both parties passing the threshold."}],"ID":12748},{"Watched":false,"Name":"Exercise 7 - Part c","Duration":"2m 54s","ChapterTopicVideoID":12266,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12266.jpeg","UploadDate":"2018-07-22T16:15:29.6270000","DurationForVideoObject":"PT2M54S","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":"00:05.745","Text":"of only the \"Grey\" party passing the threshold?"},{"Start":"00:05.745 ","End":"00:06.965","Text":"Now we have to note,"},{"Start":"00:06.965 ","End":"00:09.345","Text":"the use of the word only."},{"Start":"00:09.345 ","End":"00:11.850","Text":"What does that mean? We\u0027re looking for"},{"Start":"00:11.850 ","End":"00:15.330","Text":"the event of only the \"Grey\" party passing the threshold."},{"Start":"00:15.330 ","End":"00:20.835","Text":"That means that we\u0027re looking for the event B and not A."},{"Start":"00:20.835 ","End":"00:24.570","Text":"We\u0027re looking for this probability right here."},{"Start":"00:24.570 ","End":"00:31.469","Text":"Now, there are lots of ways to calculate this probability using Venn diagrams whatnot."},{"Start":"00:31.469 ","End":"00:34.960","Text":"Let\u0027s use the probability matrix."},{"Start":"00:36.320 ","End":"00:39.410","Text":"Here we have the probability matrix,"},{"Start":"00:39.410 ","End":"00:43.625","Text":"here in the columns we have event A and naught A,"},{"Start":"00:43.625 ","End":"00:47.150","Text":"in here the row we have B and naught B."},{"Start":"00:47.150 ","End":"00:51.800","Text":"In the margins, we have the probability of being the probability of naught B"},{"Start":"00:51.800 ","End":"00:56.345","Text":"here and the probability of naught A and the probability of A here,"},{"Start":"00:56.345 ","End":"01:00.020","Text":"all the probabilities have to add up to 1."},{"Start":"01:00.020 ","End":"01:03.950","Text":"The cells here would be the intersect."},{"Start":"01:03.950 ","End":"01:09.215","Text":"That means that we\u0027re looking for the probability of naught A and B in this cell,"},{"Start":"01:09.215 ","End":"01:12.650","Text":"the probability of A and B in this cell,"},{"Start":"01:12.650 ","End":"01:14.000","Text":"and so on and so forth."},{"Start":"01:14.000 ","End":"01:17.590","Text":"Now, let\u0027s just write down what we\u0027re given."},{"Start":"01:17.590 ","End":"01:21.705","Text":"The probability of A that\u0027s 0.08."},{"Start":"01:21.705 ","End":"01:25.680","Text":"Let\u0027s write down here 0.08."},{"Start":"01:25.680 ","End":"01:27.950","Text":"What about the probability of B?"},{"Start":"01:27.950 ","End":"01:29.845","Text":"Well, that\u0027s 0.2,"},{"Start":"01:29.845 ","End":"01:31.620","Text":"write that down here."},{"Start":"01:31.620 ","End":"01:37.965","Text":"The probability of naught A intersect naught B,"},{"Start":"01:37.965 ","End":"01:44.015","Text":"well, that\u0027s 0.76 and that\u0027s 0.76, and that\u0027s here."},{"Start":"01:44.015 ","End":"01:49.505","Text":"Now, let\u0027s just complete this matrix here."},{"Start":"01:49.505 ","End":"01:52.910","Text":"Now we know that the probabilities have to add up to 1."},{"Start":"01:52.910 ","End":"01:57.185","Text":"So 0.2 plus 1 equals 1, that\u0027s 0.8."},{"Start":"01:57.185 ","End":"02:04.555","Text":"Now 0.08 plus 1 equals to 1, but that\u0027s 0.92."},{"Start":"02:04.555 ","End":"02:09.390","Text":"0.76 plus 1 equals to 0.8,"},{"Start":"02:09.390 ","End":"02:17.160","Text":"that\u0027s 0.04 and 0.04 plus 1 equals to 0.08, that\u0027s 0.04,"},{"Start":"02:17.160 ","End":"02:22.635","Text":"and 0.04 plus what"},{"Start":"02:22.635 ","End":"02:28.200","Text":"equals to 0.2, that\u0027s 0.16."},{"Start":"02:28.200 ","End":"02:30.150","Text":"Here in the blue,"},{"Start":"02:30.150 ","End":"02:32.360","Text":"that\u0027s what we were given in the red,"},{"Start":"02:32.360 ","End":"02:33.710","Text":"that\u0027s what we calculated."},{"Start":"02:33.710 ","End":"02:38.210","Text":"Now we\u0027re asked for the probability of B intersect naught A."},{"Start":"02:38.210 ","End":"02:40.130","Text":"Well, here\u0027s B,"},{"Start":"02:40.130 ","End":"02:42.905","Text":"that\u0027s B intersect naught A."},{"Start":"02:42.905 ","End":"02:46.140","Text":"That\u0027ll be this cell right here."},{"Start":"02:46.140 ","End":"02:49.835","Text":"The probability of B intersect naught A,"},{"Start":"02:49.835 ","End":"02:52.950","Text":"that equals to 0.16."}],"ID":12749},{"Watched":false,"Name":"Exercise 8","Duration":"5m 36s","ChapterTopicVideoID":12267,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12267.jpeg","UploadDate":"2018-07-22T16:15:44.2730000","DurationForVideoObject":"PT5M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this question, we\u0027re given that 40 percent of all the employees of work are men,"},{"Start":"00:05.070 ","End":"00:08.010","Text":"20 percent are university graduates and 10 percent are"},{"Start":"00:08.010 ","End":"00:13.650","Text":"female university graduates and we\u0027re asked about the proportion of various events."},{"Start":"00:13.650 ","End":"00:16.470","Text":"Before answering these questions,"},{"Start":"00:16.470 ","End":"00:18.600","Text":"let\u0027s just define our events."},{"Start":"00:18.600 ","End":"00:28.490","Text":"Let\u0027s define A as being a man and let\u0027s define B as being a University graduate."},{"Start":"00:28.490 ","End":"00:32.165","Text":"Not A then would be a woman."},{"Start":"00:32.165 ","End":"00:33.995","Text":"If you\u0027re not a man and you\u0027re a woman,"},{"Start":"00:33.995 ","End":"00:36.410","Text":"and B is being a graduate,"},{"Start":"00:36.410 ","End":"00:43.505","Text":"so not B is not having a university degree or not being a university graduate."},{"Start":"00:43.505 ","End":"00:45.395","Text":"Now, what are we given?"},{"Start":"00:45.395 ","End":"00:49.415","Text":"We\u0027re given that the probability of A,"},{"Start":"00:49.415 ","End":"00:57.620","Text":"the probability of all the employees are men that equals to 0.4 or 40 percent,"},{"Start":"00:57.620 ","End":"01:03.740","Text":"20 percent are university graduates so the probability of B would be equal to"},{"Start":"01:03.740 ","End":"01:10.300","Text":"0.2 and 10 percent are female university graduates."},{"Start":"01:10.300 ","End":"01:12.855","Text":"That means not A,"},{"Start":"01:12.855 ","End":"01:15.025","Text":"not being a man, that\u0027s a woman,"},{"Start":"01:15.025 ","End":"01:18.160","Text":"and B being a graduate,"},{"Start":"01:18.160 ","End":"01:22.960","Text":"so the probability of this event that equals to 0.1."},{"Start":"01:22.960 ","End":"01:26.995","Text":"Now again, we\u0027re asked about the proportions of various events here,"},{"Start":"01:26.995 ","End":"01:34.090","Text":"so the best way to do that is to just create a probability matrix and fill that out."},{"Start":"01:34.090 ","End":"01:39.640","Text":"Let\u0027s get to work. Here\u0027s our probability matrix."},{"Start":"01:39.640 ","End":"01:41.980","Text":"Now let\u0027s just fill in what we are given."},{"Start":"01:41.980 ","End":"01:44.905","Text":"The probability of A that\u0027s 0.4,"},{"Start":"01:44.905 ","End":"01:49.695","Text":"so here\u0027s A and here\u0027s the probability of A that\u0027s 0.4."},{"Start":"01:49.695 ","End":"01:52.740","Text":"The probability of B, that\u0027s 0.2."},{"Start":"01:52.740 ","End":"01:59.835","Text":"Here\u0027s B and here\u0027s 0.2 and the probability of not A and B,"},{"Start":"01:59.835 ","End":"02:03.980","Text":"well here\u0027s not A and here\u0027s B so the intersect right here,"},{"Start":"02:03.980 ","End":"02:08.360","Text":"that\u0027s the probability of not A and B, that\u0027s 0.1."},{"Start":"02:08.360 ","End":"02:13.835","Text":"Let\u0027s just fill in the probability matrix."},{"Start":"02:13.835 ","End":"02:18.020","Text":"We know that the sum of all the probabilities have to add up to 1,"},{"Start":"02:18.020 ","End":"02:22.205","Text":"so 0.2 plus 0.8 that equals to 1,"},{"Start":"02:22.205 ","End":"02:26.445","Text":"0.4 plus 0.6 that equals to 1,"},{"Start":"02:26.445 ","End":"02:32.416","Text":"0.1 plus 0.1 that equals to 0.2,"},{"Start":"02:32.416 ","End":"02:38.730","Text":"0.1 plus 0.3 that equals to 0.4 and 0.1 plus"},{"Start":"02:38.730 ","End":"02:47.855","Text":"0.5 that equals to 0.6 and just to make sure 0.5 plus 0.3 that equals to 0.8,"},{"Start":"02:47.855 ","End":"02:50.810","Text":"so this is a probability matrix."},{"Start":"02:50.810 ","End":"02:53.600","Text":"Now, what does A asks us?"},{"Start":"02:53.600 ","End":"02:59.660","Text":"A asks us what proportion of the employees are male university graduates,"},{"Start":"02:59.660 ","End":"03:06.105","Text":"so we\u0027re looking for the event A male and university graduate B."},{"Start":"03:06.105 ","End":"03:10.605","Text":"What\u0027s the probability of A and B?"},{"Start":"03:10.605 ","End":"03:12.555","Text":"Well, that\u0027s right here."},{"Start":"03:12.555 ","End":"03:17.820","Text":"That equals 2 A and B."},{"Start":"03:17.820 ","End":"03:20.100","Text":"This would be the intersect right here,"},{"Start":"03:20.100 ","End":"03:24.165","Text":"that will be 0.1 or 10 percent."},{"Start":"03:24.165 ","End":"03:27.000","Text":"Now, what about Section B."},{"Start":"03:27.000 ","End":"03:28.430","Text":"Well, Section B ask us,"},{"Start":"03:28.430 ","End":"03:35.180","Text":"what\u0027s the proportion of employees that are either male or university graduates."},{"Start":"03:35.180 ","End":"03:39.600","Text":"Male, that\u0027s A or university graduates."},{"Start":"03:39.600 ","End":"03:42.375","Text":"Well, we\u0027re looking for union B,"},{"Start":"03:42.375 ","End":"03:45.785","Text":"so we\u0027re looking for the probability of A union B."},{"Start":"03:45.785 ","End":"03:49.100","Text":"Well, we can solve this in 2 ways."},{"Start":"03:49.100 ","End":"03:53.390","Text":"We know that that equals to the probability of A"},{"Start":"03:53.390 ","End":"03:59.585","Text":"plus the probability of B minus the probability of A intersect B."},{"Start":"03:59.585 ","End":"04:08.015","Text":"Now, that equals to the probability of A that equals to 0.4 plus the probability of B,"},{"Start":"04:08.015 ","End":"04:14.150","Text":"that\u0027s 0.2 minus the probability of A intersect B."},{"Start":"04:14.150 ","End":"04:22.430","Text":"Well, that\u0027s right here 0.1 and that equals to 0.5 or 50 percent."},{"Start":"04:22.430 ","End":"04:25.870","Text":"Now, we can also solve this using De Morgan\u0027s,"},{"Start":"04:25.870 ","End":"04:28.940","Text":"the probability of A union B."},{"Start":"04:28.940 ","End":"04:35.820","Text":"Well, that equals to 1 minus the probability of not A and not B."},{"Start":"04:35.820 ","End":"04:39.860","Text":"Now, that equals to 1 minus."},{"Start":"04:39.860 ","End":"04:43.370","Text":"Now what\u0027s the probability of not A intersect not B?"},{"Start":"04:43.370 ","End":"04:46.365","Text":"Well, here\u0027s not A and here\u0027s not B."},{"Start":"04:46.365 ","End":"04:48.300","Text":"The Intersect is 0.5,"},{"Start":"04:48.300 ","End":"04:53.085","Text":"so that\u0027s 0.5 and 1 minus 0 is 0.5,"},{"Start":"04:53.085 ","End":"04:55.230","Text":"or again 50 percent."},{"Start":"04:55.230 ","End":"04:59.500","Text":"We solve Section B in 2 ways."},{"Start":"04:59.500 ","End":"05:02.330","Text":"Now, what about Section C?"},{"Start":"05:02.330 ","End":"05:03.950","Text":"Well C asks us,"},{"Start":"05:03.950 ","End":"05:10.170","Text":"what proportion of the workers are male or a female non-university graduates?"},{"Start":"05:10.170 ","End":"05:11.505","Text":"They\u0027re female."},{"Start":"05:11.505 ","End":"05:16.190","Text":"That\u0027s not A and non university graduates."},{"Start":"05:16.190 ","End":"05:18.055","Text":"Well, that\u0027s not B."},{"Start":"05:18.055 ","End":"05:26.015","Text":"We\u0027re looking for the intersect of not A and not B and what\u0027s the probability of that?"},{"Start":"05:26.015 ","End":"05:29.770","Text":"Well, here\u0027s not A and here\u0027s not B,"},{"Start":"05:29.770 ","End":"05:35.300","Text":"that\u0027s 0.5 or 50 percent."}],"ID":12750},{"Watched":false,"Name":"Exercise 9 - Part a","Duration":"2m 27s","ChapterTopicVideoID":12268,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12268.jpeg","UploadDate":"2018-07-22T16:15:50.2900000","DurationForVideoObject":"PT2M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.540","Text":"In this question, we\u0027re given that the chances of share A rising on a given day are 0.5,"},{"Start":"00:06.540 ","End":"00:11.010","Text":"and the chances of share B rising on a given day are 0.4."},{"Start":"00:11.010 ","End":"00:15.780","Text":"Now, the chances of at least 1 share rising on a given day are 0.7,"},{"Start":"00:15.780 ","End":"00:17.040","Text":"and we\u0027re asked to calculate"},{"Start":"00:17.040 ","End":"00:20.550","Text":"the following probabilities for these 2 shares on a given day."},{"Start":"00:20.550 ","End":"00:25.050","Text":"Now the first probability is the probability of this event,"},{"Start":"00:25.050 ","End":"00:26.310","Text":"of both shares rising."},{"Start":"00:26.310 ","End":"00:27.930","Text":"But before we can do that,"},{"Start":"00:27.930 ","End":"00:30.540","Text":"let\u0027s just define our events."},{"Start":"00:30.540 ","End":"00:37.980","Text":"Let\u0027s define A as share A rising,"},{"Start":"00:37.980 ","End":"00:43.530","Text":"and B would be share B rising."},{"Start":"00:43.530 ","End":"00:47.745","Text":"Now, the probability of A,"},{"Start":"00:47.745 ","End":"00:50.325","Text":"well that equals to 0.5."},{"Start":"00:50.325 ","End":"00:52.125","Text":"That\u0027s given to us right here."},{"Start":"00:52.125 ","End":"00:54.350","Text":"The probability of B,"},{"Start":"00:54.350 ","End":"00:58.700","Text":"well that is 0.4, that\u0027s right here."},{"Start":"00:58.700 ","End":"01:02.720","Text":"Now, the chances of at least 1 share rising,"},{"Start":"01:02.720 ","End":"01:05.720","Text":"that\u0027s the probability of A union B."},{"Start":"01:05.720 ","End":"01:07.310","Text":"Now A union B,"},{"Start":"01:07.310 ","End":"01:09.005","Text":"why do we say that?"},{"Start":"01:09.005 ","End":"01:12.630","Text":"Well, that\u0027s because we\u0027re given that at least 1 share."},{"Start":"01:12.630 ","End":"01:16.125","Text":"That means that either A or B or both of them."},{"Start":"01:16.125 ","End":"01:18.960","Text":"Now the probability of A union B,"},{"Start":"01:18.960 ","End":"01:21.835","Text":"that equals to 0.7."},{"Start":"01:21.835 ","End":"01:23.495","Text":"This is what we\u0027re given."},{"Start":"01:23.495 ","End":"01:27.080","Text":"Now we\u0027re asked about the probability of both shares rising,"},{"Start":"01:27.080 ","End":"01:31.930","Text":"so we\u0027re looking at the event A intersect B."},{"Start":"01:31.930 ","End":"01:35.505","Text":"Now, what\u0027s the probability of that?"},{"Start":"01:35.505 ","End":"01:37.500","Text":"Well, we know,"},{"Start":"01:37.500 ","End":"01:41.280","Text":"that the probability of A union B, well,"},{"Start":"01:41.280 ","End":"01:45.530","Text":"that equals to the probability of A plus the probability"},{"Start":"01:45.530 ","End":"01:50.890","Text":"of B minus the probability of A intersect B."},{"Start":"01:50.890 ","End":"01:53.660","Text":"Let\u0027s just plug in the numbers."},{"Start":"01:53.660 ","End":"01:55.790","Text":"The probability of A union B, well,"},{"Start":"01:55.790 ","End":"01:58.205","Text":"that\u0027s right here, that\u0027s 0.7,"},{"Start":"01:58.205 ","End":"02:02.180","Text":"and that equals to the probability of A that\u0027s 0.5,"},{"Start":"02:02.180 ","End":"02:08.645","Text":"plus the probability of B that\u0027s 0.4 minus the probability of A intersect B."},{"Start":"02:08.645 ","End":"02:11.915","Text":"Now, let\u0027s just move things around here."},{"Start":"02:11.915 ","End":"02:15.530","Text":"The probability of A intersect B, well,"},{"Start":"02:15.530 ","End":"02:21.230","Text":"that equals to 0.5 plus 0.4 minus 0.7,"},{"Start":"02:21.230 ","End":"02:25.620","Text":"and that equals to 0.2."}],"ID":12751},{"Watched":false,"Name":"Exercise 9 - Part b","Duration":"58s","ChapterTopicVideoID":12269,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12269.jpeg","UploadDate":"2018-07-22T16:15:52.4670000","DurationForVideoObject":"PT58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.005","Text":"In this section we\u0027re asked for the probability of the event that neither share rises."},{"Start":"00:07.005 ","End":"00:10.000","Text":"How do we define this event?"},{"Start":"00:10.000 ","End":"00:12.690","Text":"Here again, this is what we\u0027re given in the question."},{"Start":"00:12.690 ","End":"00:15.795","Text":"This was what was calculated in section A."},{"Start":"00:15.795 ","End":"00:18.600","Text":"Now, the event neither share rises,"},{"Start":"00:18.600 ","End":"00:23.505","Text":"that means that we\u0027re looking at not A and not B."},{"Start":"00:23.505 ","End":"00:26.400","Text":"What\u0027s this probability right here?"},{"Start":"00:26.400 ","End":"00:28.680","Text":"Let\u0027s just recall De Morgan\u0027s,"},{"Start":"00:28.680 ","End":"00:33.570","Text":"De Morgan\u0027s means that the probability of not A and not B"},{"Start":"00:33.570 ","End":"00:39.640","Text":"that equals to 1 minus the probability of A union B."},{"Start":"00:39.640 ","End":"00:47.365","Text":"Well, that equals to 1 minus the probability of A union B, that\u0027s 0.7."},{"Start":"00:47.365 ","End":"00:51.570","Text":"The probability of not A and not B,"},{"Start":"00:51.570 ","End":"00:53.930","Text":"that neither share rises,"},{"Start":"00:53.930 ","End":"00:57.810","Text":"that equals to 0.3."}],"ID":12752},{"Watched":false,"Name":"Exercise 9 - Part c","Duration":"2m 36s","ChapterTopicVideoID":12270,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12270.jpeg","UploadDate":"2018-07-22T16:15:58.4400000","DurationForVideoObject":"PT2M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.355","Text":"In this section we\u0027re asked for the probability of only share A rising."},{"Start":"00:05.355 ","End":"00:07.845","Text":"Again, this is what we\u0027re given the question,"},{"Start":"00:07.845 ","End":"00:10.980","Text":"this is what was calculated in Section A and in Section B."},{"Start":"00:10.980 ","End":"00:14.850","Text":"Now, what\u0027s this event only share A rising?"},{"Start":"00:14.850 ","End":"00:20.220","Text":"Well, that means that A rises and B does not."},{"Start":"00:20.220 ","End":"00:23.385","Text":"We\u0027re looking for this probability right here."},{"Start":"00:23.385 ","End":"00:27.600","Text":"Now, the best way to do that is through the probability matrix."},{"Start":"00:27.600 ","End":"00:30.720","Text":"Let\u0027s just write the probability matrix."},{"Start":"00:30.720 ","End":"00:33.304","Text":"Here\u0027s the probability matrix."},{"Start":"00:33.304 ","End":"00:35.270","Text":"Now, let\u0027s just fill that in,"},{"Start":"00:35.270 ","End":"00:36.845","Text":"the probability of A,"},{"Start":"00:36.845 ","End":"00:39.245","Text":"that\u0027s what\u0027s given to us, that\u0027s 0.5."},{"Start":"00:39.245 ","End":"00:41.970","Text":"Here\u0027s A and here\u0027s the probability of A."},{"Start":"00:41.970 ","End":"00:45.590","Text":"That\u0027s 0.5. Now the probability of B,"},{"Start":"00:45.590 ","End":"00:47.660","Text":"that\u0027s 0.4, so here\u0027s B."},{"Start":"00:47.660 ","End":"00:54.035","Text":"Here\u0027s the probability of B that\u0027s 0.4 and the probability of A union B,"},{"Start":"00:54.035 ","End":"00:56.390","Text":"well, that equals to 0.7."},{"Start":"00:56.390 ","End":"01:03.825","Text":"But if we recall we calculated the probability of A intersect B in Section B,"},{"Start":"01:03.825 ","End":"01:05.730","Text":"and that equal to 0.2."},{"Start":"01:05.730 ","End":"01:07.440","Text":"Let\u0027s just write that in."},{"Start":"01:07.440 ","End":"01:10.085","Text":"That\u0027ll be right here, 0.2."},{"Start":"01:10.085 ","End":"01:13.205","Text":"The probability of A intersect B."},{"Start":"01:13.205 ","End":"01:14.570","Text":"Here\u0027s B, here\u0027s A,"},{"Start":"01:14.570 ","End":"01:15.950","Text":"and here\u0027s the intersection."},{"Start":"01:15.950 ","End":"01:20.090","Text":"Now, let\u0027s just fill in the rest of the matrix."},{"Start":"01:20.090 ","End":"01:23.820","Text":"We know that the sum of the probabilities have to equal to 1,"},{"Start":"01:23.820 ","End":"01:28.250","Text":"so 0.4 plus 0.6, that equals to 1."},{"Start":"01:28.250 ","End":"01:31.895","Text":"0.5 plus 0.5, that equals to 1."},{"Start":"01:31.895 ","End":"01:37.203","Text":"0.2 plus 0.3, that equals to 0.5."},{"Start":"01:37.203 ","End":"01:40.685","Text":"0.3 plus 0.3, that equals to 0.6,"},{"Start":"01:40.685 ","End":"01:42.920","Text":"and 0.2 plus 0.3,"},{"Start":"01:42.920 ","End":"01:47.600","Text":"that equals to 0.5 and just to make sure 0.2 plus 0.2,"},{"Start":"01:47.600 ","End":"01:49.730","Text":"that equals to 0.4."},{"Start":"01:49.730 ","End":"01:53.910","Text":"Now, the probability of A intersect not B, well,"},{"Start":"01:53.910 ","End":"01:57.930","Text":"here\u0027s A and here\u0027s not B,"},{"Start":"01:57.930 ","End":"02:04.710","Text":"so that\u0027s this probability right here and that equals to 0.3."},{"Start":"02:04.710 ","End":"02:11.745","Text":"Now, we can see that we\u0027ve gotten the probabilities for Section A,"},{"Start":"02:11.745 ","End":"02:13.650","Text":"and Section B as well."},{"Start":"02:13.650 ","End":"02:15.770","Text":"The probability of A intersect B,"},{"Start":"02:15.770 ","End":"02:17.465","Text":"as we said, that\u0027s 0.2,"},{"Start":"02:17.465 ","End":"02:22.430","Text":"but the probability of not A and not B, that\u0027s 0.3."},{"Start":"02:22.430 ","End":"02:23.675","Text":"Here\u0027s not A,"},{"Start":"02:23.675 ","End":"02:25.094","Text":"here\u0027s not B,"},{"Start":"02:25.094 ","End":"02:28.535","Text":"so this then is 0.3."},{"Start":"02:28.535 ","End":"02:36.570","Text":"We could have gotten all the probabilities just by using the probability matrix."}],"ID":12753},{"Watched":false,"Name":"Exercise 10","Duration":"3m 43s","ChapterTopicVideoID":12271,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12271.jpeg","UploadDate":"2018-07-22T16:16:08.3630000","DurationForVideoObject":"PT3M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this question we have a pair of dice,"},{"Start":"00:03.150 ","End":"00:04.560","Text":"1 red and 1 black."},{"Start":"00:04.560 ","End":"00:07.005","Text":"Now, we define the following events."},{"Start":"00:07.005 ","End":"00:10.305","Text":"A is that the red die is 4 and the black dice is 2."},{"Start":"00:10.305 ","End":"00:13.575","Text":"B, the sum of the 2 dice is 6 and C,"},{"Start":"00:13.575 ","End":"00:16.020","Text":"the product of the 2 dice is 10."},{"Start":"00:16.020 ","End":"00:21.420","Text":"First of all let\u0027s define all the results for each one of the events."},{"Start":"00:21.420 ","End":"00:24.705","Text":"For A, well, what would that be?"},{"Start":"00:24.705 ","End":"00:26.940","Text":"Well, the red die is 4 and the black dice 2,"},{"Start":"00:26.940 ","End":"00:29.460","Text":"so we have 4, 2."},{"Start":"00:29.460 ","End":"00:34.875","Text":"That would be the only result in event A."},{"Start":"00:34.875 ","End":"00:39.825","Text":"The red die being 4 and the black die being 2. What about B?"},{"Start":"00:39.825 ","End":"00:45.200","Text":"Well, B says that we\u0027re looking for the sum of the 2 dice is 6."},{"Start":"00:45.200 ","End":"00:49.430","Text":"Well that happens when the red die is 1 and the black die is 5."},{"Start":"00:49.430 ","End":"00:54.240","Text":"It could be when we have 2 and 4 or 3 and"},{"Start":"00:54.240 ","End":"01:00.280","Text":"3 or 4 and 2 or 5 and 1."},{"Start":"01:00.280 ","End":"01:06.440","Text":"These are all the events or the results for event B."},{"Start":"01:06.440 ","End":"01:08.775","Text":"Now what about event C?"},{"Start":"01:08.775 ","End":"01:12.770","Text":"Well, event C means that the product of the 2 dice is 10."},{"Start":"01:12.770 ","End":"01:16.730","Text":"Well that happens only when the red die is 2 and the black die is"},{"Start":"01:16.730 ","End":"01:23.130","Text":"5 or the other way around when the red die is 5 and the black die is 2."},{"Start":"01:23.130 ","End":"01:28.130","Text":"These are all the possible results of all 3 events."},{"Start":"01:28.130 ","End":"01:30.470","Text":"Now let\u0027s answer the questions here."},{"Start":"01:30.470 ","End":"01:35.090","Text":"a, are A and B mutually exclusive events?"},{"Start":"01:35.090 ","End":"01:38.015","Text":"Well, what does mutually exclusive event mean?"},{"Start":"01:38.015 ","End":"01:41.525","Text":"That means that A intersect B."},{"Start":"01:41.525 ","End":"01:44.545","Text":"Well, that equals a null set."},{"Start":"01:44.545 ","End":"01:46.550","Text":"That\u0027s an empty set."},{"Start":"01:46.550 ","End":"01:48.650","Text":"Well, let\u0027s take a look at A,"},{"Start":"01:48.650 ","End":"01:50.015","Text":"A is 4, 2."},{"Start":"01:50.015 ","End":"01:54.560","Text":"Now, do we have the result for 2 in B? Well, yes, we do."},{"Start":"01:54.560 ","End":"01:59.115","Text":"In a, the answer would be no."},{"Start":"01:59.115 ","End":"02:02.690","Text":"The A and B are not mutually exclusive"},{"Start":"02:02.690 ","End":"02:07.190","Text":"because we have something that\u0027s common to both A and B."},{"Start":"02:07.190 ","End":"02:09.455","Text":"What about section b?"},{"Start":"02:09.455 ","End":"02:14.630","Text":"Is event A a subset of event B?"},{"Start":"02:14.630 ","End":"02:19.950","Text":"Well, are all the results in A contained in B?"},{"Start":"02:19.950 ","End":"02:23.310","Text":"Well, we only have 1 result in A that\u0027s 4, 2."},{"Start":"02:23.310 ","End":"02:25.500","Text":"Now, do we have 4, 2 in B?"},{"Start":"02:25.500 ","End":"02:26.825","Text":"Well, yes, we do."},{"Start":"02:26.825 ","End":"02:31.485","Text":"Here it is. That means that the answer to b is yes."},{"Start":"02:31.485 ","End":"02:34.820","Text":"Event A is a subset of event B."},{"Start":"02:34.820 ","End":"02:37.025","Text":"Now, what about c?"},{"Start":"02:37.025 ","End":"02:39.910","Text":"Are A and C mutually exclusive?"},{"Start":"02:39.910 ","End":"02:45.685","Text":"Well, again, we\u0027re looking for A intersect C. Now,"},{"Start":"02:45.685 ","End":"02:47.560","Text":"if they\u0027re mutually exclusive,"},{"Start":"02:47.560 ","End":"02:51.710","Text":"then we should have a null set right here."},{"Start":"02:51.710 ","End":"02:54.705","Text":"What\u0027s common with A,"},{"Start":"02:54.705 ","End":"02:56.410","Text":"between A and C?"},{"Start":"02:56.410 ","End":"03:00.331","Text":"Well, we see that nothing is common so the answer here is yes,"},{"Start":"03:00.331 ","End":"03:03.735","Text":"they are mutually exclusive. What about d?"},{"Start":"03:03.735 ","End":"03:07.615","Text":"D asks, are A and C complimentary events?"},{"Start":"03:07.615 ","End":"03:09.880","Text":"Well, if they\u0027re complimentary events,"},{"Start":"03:09.880 ","End":"03:12.280","Text":"then A union C,"},{"Start":"03:12.280 ","End":"03:18.565","Text":"well that has to equal to the sample space to all the possible results."},{"Start":"03:18.565 ","End":"03:26.390","Text":"Well, we see that the answer here is no because in the sample space we can get,"},{"Start":"03:26.390 ","End":"03:29.030","Text":"let\u0027s say, a result of 1, 1."},{"Start":"03:29.030 ","End":"03:31.460","Text":"We don\u0027t have that right here."},{"Start":"03:31.460 ","End":"03:37.940","Text":"When we unify A and C we don\u0027t get all the possible results in the sample space."},{"Start":"03:37.940 ","End":"03:42.990","Text":"These then are the answers to the questions."}],"ID":12754},{"Watched":false,"Name":"Exercise 11","Duration":"2m 27s","ChapterTopicVideoID":12272,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12272.jpeg","UploadDate":"2018-07-22T16:16:15.0670000","DurationForVideoObject":"PT2M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"The following probabilities are known for events A and B,"},{"Start":"00:04.110 ","End":"00:06.270","Text":"probability of A is 0.6,"},{"Start":"00:06.270 ","End":"00:08.505","Text":"the probability of B is 0.3,"},{"Start":"00:08.505 ","End":"00:11.910","Text":"and the probability of not A and not B well,"},{"Start":"00:11.910 ","End":"00:13.470","Text":"that equals to 0.1."},{"Start":"00:13.470 ","End":"00:17.340","Text":"We\u0027re asked are A and B mutually exclusive events?"},{"Start":"00:17.340 ","End":"00:21.150","Text":"Well, in order for A and B to be mutually exclusive events,"},{"Start":"00:21.150 ","End":"00:24.945","Text":"then the probability of A intersect B,"},{"Start":"00:24.945 ","End":"00:27.120","Text":"well that has to equal to 0."},{"Start":"00:27.120 ","End":"00:28.920","Text":"This is what we\u0027re trying to find out."},{"Start":"00:28.920 ","End":"00:30.390","Text":"Now the way to do that,"},{"Start":"00:30.390 ","End":"00:33.090","Text":"or the easiest way to do that would be through"},{"Start":"00:33.090 ","End":"00:36.850","Text":"the probability matrix. Let\u0027s just draw that."},{"Start":"00:37.250 ","End":"00:39.980","Text":"Here\u0027s a probability matrix."},{"Start":"00:39.980 ","End":"00:43.985","Text":"Now the probability of A is 0.6. Here\u0027s A."},{"Start":"00:43.985 ","End":"00:47.265","Text":"The probability of A that\u0027s 0.6,"},{"Start":"00:47.265 ","End":"00:49.680","Text":"probability of B is 0.3."},{"Start":"00:49.680 ","End":"00:51.080","Text":"Well, that\u0027s right here, here\u0027s B,"},{"Start":"00:51.080 ","End":"00:55.635","Text":"Here\u0027s the probability of B, and that\u0027s 0.3."},{"Start":"00:55.635 ","End":"00:59.510","Text":"The probability of not A and not B,"},{"Start":"00:59.510 ","End":"01:03.230","Text":"well that\u0027s 0.1 and that\u0027s right here, so that\u0027s 0.1."},{"Start":"01:03.230 ","End":"01:06.545","Text":"Let\u0027s just complete the table here."},{"Start":"01:06.545 ","End":"01:09.410","Text":"We note that the sums have to add up to 1."},{"Start":"01:09.410 ","End":"01:10.820","Text":"The sum of the probabilities."},{"Start":"01:10.820 ","End":"01:15.320","Text":"So 0.3 plus 0.7,"},{"Start":"01:15.320 ","End":"01:17.990","Text":"that has to equal to 1."},{"Start":"01:17.990 ","End":"01:22.610","Text":"Now, 0.1 plus 0.6,"},{"Start":"01:22.610 ","End":"01:29.795","Text":"that has to equal to 0.7 0.6 plus 0.4 that has to equal to 1,"},{"Start":"01:29.795 ","End":"01:35.540","Text":"0.3 plus 0.1, that has to equal to 0.4."},{"Start":"01:35.540 ","End":"01:41.630","Text":"That means it\u0027s 0.3 plus 0 that has to equal to 0.3."},{"Start":"01:41.630 ","End":"01:44.390","Text":"Now let\u0027s check 0 plus 0.6."},{"Start":"01:44.390 ","End":"01:46.285","Text":"That equals 0.6."},{"Start":"01:46.285 ","End":"01:51.390","Text":"Excellent. The probability now of A intersect B,"},{"Start":"01:51.390 ","End":"01:54.930","Text":"well, here\u0027s B and here\u0027s A,"},{"Start":"01:54.930 ","End":"01:57.465","Text":"and here\u0027s the intersect right here."},{"Start":"01:57.465 ","End":"01:59.595","Text":"That equals to 0."},{"Start":"01:59.595 ","End":"02:04.640","Text":"That means that A and B are mutually exclusive."},{"Start":"02:04.640 ","End":"02:06.470","Text":"Now, what about B?"},{"Start":"02:06.470 ","End":"02:13.175","Text":"B asks us to calculate the probability of not A and B."},{"Start":"02:13.175 ","End":"02:16.430","Text":"Well, all we have to do is go to the probability matrix."},{"Start":"02:16.430 ","End":"02:20.390","Text":"Here\u0027s A and here\u0027s B right here."},{"Start":"02:20.390 ","End":"02:26.910","Text":"So the intersect is right here and that equals to 0.3."}],"ID":12755},{"Watched":false,"Name":"Exercise 12","Duration":"2m 55s","ChapterTopicVideoID":12273,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12273.jpeg","UploadDate":"2018-07-22T16:16:22.6330000","DurationForVideoObject":"PT2M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.160","Text":"In this question, a coin is tossed twice."},{"Start":"00:03.160 ","End":"00:05.080","Text":"Now, we defined the following events."},{"Start":"00:05.080 ","End":"00:07.405","Text":"A is the first toss is heads,"},{"Start":"00:07.405 ","End":"00:10.795","Text":"and B at least 1 of the 2 tosses is heads."},{"Start":"00:10.795 ","End":"00:14.990","Text":"Now, we\u0027re asked to state which of the following statements is correct."},{"Start":"00:14.990 ","End":"00:18.190","Text":"First of all, let\u0027s define A and B,"},{"Start":"00:18.190 ","End":"00:23.020","Text":"or at least find out what the results of A and B are."},{"Start":"00:23.020 ","End":"00:26.440","Text":"A says that the first toss is heads,"},{"Start":"00:26.440 ","End":"00:31.300","Text":"so we can either have heads and tails,"},{"Start":"00:31.300 ","End":"00:33.460","Text":"or heads and heads,"},{"Start":"00:33.460 ","End":"00:35.054","Text":"where this is the first toss,"},{"Start":"00:35.054 ","End":"00:37.330","Text":"and this is the second toss."},{"Start":"00:37.330 ","End":"00:41.110","Text":"These are the 2 results for A."},{"Start":"00:41.110 ","End":"00:43.190","Text":"Now, what about B?"},{"Start":"00:43.190 ","End":"00:46.745","Text":"B says that at least 1 of the 2 tosses is heads."},{"Start":"00:46.745 ","End":"00:48.440","Text":"Well again, this is the first toss,"},{"Start":"00:48.440 ","End":"00:49.910","Text":"this is the second toss."},{"Start":"00:49.910 ","End":"00:52.100","Text":"If at least 1 of the 2 tosses,"},{"Start":"00:52.100 ","End":"00:54.275","Text":"it says, we can have heads and tails,"},{"Start":"00:54.275 ","End":"00:56.420","Text":"we can have tails and heads,"},{"Start":"00:56.420 ","End":"00:59.805","Text":"and we can have heads and heads."},{"Start":"00:59.805 ","End":"01:02.915","Text":"These are the results for B."},{"Start":"01:02.915 ","End":"01:05.350","Text":"Now, let\u0027s take a look at A."},{"Start":"01:05.350 ","End":"01:10.475","Text":"Section A states A and B are mutually exclusive events."},{"Start":"01:10.475 ","End":"01:14.090","Text":"Well, for A and B to be mutually exclusive,"},{"Start":"01:14.090 ","End":"01:16.475","Text":"then A intersect B."},{"Start":"01:16.475 ","End":"01:19.370","Text":"Well, that has to be an empty set."},{"Start":"01:19.370 ","End":"01:24.890","Text":"Well, do we have anything in common between A and B?"},{"Start":"01:24.890 ","End":"01:26.330","Text":"Well, yes, we do."},{"Start":"01:26.330 ","End":"01:28.760","Text":"We have heads and tails."},{"Start":"01:28.760 ","End":"01:32.665","Text":"That means that A is wrong."},{"Start":"01:32.665 ","End":"01:37.980","Text":"What about B? A and B are complementary events."},{"Start":"01:37.980 ","End":"01:43.410","Text":"Well, in order for A and B to be complementary events,"},{"Start":"01:43.410 ","End":"01:45.965","Text":"that means that a union B,"},{"Start":"01:45.965 ","End":"01:49.385","Text":"well, that has to give me the whole sample space."},{"Start":"01:49.385 ","End":"01:55.585","Text":"That\u0027s not true because we\u0027re missing the results tails and tails."},{"Start":"01:55.585 ","End":"02:00.680","Text":"That means since we don\u0027t have that in A union B,"},{"Start":"02:00.680 ","End":"02:03.470","Text":"then that means that B is wrong."},{"Start":"02:03.470 ","End":"02:05.585","Text":"Now, what about C,"},{"Start":"02:05.585 ","End":"02:10.445","Text":"A is a subset of B."},{"Start":"02:10.445 ","End":"02:16.655","Text":"Well, that means that all the results in A have to be contained in B."},{"Start":"02:16.655 ","End":"02:18.830","Text":"Well, we have heads and tails and A,"},{"Start":"02:18.830 ","End":"02:21.275","Text":"and here we have heads and tails in B."},{"Start":"02:21.275 ","End":"02:23.540","Text":"We have heads and heads in A,"},{"Start":"02:23.540 ","End":"02:25.820","Text":"and we have heads and heads in B."},{"Start":"02:25.820 ","End":"02:29.360","Text":"All the results in A are contained in B,"},{"Start":"02:29.360 ","End":"02:31.250","Text":"so that\u0027s true,"},{"Start":"02:31.250 ","End":"02:33.545","Text":"C is the correct answer."},{"Start":"02:33.545 ","End":"02:35.930","Text":"Now, if A is contained in B,"},{"Start":"02:35.930 ","End":"02:38.360","Text":"that doesn\u0027t mean that B is contained in A."},{"Start":"02:38.360 ","End":"02:42.095","Text":"B is not a subset of A. Why is that?"},{"Start":"02:42.095 ","End":"02:46.513","Text":"Well, because B has tails and heads,"},{"Start":"02:46.513 ","End":"02:49.340","Text":"and A does not have that result,"},{"Start":"02:49.340 ","End":"02:51.245","Text":"so D would be wrong,"},{"Start":"02:51.245 ","End":"02:55.020","Text":"and C then is the only correct answer."}],"ID":12756},{"Watched":false,"Name":"Exercise 13","Duration":"4m 21s","ChapterTopicVideoID":12274,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12274.jpeg","UploadDate":"2018-07-22T16:16:34.8630000","DurationForVideoObject":"PT4M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this question, a lottery is held with a 100 tickets being sold."},{"Start":"00:04.320 ","End":"00:06.869","Text":"Now, 3 of the tickets are for vacation,"},{"Start":"00:06.869 ","End":"00:09.540","Text":"2 are for computers and the rest are empty."},{"Start":"00:09.540 ","End":"00:13.020","Text":"A person randomly buys 1 ticket, and we\u0027re asked,"},{"Start":"00:13.020 ","End":"00:15.240","Text":"what are the chances of winning a vacation or"},{"Start":"00:15.240 ","End":"00:19.395","Text":"a computer and are these mutually exclusive events?"},{"Start":"00:19.395 ","End":"00:23.770","Text":"Well, let\u0027s first write down the information that we\u0027re given."},{"Start":"00:23.900 ","End":"00:27.285","Text":"As we said, we have 100 tickets,"},{"Start":"00:27.285 ","End":"00:29.010","Text":"3 win you a vacation,"},{"Start":"00:29.010 ","End":"00:30.645","Text":"3 tickets win you a vacation,"},{"Start":"00:30.645 ","End":"00:32.490","Text":"2 tickets win you a computer,"},{"Start":"00:32.490 ","End":"00:36.630","Text":"and 95 tickets don\u0027t win you anything."},{"Start":"00:36.630 ","End":"00:40.840","Text":"Now, a person bought 1 ticket,"},{"Start":"00:40.910 ","End":"00:43.310","Text":"and we\u0027re asked, first of all,"},{"Start":"00:43.310 ","End":"00:45.380","Text":"are these mutually exclusive events?"},{"Start":"00:45.380 ","End":"00:46.565","Text":"Well, what are the events?"},{"Start":"00:46.565 ","End":"00:48.095","Text":"Well, the event A,"},{"Start":"00:48.095 ","End":"00:52.025","Text":"let\u0027s define event A as winning a vacation,"},{"Start":"00:52.025 ","End":"00:55.330","Text":"and event B as winning a computer,"},{"Start":"00:55.330 ","End":"01:01.415","Text":"so we\u0027re asked for the event of winning a vacation A,"},{"Start":"01:01.415 ","End":"01:03.620","Text":"or winning a computer,"},{"Start":"01:03.620 ","End":"01:05.120","Text":"that\u0027s Union B,"},{"Start":"01:05.120 ","End":"01:08.195","Text":"so what\u0027s this probability right here?"},{"Start":"01:08.195 ","End":"01:13.520","Text":"Well, we know that equals to the probability of A plus"},{"Start":"01:13.520 ","End":"01:19.405","Text":"the probability of B minus the probability of A intersect B."},{"Start":"01:19.405 ","End":"01:23.295","Text":"Well, let\u0027s go to the second question here first,"},{"Start":"01:23.295 ","End":"01:25.955","Text":"are these mutually exclusive events?"},{"Start":"01:25.955 ","End":"01:31.275","Text":"Are the events winning a vacation and winning a computer,"},{"Start":"01:31.275 ","End":"01:32.930","Text":"are they mutually exclusive?"},{"Start":"01:32.930 ","End":"01:35.510","Text":"Well, the answer to that is yes,"},{"Start":"01:35.510 ","End":"01:39.770","Text":"because on a ticket you can only win 1 price,"},{"Start":"01:39.770 ","End":"01:42.755","Text":"it\u0027s either a vacation or a computer,"},{"Start":"01:42.755 ","End":"01:48.830","Text":"so that means that the probability of A intersect B,"},{"Start":"01:48.830 ","End":"01:52.055","Text":"but that has to be equal to 0, so now,"},{"Start":"01:52.055 ","End":"01:53.480","Text":"in answering the question,"},{"Start":"01:53.480 ","End":"01:56.270","Text":"what are the chances of winning a vacation or a computer?"},{"Start":"01:56.270 ","End":"01:58.825","Text":"Well, we know since A and B,"},{"Start":"01:58.825 ","End":"02:03.020","Text":"winning a vacation and winning a computer are mutually exclusive events."},{"Start":"02:03.020 ","End":"02:06.620","Text":"Well, we know that the probability of A and B,"},{"Start":"02:06.620 ","End":"02:08.345","Text":"well that equals to 0,"},{"Start":"02:08.345 ","End":"02:12.020","Text":"so the probability of a union B,"},{"Start":"02:12.020 ","End":"02:16.500","Text":"well that equals to the probability of A plus the probability of B."},{"Start":"02:16.500 ","End":"02:18.845","Text":"Now what is the probability of A?"},{"Start":"02:18.845 ","End":"02:21.930","Text":"Well, the probability of A,"},{"Start":"02:21.930 ","End":"02:27.515","Text":"that equals to the size of A divided by the size of Omega."},{"Start":"02:27.515 ","End":"02:30.030","Text":"Now A, the size of A, well,"},{"Start":"02:30.030 ","End":"02:31.400","Text":"we have 3 tickets,"},{"Start":"02:31.400 ","End":"02:33.605","Text":"and how many tickets do we have in total?"},{"Start":"02:33.605 ","End":"02:37.580","Text":"Well, that\u0027s 100, and what about the probability of B?"},{"Start":"02:37.580 ","End":"02:41.645","Text":"Well, that\u0027s the size of B divided by the size of Omega."},{"Start":"02:41.645 ","End":"02:43.410","Text":"Now the size of B,"},{"Start":"02:43.410 ","End":"02:46.440","Text":"that\u0027s 2 divided by 100,"},{"Start":"02:46.440 ","End":"02:51.860","Text":"so the probability of A union B,"},{"Start":"02:51.860 ","End":"02:55.400","Text":"the chances of winning a vacation or a computer,"},{"Start":"02:55.400 ","End":"03:01.775","Text":"well that equals to 3 divided by 100 plus 2 divided by 100,"},{"Start":"03:01.775 ","End":"03:05.780","Text":"and that equals to 5 divided by 100,"},{"Start":"03:05.780 ","End":"03:11.230","Text":"and that equals to 0.05 or 5 percent."},{"Start":"03:11.230 ","End":"03:13.340","Text":"Section B asks us,"},{"Start":"03:13.340 ","End":"03:16.670","Text":"what\u0027s the probability of not winning a prize?"},{"Start":"03:16.670 ","End":"03:18.730","Text":"Well, not winning a prize,"},{"Start":"03:18.730 ","End":"03:23.035","Text":"that means that we have winning at least 1 prize,"},{"Start":"03:23.035 ","End":"03:25.865","Text":"and taking the complimentary set of that."},{"Start":"03:25.865 ","End":"03:28.670","Text":"Well, what\u0027s the probability of that?"},{"Start":"03:28.670 ","End":"03:35.850","Text":"Well, that equals to 1 minus the probability of A union B."},{"Start":"03:35.850 ","End":"03:38.145","Text":"Now, that equals to 1 minus,"},{"Start":"03:38.145 ","End":"03:40.440","Text":"what\u0027s the probability of A union B?"},{"Start":"03:40.440 ","End":"03:41.790","Text":"We calculated that."},{"Start":"03:41.790 ","End":"03:44.260","Text":"That\u0027s 0.05."},{"Start":"03:44.260 ","End":"03:50.240","Text":"That equals to 0.95."},{"Start":"03:50.240 ","End":"03:53.705","Text":"Now, we could have calculated this directly."},{"Start":"03:53.705 ","End":"03:57.900","Text":"The chances of a no win,"},{"Start":"03:57.950 ","End":"04:01.665","Text":"let\u0027s call that event C. Well,"},{"Start":"04:01.665 ","End":"04:04.370","Text":"the probability of C, well,"},{"Start":"04:04.370 ","End":"04:10.369","Text":"that would be equal to the size of C divided by the size of Omega."},{"Start":"04:10.369 ","End":"04:14.225","Text":"Now the size of C would be 95 divided by 100,"},{"Start":"04:14.225 ","End":"04:20.250","Text":"and that equals to 0.95 or 95 percent."}],"ID":12757},{"Watched":false,"Name":"Exercise 14","Duration":"6m 7s","ChapterTopicVideoID":12275,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12275.jpeg","UploadDate":"2018-07-22T16:16:50.1730000","DurationForVideoObject":"PT6M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this question, we\u0027re given that the probability of event A is 0.3,"},{"Start":"00:04.740 ","End":"00:06.840","Text":"the probability of event B,"},{"Start":"00:06.840 ","End":"00:10.905","Text":"that\u0027s 0.25, and the probability of the union of 2 events,"},{"Start":"00:10.905 ","End":"00:13.110","Text":"A and B, well, that equals to 0.49."},{"Start":"00:13.110 ","End":"00:18.090","Text":"We\u0027re asked to calculate the probability of A intersect B."},{"Start":"00:18.090 ","End":"00:23.385","Text":"We know that the probability of A union B,"},{"Start":"00:23.385 ","End":"00:27.630","Text":"that equals to the probability of A plus the probability"},{"Start":"00:27.630 ","End":"00:32.370","Text":"of B minus the probability of A intersect B."},{"Start":"00:32.370 ","End":"00:35.640","Text":"Now, all we have to do is just plug in the numbers."},{"Start":"00:35.640 ","End":"00:38.400","Text":"The probability of A union B, well,"},{"Start":"00:38.400 ","End":"00:40.860","Text":"that\u0027s here, that\u0027s 0.49,"},{"Start":"00:40.860 ","End":"00:42.175","Text":"that\u0027s given to us."},{"Start":"00:42.175 ","End":"00:45.905","Text":"That equals to the probability of A, that\u0027s 0.3,"},{"Start":"00:45.905 ","End":"00:47.669","Text":"plus the probability of B,"},{"Start":"00:47.669 ","End":"00:53.135","Text":"that\u0027s 0.25, minus the probability of A intersect B."},{"Start":"00:53.135 ","End":"00:55.340","Text":"Now, let\u0027s just switch things around."},{"Start":"00:55.340 ","End":"00:58.058","Text":"The probability of A intersect B, well,"},{"Start":"00:58.058 ","End":"01:05.690","Text":"that equals to 0.3 plus 0.25 minus 0.49,"},{"Start":"01:05.690 ","End":"01:09.275","Text":"and that equals to 0.06."},{"Start":"01:09.275 ","End":"01:12.015","Text":"What about B? Well,"},{"Start":"01:12.015 ","End":"01:13.915","Text":"B asks us,"},{"Start":"01:13.915 ","End":"01:17.226","Text":"are A and B mutually exclusive events?"},{"Start":"01:17.226 ","End":"01:20.615","Text":"Well, in order for A and B to be mutually exclusive events,"},{"Start":"01:20.615 ","End":"01:24.060","Text":"then the probability of A intersect B,"},{"Start":"01:24.060 ","End":"01:26.895","Text":"well, that has to be equal to 0."},{"Start":"01:26.895 ","End":"01:30.435","Text":"We saw that it wasn\u0027t, it\u0027s 0.06."},{"Start":"01:30.435 ","End":"01:33.510","Text":"The answer to B is no."},{"Start":"01:33.510 ","End":"01:39.755","Text":"A and B are not mutually exclusive. Now, what about C?"},{"Start":"01:39.755 ","End":"01:46.134","Text":"C asks us, what\u0027s the probability of only A or only B occurring?"},{"Start":"01:46.134 ","End":"01:48.305","Text":"In order to answer that,"},{"Start":"01:48.305 ","End":"01:51.185","Text":"let\u0027s define what does that mean?"},{"Start":"01:51.185 ","End":"01:53.675","Text":"Let\u0027s define the event here."},{"Start":"01:53.675 ","End":"02:00.210","Text":"We\u0027re looking for the event of only A occurring or only B occurring."},{"Start":"02:00.210 ","End":"02:05.618","Text":"Only A occurring means that we have A and we have to take away B."},{"Start":"02:05.618 ","End":"02:08.765","Text":"It\u0027s A and not B."},{"Start":"02:08.765 ","End":"02:11.165","Text":"Now, what\u0027s only B occurring?"},{"Start":"02:11.165 ","End":"02:14.795","Text":"That\u0027s B and not A."},{"Start":"02:14.795 ","End":"02:24.660","Text":"Now, we were asked about the probability of this occurring or this occurring."},{"Start":"02:24.660 ","End":"02:28.415","Text":"This is the probability that we have to calculate."},{"Start":"02:28.415 ","End":"02:32.800","Text":"Now, let\u0027s do that using the Venn diagram."},{"Start":"02:32.800 ","End":"02:34.710","Text":"Here\u0027s a Venn diagram,"},{"Start":"02:34.710 ","End":"02:36.830","Text":"Here\u0027s event A and here is event B,"},{"Start":"02:36.830 ","End":"02:38.989","Text":"and they\u0027re both in the sample space."},{"Start":"02:38.989 ","End":"02:41.870","Text":"Now, what is this event right here,"},{"Start":"02:41.870 ","End":"02:43.685","Text":"A and not B?"},{"Start":"02:43.685 ","End":"02:49.024","Text":"Well, let\u0027s draw that area on the Venn diagram."},{"Start":"02:49.024 ","End":"02:51.485","Text":"Here it is. That\u0027s the area in yellow."},{"Start":"02:51.485 ","End":"02:55.145","Text":"Now, what about B and not A."},{"Start":"02:55.145 ","End":"02:57.320","Text":"That means that we have only B."},{"Start":"02:57.320 ","End":"03:00.655","Text":"Now, how does that look on the Venn diagram?"},{"Start":"03:00.655 ","End":"03:06.035","Text":"Excellent. Here we have both of these events right here,"},{"Start":"03:06.035 ","End":"03:08.390","Text":"A and not B, that\u0027s in yellow,"},{"Start":"03:08.390 ","End":"03:11.630","Text":"and B and not A; that\u0027s in green."},{"Start":"03:11.630 ","End":"03:14.965","Text":"Now, what are we asking for?"},{"Start":"03:14.965 ","End":"03:19.375","Text":"We know that the intersect of A and B,"},{"Start":"03:19.375 ","End":"03:22.160","Text":"that equals to 0.06,"},{"Start":"03:22.160 ","End":"03:25.990","Text":"that\u0027s right here. That\u0027s 0.06."},{"Start":"03:25.990 ","End":"03:30.104","Text":"We know also that the probability of A,"},{"Start":"03:30.104 ","End":"03:31.865","Text":"well, that equals to 0.3."},{"Start":"03:31.865 ","End":"03:33.425","Text":"All the circle here,"},{"Start":"03:33.425 ","End":"03:36.830","Text":"all A here, that equals to 0.3."},{"Start":"03:36.830 ","End":"03:38.750","Text":"The probability of B,"},{"Start":"03:38.750 ","End":"03:40.400","Text":"all the circle here,"},{"Start":"03:40.400 ","End":"03:44.310","Text":"that equals to 0.25."},{"Start":"03:44.310 ","End":"03:48.130","Text":"What\u0027s the probability of the yellow area?"},{"Start":"03:48.130 ","End":"03:52.490","Text":"Well, that will be 0.3 minus the intersect,"},{"Start":"03:52.490 ","End":"03:57.355","Text":"0.06, and that equals to 0.24."},{"Start":"03:57.355 ","End":"04:01.095","Text":"What about this area right here?"},{"Start":"04:01.095 ","End":"04:02.648","Text":"All of B,"},{"Start":"04:02.648 ","End":"04:09.674","Text":"that equals to 0.25 and we have to take away the intersect here, minus 0.06."},{"Start":"04:09.674 ","End":"04:13.315","Text":"That equals to 0.19."},{"Start":"04:13.315 ","End":"04:20.010","Text":"This then would be the probabilities of the yellow area and the green area."},{"Start":"04:20.010 ","End":"04:21.625","Text":"Getting back to here,"},{"Start":"04:21.625 ","End":"04:27.020","Text":"this then would be the probability of A intersect"},{"Start":"04:27.020 ","End":"04:34.220","Text":"not B plus the probability of B intersect not A."},{"Start":"04:34.220 ","End":"04:43.320","Text":"Now we\u0027ve calculated that the probability of A intersect not B, well, that\u0027s 0.24."},{"Start":"04:43.320 ","End":"04:46.520","Text":"The probability of B intersect not A,"},{"Start":"04:46.520 ","End":"04:53.450","Text":"well that equals to 0.19 and that equals to 0.43."},{"Start":"04:53.450 ","End":"04:57.050","Text":"Now, there\u0027s another way of calculating this event right here,"},{"Start":"04:57.050 ","End":"04:58.965","Text":"this probability right here."},{"Start":"04:58.965 ","End":"05:02.935","Text":"This then equals to the following."},{"Start":"05:02.935 ","End":"05:08.375","Text":"We\u0027re taking the probability of A union B."},{"Start":"05:08.375 ","End":"05:12.010","Text":"That means that either A occurs,"},{"Start":"05:12.010 ","End":"05:14.350","Text":"or B occurs, or both of them occurs."},{"Start":"05:14.350 ","End":"05:17.500","Text":"That\u0027s the definition of A union B."},{"Start":"05:17.500 ","End":"05:23.710","Text":"But we\u0027re taking away from that the probability of A intersect B,"},{"Start":"05:23.710 ","End":"05:26.950","Text":"the probability of both of them occurring."},{"Start":"05:26.950 ","End":"05:34.565","Text":"That\u0027s the definition of A and not B or B and not A."},{"Start":"05:34.565 ","End":"05:38.860","Text":"That\u0027s the definition of only A occurring or only B occurring."},{"Start":"05:38.860 ","End":"05:41.224","Text":"Now, A union B, well,"},{"Start":"05:41.224 ","End":"05:43.985","Text":"that equals to 0.49."},{"Start":"05:43.985 ","End":"05:49.440","Text":"We\u0027re given that. The probability of A intersect B,"},{"Start":"05:49.440 ","End":"05:51.930","Text":"well, we\u0027ve calculated that right here."},{"Start":"05:51.930 ","End":"05:54.150","Text":"That equals to 0.06,"},{"Start":"05:54.150 ","End":"05:57.750","Text":"and that equals to 0.43."},{"Start":"05:57.750 ","End":"06:02.180","Text":"We can see that we\u0027ve calculated the probability of only"},{"Start":"06:02.180 ","End":"06:07.530","Text":"A occurring or only B occurring in 2 different ways."}],"ID":12758},{"Watched":false,"Name":"Exercise 16 - Part a","Duration":"35s","ChapterTopicVideoID":12277,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12277.jpeg","UploadDate":"2019-01-31T23:25:56.2370000","DurationForVideoObject":"PT35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this question, we\u0027re asked to determine which of the following assertions are correct."},{"Start":"00:04.440 ","End":"00:08.340","Text":"Well, the first assertion is that A intersect B,"},{"Start":"00:08.340 ","End":"00:10.800","Text":"equals to B intersect A."},{"Start":"00:10.800 ","End":"00:16.095","Text":"Well, A intersect B is the area that\u0027s common to A and B."},{"Start":"00:16.095 ","End":"00:20.490","Text":"It\u0027s right here. What\u0027s B intersect A?"},{"Start":"00:20.490 ","End":"00:23.115","Text":"Well, that\u0027s the area that\u0027s common to B and A."},{"Start":"00:23.115 ","End":"00:26.580","Text":"Now it really doesn\u0027t matter how you say it,"},{"Start":"00:26.580 ","End":"00:30.555","Text":"we\u0027re looking for the common area between A and B,"},{"Start":"00:30.555 ","End":"00:34.690","Text":"so this statement is true."}],"ID":12760},{"Watched":false,"Name":"Exercise 16 - Part d","Duration":"2m 36s","ChapterTopicVideoID":12278,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12278.jpeg","UploadDate":"2019-01-31T23:26:09.8770000","DurationForVideoObject":"PT2M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In Section D, we have this;"},{"Start":"00:02.760 ","End":"00:09.830","Text":"the complimentary set of the intersect of A and B and C. That equals to,"},{"Start":"00:09.830 ","End":"00:14.970","Text":"not A union, not B union not C. Let\u0027s see if this is true;"},{"Start":"00:14.970 ","End":"00:17.565","Text":"A intersect B, intersect C,"},{"Start":"00:17.565 ","End":"00:19.620","Text":"that\u0027s this area right here,"},{"Start":"00:19.620 ","End":"00:21.810","Text":"that\u0027s what\u0027s common to A, B,"},{"Start":"00:21.810 ","End":"00:32.325","Text":"and C. So the complimentary set of that would be all the area outside of the intersect."},{"Start":"00:32.325 ","End":"00:35.460","Text":"Complimentary set of A intersect B,"},{"Start":"00:35.460 ","End":"00:38.295","Text":"intersect C, that\u0027s the area in red."},{"Start":"00:38.295 ","End":"00:42.585","Text":"Let\u0027s take a look at not A union not B, union not C. Well,"},{"Start":"00:42.585 ","End":"00:48.830","Text":"not A means that I\u0027m taking all the area outside of A;"},{"Start":"00:48.830 ","End":"00:52.895","Text":"so that\u0027s this area right here in blue."},{"Start":"00:52.895 ","End":"00:54.905","Text":"Now what about not B?"},{"Start":"00:54.905 ","End":"01:00.840","Text":"Not B is all the area outside of B,"},{"Start":"01:00.840 ","End":"01:04.290","Text":"and that\u0027ll be the area in red."},{"Start":"01:04.290 ","End":"01:05.970","Text":"What about not C?"},{"Start":"01:05.970 ","End":"01:11.005","Text":"Well, not C is all the area outside of C,"},{"Start":"01:11.005 ","End":"01:16.070","Text":"and that\u0027ll be the area in green right here."},{"Start":"01:16.070 ","End":"01:24.710","Text":"Now, here, the only area that\u0027s not colored is this area right here outlined in black,"},{"Start":"01:24.710 ","End":"01:27.365","Text":"or colored in black, right here."},{"Start":"01:27.365 ","End":"01:32.690","Text":"We can see that the colored area outside in red,"},{"Start":"01:32.690 ","End":"01:34.025","Text":"green, and blue,"},{"Start":"01:34.025 ","End":"01:37.295","Text":"that equals to this area that\u0027s colored in red."},{"Start":"01:37.295 ","End":"01:39.940","Text":"So that means that both areas are the same,"},{"Start":"01:39.940 ","End":"01:44.435","Text":"so that means that this expression is true."},{"Start":"01:44.435 ","End":"01:46.580","Text":"Now, if we take a look at this,"},{"Start":"01:46.580 ","End":"01:49.880","Text":"we see that this is De Morgan\u0027s rule for 3 events."},{"Start":"01:49.880 ","End":"01:52.505","Text":"Now if we recall, for 2 events,"},{"Start":"01:52.505 ","End":"01:56.360","Text":"it was this, it was A intersect B,"},{"Start":"01:56.360 ","End":"01:59.615","Text":"we take the complimentary set of A intersect B,"},{"Start":"01:59.615 ","End":"02:04.055","Text":"and that would be equal to not A union not B."},{"Start":"02:04.055 ","End":"02:08.990","Text":"Now, if we generalize De Morgan\u0027s rule into N events,"},{"Start":"02:08.990 ","End":"02:10.850","Text":"then this is what we get."},{"Start":"02:10.850 ","End":"02:13.085","Text":"We have, let\u0027s say N events,"},{"Start":"02:13.085 ","End":"02:14.690","Text":"I goes from 1 to N,"},{"Start":"02:14.690 ","End":"02:17.245","Text":"and we take the intersect"},{"Start":"02:17.245 ","End":"02:21.590","Text":"of these N events and then we take the complimentary set, well,"},{"Start":"02:21.590 ","End":"02:30.415","Text":"that will be equal to the union of the complimentary sets of N events."},{"Start":"02:30.415 ","End":"02:35.790","Text":"So this then is the generalized De Morgan\u0027s rule."}],"ID":12761},{"Watched":false,"Name":"Exercise 16 - Parts b-c","Duration":"1m 56s","ChapterTopicVideoID":12279,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12279.jpeg","UploadDate":"2019-01-31T23:26:16.0900000","DurationForVideoObject":"PT1M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.640","Text":"This assertion states that the complementary set of A Union B equals to A intersect B."},{"Start":"00:08.640 ","End":"00:13.065","Text":"What\u0027s the complementary set of A Union B?"},{"Start":"00:13.065 ","End":"00:20.234","Text":"A Union B is what is encased in the 2 circles combined."},{"Start":"00:20.234 ","End":"00:27.795","Text":"So the complimentary set is everything outside of both of the circles."},{"Start":"00:27.795 ","End":"00:30.645","Text":"That\u0027s this area right here."},{"Start":"00:30.645 ","End":"00:33.645","Text":"Now, what\u0027s A intersect B?"},{"Start":"00:33.645 ","End":"00:36.180","Text":"That\u0027s this area right here."},{"Start":"00:36.180 ","End":"00:43.880","Text":"We can see that these 2 areas are not the same so this assertion is false."},{"Start":"00:43.880 ","End":"00:48.330","Text":"This statement asserts that A intersects B intersect C. That"},{"Start":"00:48.330 ","End":"00:53.325","Text":"equals to A intersect B intersect C Union B."},{"Start":"00:53.325 ","End":"00:58.440","Text":"First of all, let\u0027s see what A intersect B intersect C is."},{"Start":"00:58.440 ","End":"01:00.720","Text":"That\u0027s the common area of A,"},{"Start":"01:00.720 ","End":"01:03.135","Text":"B, and C, and what\u0027s that?"},{"Start":"01:03.135 ","End":"01:06.190","Text":"That\u0027s this area right here."},{"Start":"01:06.190 ","End":"01:12.080","Text":"We\u0027re looking at the intersect of all 3 circles."},{"Start":"01:12.080 ","End":"01:16.680","Text":"What\u0027s the other side of the equation?"},{"Start":"01:16.680 ","End":"01:21.360","Text":"That\u0027s A intersect B intersect C Union B."},{"Start":"01:21.360 ","End":"01:23.820","Text":"What\u0027s C Union B here?"},{"Start":"01:23.820 ","End":"01:30.300","Text":"Here we see that C Union B is the yellow area right here."},{"Start":"01:30.300 ","End":"01:35.325","Text":"A intersect B intersects the yellow area right here."},{"Start":"01:35.325 ","End":"01:38.250","Text":"That\u0027s this area right here."},{"Start":"01:38.250 ","End":"01:42.230","Text":"A intersect B, intersect the yellow area,"},{"Start":"01:42.230 ","End":"01:44.345","Text":"that\u0027s this area right here."},{"Start":"01:44.345 ","End":"01:49.490","Text":"We see that that\u0027s not equal to A intersect B, intersect C,"},{"Start":"01:49.490 ","End":"01:55.710","Text":"the area in red so this statement is false."}],"ID":12762},{"Watched":false,"Name":"Exercise 17 - Part a","Duration":"1m 50s","ChapterTopicVideoID":12280,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12280.jpeg","UploadDate":"2019-01-31T23:26:21.4330000","DurationForVideoObject":"PT1M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"In this question, we have 2 events in a sample space."},{"Start":"00:03.420 ","End":"00:07.170","Text":"Now we\u0027re given that the probability of A equals 0.3"},{"Start":"00:07.170 ","End":"00:11.205","Text":"and the probability of B equals 0.2, and we\u0027re asked,"},{"Start":"00:11.205 ","End":"00:16.380","Text":"is it possible that the probability of the union of A and B,"},{"Start":"00:16.380 ","End":"00:21.420","Text":"would that be possible to be equal to 0.4 and under which conditions?"},{"Start":"00:21.420 ","End":"00:25.875","Text":"So let\u0027s just take a look at a Venn diagram right here."},{"Start":"00:25.875 ","End":"00:29.730","Text":"This is our Venn diagram and we see here the sample space,"},{"Start":"00:29.730 ","End":"00:31.740","Text":"and here\u0027s event A and event B."},{"Start":"00:31.740 ","End":"00:34.800","Text":"The probability here for event A that will be"},{"Start":"00:34.800 ","End":"00:39.025","Text":"0.3 and the probability for B, that\u0027ll be 0.2."},{"Start":"00:39.025 ","End":"00:44.040","Text":"Now we know that the probability of A union B,"},{"Start":"00:44.040 ","End":"00:47.540","Text":"well that equals to the probability of A plus"},{"Start":"00:47.540 ","End":"00:53.090","Text":"the probability of B minus the probability of A intersect B."},{"Start":"00:53.090 ","End":"00:59.750","Text":"We want to know if it\u0027s possible that this be equal to 0.4."},{"Start":"00:59.750 ","End":"01:02.060","Text":"So let\u0027s just plug in some numbers here."},{"Start":"01:02.060 ","End":"01:05.645","Text":"0.4, that would be equal to what?"},{"Start":"01:05.645 ","End":"01:09.980","Text":"Well, the probability of A that\u0027s 0.3 and the probability of B that\u0027s"},{"Start":"01:09.980 ","End":"01:14.660","Text":"0.2 minus the probability of A intersect B."},{"Start":"01:14.660 ","End":"01:18.800","Text":"Now, let\u0027s switch things around here and we have the probability of"},{"Start":"01:18.800 ","End":"01:25.775","Text":"A intersect B that equals to 0.3 plus 0.2 minus 0.4,"},{"Start":"01:25.775 ","End":"01:28.145","Text":"and that equals to 0.1."},{"Start":"01:28.145 ","End":"01:29.690","Text":"So to answer the question,"},{"Start":"01:29.690 ","End":"01:35.030","Text":"is it possible that the probability of A union B be equal to 0.4."},{"Start":"01:35.030 ","End":"01:37.310","Text":"Well, the answer to that is yes."},{"Start":"01:37.310 ","End":"01:43.130","Text":"If the probability of A intersect B equals to 0.1,"},{"Start":"01:43.130 ","End":"01:45.380","Text":"that means that this area right here,"},{"Start":"01:45.380 ","End":"01:49.530","Text":"that equals to 0.1."}],"ID":12763},{"Watched":false,"Name":"Exercise 17 - Part b","Duration":"1m 32s","ChapterTopicVideoID":12281,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12281.jpeg","UploadDate":"2019-01-31T23:26:25.3730000","DurationForVideoObject":"PT1M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In Section B, we\u0027re asked,"},{"Start":"00:02.160 ","End":"00:07.665","Text":"is it possible for the probability of the union of A and B be equal to 0.6?"},{"Start":"00:07.665 ","End":"00:11.610","Text":"Well again, the probability of A union B,"},{"Start":"00:11.610 ","End":"00:14.430","Text":"that equals to the probability of A,"},{"Start":"00:14.430 ","End":"00:16.440","Text":"plus the probability of B,"},{"Start":"00:16.440 ","End":"00:20.220","Text":"minus the probability of A intersect B."},{"Start":"00:20.220 ","End":"00:22.665","Text":"Now, the question is,"},{"Start":"00:22.665 ","End":"00:27.390","Text":"what is the maximum value that the probability of A union B,"},{"Start":"00:27.390 ","End":"00:29.995","Text":"what is the maximum value that this can have?"},{"Start":"00:29.995 ","End":"00:36.350","Text":"Well, this will have a maximum value when the probability of A intersect B equal to 0."},{"Start":"00:36.350 ","End":"00:40.955","Text":"Now, when does the probability of A intersect B equal to 0?"},{"Start":"00:40.955 ","End":"00:45.550","Text":"When A and B are mutually exclusive events."},{"Start":"00:45.550 ","End":"00:48.950","Text":"Here we have mutually exclusive events,"},{"Start":"00:48.950 ","End":"00:54.560","Text":"and the probability of A that\u0027s 0.3 and the probability of B that\u0027s 0.2."},{"Start":"00:54.560 ","End":"00:57.485","Text":"This will have a maximum value,"},{"Start":"00:57.485 ","End":"00:59.805","Text":"P of A union B,"},{"Start":"00:59.805 ","End":"01:09.185","Text":"when the intersect equals to 0 and there\u0027s no intersect and that equals to 0.3 plus 0.2,"},{"Start":"01:09.185 ","End":"01:11.855","Text":"which is equal to 0.5."},{"Start":"01:11.855 ","End":"01:16.595","Text":"Now, 0.5 is obviously less than 0.6."},{"Start":"01:16.595 ","End":"01:18.125","Text":"To answer the question,"},{"Start":"01:18.125 ","End":"01:22.895","Text":"is it possible that the probability of A union B equal to 0.6?"},{"Start":"01:22.895 ","End":"01:25.160","Text":"Well, the answer to that is, no,"},{"Start":"01:25.160 ","End":"01:31.560","Text":"because the maximum value that this probability can have is 0.5."}],"ID":12764},{"Watched":false,"Name":"Exercise 17 - Parts c-d","Duration":"1m 40s","ChapterTopicVideoID":12282,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12282.jpeg","UploadDate":"2019-01-31T23:26:30.0900000","DurationForVideoObject":"PT1M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this section we\u0027re asked,"},{"Start":"00:02.070 ","End":"00:04.260","Text":"if A and B are mutually exclusive events,"},{"Start":"00:04.260 ","End":"00:07.680","Text":"what\u0027s the probability of A union B?"},{"Start":"00:07.680 ","End":"00:09.900","Text":"Well, we\u0027ve answered that in the last section."},{"Start":"00:09.900 ","End":"00:14.790","Text":"The probability of A that\u0027s 0.3 and the probability of B that\u0027s 0.2."},{"Start":"00:14.790 ","End":"00:18.435","Text":"Now, the probability of A union B,"},{"Start":"00:18.435 ","End":"00:24.075","Text":"that equals to the probability of A plus the probability of B,"},{"Start":"00:24.075 ","End":"00:26.325","Text":"and that is it, why is that?"},{"Start":"00:26.325 ","End":"00:28.970","Text":"Well, because A and B are"},{"Start":"00:28.970 ","End":"00:34.715","Text":"mutually exclusive events and the probability of A intersect B that equals to 0."},{"Start":"00:34.715 ","End":"00:39.155","Text":"The probability of A union B, again,"},{"Start":"00:39.155 ","End":"00:45.425","Text":"that equals to the probability of A that\u0027s 0.3 plus the probability of B that\u0027s 0.2,"},{"Start":"00:45.425 ","End":"00:48.875","Text":"and that equals to 0.5."},{"Start":"00:48.875 ","End":"00:50.630","Text":"In section d we\u0027re asked,"},{"Start":"00:50.630 ","End":"00:52.490","Text":"if B is a subset of A,"},{"Start":"00:52.490 ","End":"00:55.370","Text":"what\u0027s the probability of A union B?"},{"Start":"00:55.370 ","End":"00:59.405","Text":"We\u0027re given that B is a subset of A,"},{"Start":"00:59.405 ","End":"01:01.940","Text":"now how would that look on a Venn diagram?"},{"Start":"01:01.940 ","End":"01:04.800","Text":"Here\u0027s our Venn diagram."},{"Start":"01:04.800 ","End":"01:08.155","Text":"We see that B is a subset of A."},{"Start":"01:08.155 ","End":"01:12.470","Text":"When we combine the values of B with"},{"Start":"01:12.470 ","End":"01:18.260","Text":"the values of A then all we\u0027re left with are the values of A."},{"Start":"01:18.260 ","End":"01:26.480","Text":"The probability of A union B that equals to the probability of A,"},{"Start":"01:26.480 ","End":"01:31.805","Text":"given that the probability of A is 0.3 and the probability of B is 0.2."},{"Start":"01:31.805 ","End":"01:36.680","Text":"But since the probability of A union B equals to the probability of A,"},{"Start":"01:36.680 ","End":"01:40.140","Text":"but that equals to 0.3."}],"ID":12765},{"Watched":false,"Name":"Exercise 18 - Part a","Duration":"6m 46s","ChapterTopicVideoID":12283,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12283.jpeg","UploadDate":"2019-01-31T23:26:52.3730000","DurationForVideoObject":"PT6M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this question, we\u0027re told that 30 percent of the people"},{"Start":"00:03.150 ","End":"00:05.925","Text":"in the community have accounts at Bank A,"},{"Start":"00:05.925 ","End":"00:08.130","Text":"28 percent have accounts at Bank B,"},{"Start":"00:08.130 ","End":"00:11.535","Text":"and 15 percent have accounts at Bank C. Now,"},{"Start":"00:11.535 ","End":"00:15.450","Text":"6 percent of the people have accounts at both Bank A and Bank B,"},{"Start":"00:15.450 ","End":"00:18.930","Text":"5 percent have accounts at both Bank A and Bank C,"},{"Start":"00:18.930 ","End":"00:24.270","Text":"and 4 percent have accounts at both Bank B and Bank C. Now,"},{"Start":"00:24.270 ","End":"00:29.025","Text":"1 percent of the people have accounts at all 3 banks."},{"Start":"00:29.025 ","End":"00:31.350","Text":"Now, before we answer any question,"},{"Start":"00:31.350 ","End":"00:37.630","Text":"let\u0027s just define our events and write all this information down."},{"Start":"00:38.420 ","End":"00:42.100","Text":"Here, we defined 3 events."},{"Start":"00:42.100 ","End":"00:45.160","Text":"A is having an account at Bank A,"},{"Start":"00:45.160 ","End":"00:51.515","Text":"B is having an account at Bank B, and C is having an account at Bank C. Now,"},{"Start":"00:51.515 ","End":"00:53.029","Text":"here are the probabilities."},{"Start":"00:53.029 ","End":"00:55.400","Text":"The probability of having an account at Bank A,"},{"Start":"00:55.400 ","End":"00:57.560","Text":"that\u0027s 30 percent, that\u0027s right here."},{"Start":"00:57.560 ","End":"00:59.975","Text":"At Bank B, that\u0027s 28 percent,"},{"Start":"00:59.975 ","End":"01:03.040","Text":"and at Bank C, that\u0027s 15 percent."},{"Start":"01:03.040 ","End":"01:09.420","Text":"6 percent have accounts at Bank A and B. That\u0027s right here."},{"Start":"01:09.420 ","End":"01:13.650","Text":"5 percent have a accounts Bank A and C,"},{"Start":"01:13.650 ","End":"01:19.190","Text":"and 4 percent have accounts at Bank B and C, that\u0027s right here."},{"Start":"01:19.190 ","End":"01:23.735","Text":"Now, 1 percent have accounts at all 3 Banks,"},{"Start":"01:23.735 ","End":"01:26.410","Text":"A, B, and C. Now,"},{"Start":"01:26.410 ","End":"01:30.435","Text":"as we said, before we answer any questions here,"},{"Start":"01:30.435 ","End":"01:35.565","Text":"let\u0027s first draw a Venn diagram and put all these information on it."},{"Start":"01:35.565 ","End":"01:39.665","Text":"Now, why Venn diagram and not a probability matrix?"},{"Start":"01:39.665 ","End":"01:43.400","Text":"Well, a probability matrix works on 2 events,"},{"Start":"01:43.400 ","End":"01:45.325","Text":"and here we have 3 events."},{"Start":"01:45.325 ","End":"01:49.360","Text":"That\u0027s why we\u0027ll be using the Venn diagram."},{"Start":"01:49.700 ","End":"01:52.670","Text":"Here we have our Venn diagram."},{"Start":"01:52.670 ","End":"01:56.630","Text":"Now, let\u0027s start adding on the probabilities."},{"Start":"01:56.630 ","End":"01:59.670","Text":"The probability of A and B and C,"},{"Start":"01:59.670 ","End":"02:03.930","Text":"that\u0027s the intersection of all 3 events, that\u0027s 0.01."},{"Start":"02:03.930 ","End":"02:05.400","Text":"We\u0027ll put that right here."},{"Start":"02:05.400 ","End":"02:09.680","Text":"This is the intersect of all 3 events, that\u0027s 0.01."},{"Start":"02:09.680 ","End":"02:13.730","Text":"Now, what about the probability of B and C?"},{"Start":"02:13.730 ","End":"02:15.700","Text":"Well, B and C, well,"},{"Start":"02:15.700 ","End":"02:18.050","Text":"that\u0027s this area right here."},{"Start":"02:18.050 ","End":"02:19.865","Text":"Now, all of this area,"},{"Start":"02:19.865 ","End":"02:23.525","Text":"the probability of all this area is 0.04,"},{"Start":"02:23.525 ","End":"02:25.580","Text":"but we have 0.01,"},{"Start":"02:25.580 ","End":"02:28.265","Text":"that\u0027s already in the intersect of all 3."},{"Start":"02:28.265 ","End":"02:31.475","Text":"Here, that\u0027ll be 0.03."},{"Start":"02:31.475 ","End":"02:32.885","Text":"Again, why is that?"},{"Start":"02:32.885 ","End":"02:36.440","Text":"Because 0.03 plus 0.01,"},{"Start":"02:36.440 ","End":"02:42.010","Text":"that equals to 0.04 and that\u0027s the probability of all of this area."},{"Start":"02:42.010 ","End":"02:44.160","Text":"Now, what about A and C?"},{"Start":"02:44.160 ","End":"02:45.920","Text":"The probability of A and C?"},{"Start":"02:45.920 ","End":"02:47.090","Text":"Well, that\u0027s right here,"},{"Start":"02:47.090 ","End":"02:49.265","Text":"that\u0027s this area right here."},{"Start":"02:49.265 ","End":"02:52.130","Text":"That equals to 0.05."},{"Start":"02:52.130 ","End":"02:58.130","Text":"If all this area equals to 0.05 and we already have 0.01 here,"},{"Start":"02:58.130 ","End":"03:02.205","Text":"then here, that\u0027ll be 0.04."},{"Start":"03:02.205 ","End":"03:05.760","Text":"Again, 0.01 plus 0.04,"},{"Start":"03:05.760 ","End":"03:08.595","Text":"that equals to 0.05."},{"Start":"03:08.595 ","End":"03:10.950","Text":"Now, A and B."},{"Start":"03:10.950 ","End":"03:13.425","Text":"Well, the probability of A and B,"},{"Start":"03:13.425 ","End":"03:15.975","Text":"that equals to 0.06."},{"Start":"03:15.975 ","End":"03:18.510","Text":"We already have 0.01 here,"},{"Start":"03:18.510 ","End":"03:21.300","Text":"so here that\u0027ll be 0.05."},{"Start":"03:21.300 ","End":"03:24.940","Text":"Why is that? Because 0.05 plus 0.01,"},{"Start":"03:24.940 ","End":"03:27.955","Text":"that equals to 0.06 and that\u0027s here, right here."},{"Start":"03:27.955 ","End":"03:31.555","Text":"Now, what about the rest of the probabilities?"},{"Start":"03:31.555 ","End":"03:35.170","Text":"Well, we know that the total probability of A,"},{"Start":"03:35.170 ","End":"03:37.735","Text":"that equals to 0.3."},{"Start":"03:37.735 ","End":"03:44.350","Text":"Now, what\u0027s the probability of this area right here outside of the intersects?"},{"Start":"03:44.350 ","End":"03:50.582","Text":"Well, that\u0027ll be 0.3 minus the sum of these 3 intersects,"},{"Start":"03:50.582 ","End":"03:54.690","Text":"0.05 plus 0.01 plus 0.04."},{"Start":"03:54.690 ","End":"03:56.445","Text":"Now, 5 plus 1 plus 4,"},{"Start":"03:56.445 ","End":"03:58.605","Text":"that equals to 0.1."},{"Start":"03:58.605 ","End":"04:04.065","Text":"0.3 minus 0.1, that equals to 0.2."},{"Start":"04:04.065 ","End":"04:10.995","Text":"Now, if we add up 0.2 plus 0.05 plus 0.01 plus 0.04,"},{"Start":"04:10.995 ","End":"04:13.365","Text":"that\u0027ll give us 0.3,"},{"Start":"04:13.365 ","End":"04:16.915","Text":"the total probability of event A."},{"Start":"04:16.915 ","End":"04:23.660","Text":"Now, we\u0027ll do the same thing for B and C. The probability of B is 0.28."},{"Start":"04:23.660 ","End":"04:28.580","Text":"We take away from that the probabilities of the intersects."},{"Start":"04:28.580 ","End":"04:35.630","Text":"That\u0027ll be 0.28 minus 0.05 minus 0.01 minus 0.03,"},{"Start":"04:35.630 ","End":"04:38.670","Text":"and that equals to 0.19."},{"Start":"04:38.750 ","End":"04:44.795","Text":"Here, the total probability of event C is 0.15."},{"Start":"04:44.795 ","End":"04:50.240","Text":"Let\u0027s take a way the probabilities of the intersect,"},{"Start":"04:50.240 ","End":"04:58.475","Text":"so that\u0027ll be 0.15 minus 0.03 minus 0.01 minus 0.04,"},{"Start":"04:58.475 ","End":"05:02.045","Text":"and that equals to 0.07."},{"Start":"05:02.045 ","End":"05:06.260","Text":"Now, what about the rest of the area?"},{"Start":"05:06.260 ","End":"05:08.990","Text":"Well, we know that the area,"},{"Start":"05:08.990 ","End":"05:11.330","Text":"the probability of the sample space,"},{"Start":"05:11.330 ","End":"05:13.190","Text":"that equals to 1."},{"Start":"05:13.190 ","End":"05:16.940","Text":"All the area outside of the 3 events,"},{"Start":"05:16.940 ","End":"05:21.295","Text":"well that\u0027s 1 minus all these probabilities right here,"},{"Start":"05:21.295 ","End":"05:26.115","Text":"and that equals to 0.41."},{"Start":"05:26.115 ","End":"05:32.540","Text":"Up to now, we\u0027ve just prepared the data in order to answer the question."},{"Start":"05:32.540 ","End":"05:34.560","Text":"Now, question A here,"},{"Start":"05:34.560 ","End":"05:35.990","Text":"section A asks,"},{"Start":"05:35.990 ","End":"05:41.015","Text":"what proportion of the adults have an account only at Bank B?"},{"Start":"05:41.015 ","End":"05:45.440","Text":"Well, that means that we\u0027re looking for the probability of"},{"Start":"05:45.440 ","End":"05:52.755","Text":"B and not A and not C. Well,"},{"Start":"05:52.755 ","End":"05:56.140","Text":"what is that area right here?"},{"Start":"05:56.270 ","End":"06:02.465","Text":"Let\u0027s take a look at this event on a fresh Venn diagram here,"},{"Start":"06:02.465 ","End":"06:05.390","Text":"B and not A and not C,"},{"Start":"06:05.390 ","End":"06:07.640","Text":"well, that\u0027s the area in yellow right here."},{"Start":"06:07.640 ","End":"06:12.510","Text":"I took the event B and I took away from that"},{"Start":"06:12.510 ","End":"06:19.805","Text":"all the intersects with A and all the intersects with C and that\u0027s what I was left with,"},{"Start":"06:19.805 ","End":"06:21.740","Text":"the yellow area right here."},{"Start":"06:21.740 ","End":"06:28.170","Text":"Now, if we take a look at the Venn diagram and where we prepared the probabilities,"},{"Start":"06:28.170 ","End":"06:30.740","Text":"the yellow area right here, well,"},{"Start":"06:30.740 ","End":"06:34.440","Text":"that equals to 0.19."},{"Start":"06:36.560 ","End":"06:39.390","Text":"That\u0027s a percentage of adults,"},{"Start":"06:39.390 ","End":"06:45.820","Text":"19 percent, who have accounts only at Bank B."}],"ID":12766},{"Watched":false,"Name":"Exercise 18 - Part b","Duration":"1m 4s","ChapterTopicVideoID":12284,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12284.jpeg","UploadDate":"2019-01-31T23:26:55.4970000","DurationForVideoObject":"PT1M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"In this section we\u0027re asked,"},{"Start":"00:01.860 ","End":"00:06.600","Text":"what\u0027s the probability of a given person having accounts at Bank A and Bank B,"},{"Start":"00:06.600 ","End":"00:09.930","Text":"but not at Bank C? What\u0027s the event?"},{"Start":"00:09.930 ","End":"00:15.825","Text":"The event is having an account at Bank A and at Bank B,"},{"Start":"00:15.825 ","End":"00:21.390","Text":"and not at C. We\u0027re looking for this probability right here."},{"Start":"00:21.390 ","End":"00:26.880","Text":"Well, let\u0027s take a look at this event on this Venn diagram."},{"Start":"00:26.880 ","End":"00:31.020","Text":"Here we see in yellow this event."},{"Start":"00:31.020 ","End":"00:34.140","Text":"We take the intersect of A and B,"},{"Start":"00:34.140 ","End":"00:36.510","Text":"that\u0027s this whole area right here,"},{"Start":"00:36.510 ","End":"00:39.690","Text":"but we take away event C,"},{"Start":"00:39.690 ","End":"00:41.570","Text":"and this is what we\u0027re left with,"},{"Start":"00:41.570 ","End":"00:42.875","Text":"the area in yellow."},{"Start":"00:42.875 ","End":"00:47.540","Text":"Now, let\u0027s take a look at the probability of this area right here."},{"Start":"00:47.540 ","End":"00:51.755","Text":"This is what we calculated in Section A when we prepared the data,"},{"Start":"00:51.755 ","End":"00:56.200","Text":"and that equals to 0.05."},{"Start":"00:56.200 ","End":"01:01.850","Text":"5 percent of the people have accounts at Bank A and at Bank B,"},{"Start":"01:01.850 ","End":"01:04.440","Text":"but not at Bank C."}],"ID":12767},{"Watched":false,"Name":"Exercise 18 - Part c","Duration":"1m 31s","ChapterTopicVideoID":12285,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12285.jpeg","UploadDate":"2019-01-31T23:26:59.8130000","DurationForVideoObject":"PT1M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:03.930","Text":"In this section, we\u0027re asked what\u0027s the probability of a given person"},{"Start":"00:03.930 ","End":"00:07.230","Text":"having an account at Bank A or Bank C,"},{"Start":"00:07.230 ","End":"00:09.705","Text":"but not at Bank B?"},{"Start":"00:09.705 ","End":"00:11.835","Text":"What\u0027s the event here?"},{"Start":"00:11.835 ","End":"00:19.845","Text":"That\u0027ll be A or C and not B,"},{"Start":"00:19.845 ","End":"00:23.600","Text":"and we\u0027re looking for this probability right here."},{"Start":"00:23.600 ","End":"00:31.560","Text":"Well again, let\u0027s take a look at this event on this Venn diagram. Here we go."},{"Start":"00:31.560 ","End":"00:35.835","Text":"A or C would be all A and all C,"},{"Start":"00:35.835 ","End":"00:38.745","Text":"but we\u0027re taking away B here."},{"Start":"00:38.745 ","End":"00:42.125","Text":"What\u0027s left is the area in yellow."},{"Start":"00:42.125 ","End":"00:45.740","Text":"Now, let\u0027s take a look at the probabilities here."},{"Start":"00:45.740 ","End":"00:56.370","Text":"Well, the probability of this area right here, that\u0027s 0.2."},{"Start":"00:58.700 ","End":"01:03.840","Text":"The probability of this area would be 0.04,"},{"Start":"01:04.790 ","End":"01:09.470","Text":"and the probability of this area right here,"},{"Start":"01:09.470 ","End":"01:13.460","Text":"well, that equals to 0.07."},{"Start":"01:13.460 ","End":"01:15.365","Text":"When we add all these up,"},{"Start":"01:15.365 ","End":"01:22.320","Text":"we get 0.31 or 31 percent."},{"Start":"01:22.320 ","End":"01:28.850","Text":"That\u0027s the probability of a given person having an account at Bank A or Bank C,"},{"Start":"01:28.850 ","End":"01:31.650","Text":"but not at Bank B."}],"ID":12768},{"Watched":false,"Name":"Exercise 18 - Part d","Duration":"2m 35s","ChapterTopicVideoID":12286,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12286.jpeg","UploadDate":"2019-01-31T23:27:07.5630000","DurationForVideoObject":"PT2M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"In this section, we\u0027re asked what proportion of people have accounts at only 1 bank."},{"Start":"00:05.400 ","End":"00:07.170","Text":"Now, we\u0027re not told which bank."},{"Start":"00:07.170 ","End":"00:11.910","Text":"It could be either Bank A and not B or C,"},{"Start":"00:11.910 ","End":"00:15.450","Text":"or Bank B and not A and C,"},{"Start":"00:15.450 ","End":"00:18.630","Text":"or Bank C and not A and B."},{"Start":"00:18.630 ","End":"00:20.700","Text":"Let\u0027s write this event."},{"Start":"00:20.700 ","End":"00:26.970","Text":"The event is A and not B"},{"Start":"00:26.970 ","End":"00:34.950","Text":"and not C or not A and B,"},{"Start":"00:34.950 ","End":"00:41.385","Text":"and not C or not A"},{"Start":"00:41.385 ","End":"00:49.685","Text":"and not B and C. We\u0027re looking for this probability right here."},{"Start":"00:49.685 ","End":"00:54.090","Text":"How would that look on the Venn diagram?"},{"Start":"00:54.680 ","End":"01:01.860","Text":"Here we go, A and not B and not C while we color A,"},{"Start":"01:01.860 ","End":"01:07.275","Text":"but we take away B and C. We\u0027re left with this area right here."},{"Start":"01:07.275 ","End":"01:11.300","Text":"We do the same thing for B where we take B and"},{"Start":"01:11.300 ","End":"01:15.135","Text":"we take away A and C. That\u0027s what we\u0027re left with,"},{"Start":"01:15.135 ","End":"01:17.205","Text":"that\u0027s this area right here."},{"Start":"01:17.205 ","End":"01:22.605","Text":"For C, we take C and we take away A and B."},{"Start":"01:22.605 ","End":"01:24.795","Text":"This is what we\u0027re left with."},{"Start":"01:24.795 ","End":"01:31.595","Text":"Because these events are mutually exclusive,"},{"Start":"01:31.595 ","End":"01:34.740","Text":"then we can add all these probabilities up."},{"Start":"01:34.740 ","End":"01:40.530","Text":"That\u0027ll be the probability of a and not B and not"},{"Start":"01:40.530 ","End":"01:47.795","Text":"C plus the probability of not A and B and not C,"},{"Start":"01:47.795 ","End":"01:51.995","Text":"plus the probability of not A and"},{"Start":"01:51.995 ","End":"01:58.260","Text":"not B and C. What are these probabilities?"},{"Start":"01:58.260 ","End":"02:00.950","Text":"The probability of A and not B and not C,"},{"Start":"02:00.950 ","End":"02:03.335","Text":"that\u0027s this area right here."},{"Start":"02:03.335 ","End":"02:11.605","Text":"That equals to 0.2 plus the probability of not A and B and not C,"},{"Start":"02:11.605 ","End":"02:19.025","Text":"that\u0027s this area right here and that equals to 0.19 plus the probability"},{"Start":"02:19.025 ","End":"02:26.955","Text":"of not A and not B and C with and that\u0027s this area right here and that equals to 0.07."},{"Start":"02:26.955 ","End":"02:34.810","Text":"That equals to 0.46 or 46 percent."}],"ID":12769},{"Watched":false,"Name":"Exercise 18 - Part e","Duration":"2m 23s","ChapterTopicVideoID":12287,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12287.jpeg","UploadDate":"2019-01-31T23:27:14.3030000","DurationForVideoObject":"PT2M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.740","Text":"In this section we\u0027re asked,"},{"Start":"00:01.740 ","End":"00:05.700","Text":"what proportion of people have a accounts at only 2 banks?"},{"Start":"00:05.700 ","End":"00:09.870","Text":"That means that they could have accounts at bank A and B and not C,"},{"Start":"00:09.870 ","End":"00:13.815","Text":"A and C and not B or B and C and not A."},{"Start":"00:13.815 ","End":"00:16.740","Text":"How do we define this event?"},{"Start":"00:16.740 ","End":"00:26.670","Text":"Well, again, it could be A and B and not C or"},{"Start":"00:26.670 ","End":"00:33.675","Text":"could have A and C and not B or"},{"Start":"00:33.675 ","End":"00:42.995","Text":"B and C and not A and we\u0027re looking for this probability right here."},{"Start":"00:42.995 ","End":"00:47.249","Text":"Well, how does this look on the Venn diagram?"},{"Start":"00:47.720 ","End":"00:52.650","Text":"Here we go, what\u0027s A and B and not C?"},{"Start":"00:52.650 ","End":"00:55.020","Text":"Well, we take the intersect of A and B,"},{"Start":"00:55.020 ","End":"00:58.800","Text":"that\u0027s this area right here and we take away C,"},{"Start":"00:58.800 ","End":"01:02.210","Text":"so we\u0027re left with this area in yellow here."},{"Start":"01:02.210 ","End":"01:06.845","Text":"The area A intersect C and not B."},{"Start":"01:06.845 ","End":"01:09.410","Text":"Well, that\u0027s this area right here,"},{"Start":"01:09.410 ","End":"01:12.920","Text":"A intersect C is this area right here and we\u0027re taking away B,"},{"Start":"01:12.920 ","End":"01:19.420","Text":"so we\u0027re left with this yellow area and B intersect C intersect not A."},{"Start":"01:19.420 ","End":"01:23.720","Text":"Well, that\u0027s this area right here and we\u0027re taking away A,"},{"Start":"01:23.720 ","End":"01:25.715","Text":"so we\u0027re left with this yellow area."},{"Start":"01:25.715 ","End":"01:30.200","Text":"Now, what are the probabilities of the areas in yellow?"},{"Start":"01:30.200 ","End":"01:33.990","Text":"Well, then B, A and B and not C,"},{"Start":"01:33.990 ","End":"01:35.850","Text":"that\u0027s this thing right here."},{"Start":"01:35.850 ","End":"01:44.295","Text":"That\u0027s 0.05, that\u0027s right here plus this area right here,"},{"Start":"01:44.295 ","End":"01:53.630","Text":"A and C and not B that\u0027s 0.04 plus this area right here, that\u0027s 0.03."},{"Start":"01:53.630 ","End":"01:57.215","Text":"Now why can we add these probabilities?"},{"Start":"01:57.215 ","End":"02:02.315","Text":"Because these areas right here these events are mutually exclusive,"},{"Start":"02:02.315 ","End":"02:07.250","Text":"so by the fact that they\u0027re mutually exclusive,"},{"Start":"02:07.250 ","End":"02:10.235","Text":"I can add these things up these probability self,"},{"Start":"02:10.235 ","End":"02:17.790","Text":"and that comes out to 0.12 or 12 percent,"},{"Start":"02:17.790 ","End":"02:23.580","Text":"so that\u0027s the proportion of people who have accounts at only 2 banks."}],"ID":12770},{"Watched":false,"Name":"Exercise 18 - Parts f-g","Duration":"2m 37s","ChapterTopicVideoID":12288,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12288.jpeg","UploadDate":"2019-01-31T23:27:22.0100000","DurationForVideoObject":"PT2M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this section, we\u0027re asked what\u0027s the probability of an adult"},{"Start":"00:03.780 ","End":"00:07.170","Text":"having an account at none of the street banks?"},{"Start":"00:07.170 ","End":"00:09.330","Text":"What would be the event here?"},{"Start":"00:09.330 ","End":"00:11.640","Text":"They\u0027ll be not A;"},{"Start":"00:11.640 ","End":"00:16.770","Text":"a person doesn\u0027t have an account at Bank A and not B;"},{"Start":"00:16.770 ","End":"00:19.005","Text":"a person doesn\u0027t have an account at Bank B,"},{"Start":"00:19.005 ","End":"00:20.760","Text":"and not C;"},{"Start":"00:20.760 ","End":"00:26.385","Text":"a person does not have an account at Bank C. We\u0027re looking for this probability."},{"Start":"00:26.385 ","End":"00:30.850","Text":"Now how does this event look on a Venn diagram?"},{"Start":"00:31.910 ","End":"00:37.960","Text":"If a person doesn\u0027t have an account at any 1 of the banks,"},{"Start":"00:37.960 ","End":"00:43.185","Text":"then that means that it is outside of all 3 circles."},{"Start":"00:43.185 ","End":"00:51.905","Text":"Now what\u0027s the probability of the area outside of all the 3 circles that equals to 0.41?"},{"Start":"00:51.905 ","End":"00:54.680","Text":"In this section, we\u0027re asked what proportion of people"},{"Start":"00:54.680 ","End":"00:57.470","Text":"have an account in at least 1 of the banks?"},{"Start":"00:57.470 ","End":"01:01.790","Text":"What we\u0027re looking for the event A, union B,"},{"Start":"01:01.790 ","End":"01:07.970","Text":"union C. That\u0027s the event of a person having an account in at least 1 of the banks,"},{"Start":"01:07.970 ","End":"01:10.910","Text":"so we\u0027re looking for this probability right here."},{"Start":"01:10.910 ","End":"01:15.970","Text":"Well how does this event look on this Venn diagram?"},{"Start":"01:15.970 ","End":"01:20.660","Text":"Here we go. This is the area in yellow right here."},{"Start":"01:20.660 ","End":"01:22.910","Text":"That\u0027s the union of all 3 events,"},{"Start":"01:22.910 ","End":"01:30.175","Text":"so what\u0027s the probability of the union of these 3 events?"},{"Start":"01:30.175 ","End":"01:34.550","Text":"Well I can add all the probabilities here,"},{"Start":"01:34.550 ","End":"01:44.735","Text":"or I can say that this equals to 1 minus the probability outside of the circles; 0.41."},{"Start":"01:44.735 ","End":"01:50.465","Text":"Now that equals to 0.59 or 59 percent."},{"Start":"01:50.465 ","End":"01:53.930","Text":"Now I want to remind you what we did in the last section;"},{"Start":"01:53.930 ","End":"01:57.620","Text":"Section F. In Section F,"},{"Start":"01:57.620 ","End":"02:05.045","Text":"we calculated the probability of not A and not B and not C. This is the area in yellow."},{"Start":"02:05.045 ","End":"02:08.120","Text":"Now if we compare these 2 Venn diagrams,"},{"Start":"02:08.120 ","End":"02:11.660","Text":"we see that we have complimentary sets here."},{"Start":"02:11.660 ","End":"02:14.585","Text":"That means that we have this."},{"Start":"02:14.585 ","End":"02:20.580","Text":"We have the probability of A union B, union C,"},{"Start":"02:20.580 ","End":"02:27.635","Text":"that equals to 1 minus the probability of not A intersect not B,"},{"Start":"02:27.635 ","End":"02:31.640","Text":"intersect not C. Well this is exactly De Morgan\u0027s rule."},{"Start":"02:31.640 ","End":"02:34.340","Text":"Here we see that in action right here,"},{"Start":"02:34.340 ","End":"02:38.890","Text":"we see that visually in action."}],"ID":12771},{"Watched":false,"Name":"Exercise 19 - Part a","Duration":"7m 22s","ChapterTopicVideoID":12289,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12289.jpeg","UploadDate":"2019-01-31T23:27:45.8670000","DurationForVideoObject":"PT7M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this question, we\u0027re told that a company published the following information about"},{"Start":"00:04.350 ","End":"00:08.505","Text":"people over 21: 40 percent of the people have a Visa card,"},{"Start":"00:08.505 ","End":"00:11.010","Text":"52 percent have a Mastercard,"},{"Start":"00:11.010 ","End":"00:13.605","Text":"and 20 percent have an American Express card."},{"Start":"00:13.605 ","End":"00:18.660","Text":"Now, 50 percent have both a Visa card and a Mastercard,"},{"Start":"00:18.660 ","End":"00:23.355","Text":"8 percent have both a Mastercard and an American Express card,"},{"Start":"00:23.355 ","End":"00:28.980","Text":"and 7 percent have both a Visa card and an American Express card,"},{"Start":"00:28.980 ","End":"00:31.560","Text":"13 percent have none of these cards."},{"Start":"00:31.560 ","End":"00:36.615","Text":"We\u0027re asked what proportion of the people have all 3 cards?"},{"Start":"00:36.615 ","End":"00:38.835","Text":"Before we answer this question,"},{"Start":"00:38.835 ","End":"00:42.770","Text":"let\u0027s just define our events and write down the information that"},{"Start":"00:42.770 ","End":"00:47.435","Text":"was given to us. Here we go."},{"Start":"00:47.435 ","End":"00:51.080","Text":"A is the event of a person having a Visa card,"},{"Start":"00:51.080 ","End":"00:53.690","Text":"B is an event of having a Mastercard,"},{"Start":"00:53.690 ","End":"00:57.425","Text":"and C is the event of having an American Express."},{"Start":"00:57.425 ","End":"00:59.810","Text":"Now, the probability of A,"},{"Start":"00:59.810 ","End":"01:03.995","Text":"probability of having a Visa card that\u0027s 40 percent, that\u0027s right here."},{"Start":"01:03.995 ","End":"01:08.465","Text":"The probability of having a Mastercard that\u0027s 52 percent,"},{"Start":"01:08.465 ","End":"01:14.570","Text":"and the probability of having an American Express is 20 percent,"},{"Start":"01:14.570 ","End":"01:21.380","Text":"and the probability of having both a Visa card and a Mastercard,"},{"Start":"01:21.380 ","End":"01:24.215","Text":"well, that\u0027s 15 percent."},{"Start":"01:24.215 ","End":"01:30.800","Text":"The probability of having a Mastercard and an American Express that\u0027s 8 percent,"},{"Start":"01:30.800 ","End":"01:38.675","Text":"and the probability of having a Visa and American Express, that\u0027s 7 percent."},{"Start":"01:38.675 ","End":"01:45.830","Text":"Now, here\u0027s a trick. Thirteen percent of the people have none of these cards,"},{"Start":"01:45.830 ","End":"01:51.230","Text":"so the probability of not A and not B and not C,"},{"Start":"01:51.230 ","End":"01:54.050","Text":"that equals to 0.13."},{"Start":"01:54.050 ","End":"01:58.360","Text":"Here we completed writing everything down."},{"Start":"01:58.360 ","End":"02:01.595","Text":"Let\u0027s see if I can simplify this a little bit."},{"Start":"02:01.595 ","End":"02:08.590","Text":"The probability of not A and not B and not C,"},{"Start":"02:08.590 ","End":"02:17.685","Text":"that equals to 1 minus the probability of A union B union C. That\u0027s De Morgan\u0027s rule."},{"Start":"02:17.685 ","End":"02:19.770","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"02:19.770 ","End":"02:24.235","Text":"0.13 that\u0027s this probability right here where we\u0027re given that."},{"Start":"02:24.235 ","End":"02:30.335","Text":"That equals to 1 minus the probability of A union B, union C. Now,"},{"Start":"02:30.335 ","End":"02:31.880","Text":"let\u0027s just switch things around."},{"Start":"02:31.880 ","End":"02:36.095","Text":"The probability of A union B, union C,"},{"Start":"02:36.095 ","End":"02:41.900","Text":"that equals to 1 minus 0.13 and that equals to 0.87."},{"Start":"02:41.900 ","End":"02:43.985","Text":"Now, how does that help us?"},{"Start":"02:43.985 ","End":"02:47.495","Text":"Well, let\u0027s just recall something a little bit simpler."},{"Start":"02:47.495 ","End":"02:50.690","Text":"The probability of A union B,"},{"Start":"02:50.690 ","End":"02:55.010","Text":"that equals to the probability of A plus"},{"Start":"02:55.010 ","End":"03:01.445","Text":"the probability of B minus the probability of A intersect B."},{"Start":"03:01.445 ","End":"03:07.430","Text":"Now, this equation is true when we have 2 events,"},{"Start":"03:07.430 ","End":"03:10.145","Text":"but what happens when we have 3 events?"},{"Start":"03:10.145 ","End":"03:14.540","Text":"We want to calculate the proportion of people who have all 3 cards."},{"Start":"03:14.540 ","End":"03:20.080","Text":"That means that we\u0027re looking for the probability of A intersect B,"},{"Start":"03:20.080 ","End":"03:29.390","Text":"intersect C. If we expand this type of equation for 3 events,"},{"Start":"03:29.390 ","End":"03:33.230","Text":"then we can get this element right here."},{"Start":"03:33.230 ","End":"03:35.875","Text":"Let\u0027s do that."},{"Start":"03:35.875 ","End":"03:38.645","Text":"Here we have a Venn diagram."},{"Start":"03:38.645 ","End":"03:47.540","Text":"Now, we want to calculate the probability of A union B union C. Now, what would that be?"},{"Start":"03:47.540 ","End":"03:53.390","Text":"First of all, let\u0027s count the probabilities of A plus"},{"Start":"03:53.390 ","End":"03:59.560","Text":"the probability of B plus the probability of C. But hold on,"},{"Start":"03:59.560 ","End":"04:04.265","Text":"when we counted the probability of A plus the probability of B,"},{"Start":"04:04.265 ","End":"04:07.965","Text":"we counted this area twice."},{"Start":"04:07.965 ","End":"04:11.599","Text":"We counted this area and then we counted this area."},{"Start":"04:11.599 ","End":"04:14.600","Text":"This area right here in red was counted twice,"},{"Start":"04:14.600 ","End":"04:21.065","Text":"so we have to take away the probability of A intersect B."},{"Start":"04:21.065 ","End":"04:26.280","Text":"But we also have to take away this area right here."},{"Start":"04:26.530 ","End":"04:36.645","Text":"The probability of A intersect C because we counted this area,"},{"Start":"04:36.645 ","End":"04:40.310","Text":"this is all of A and then we counted all of C."},{"Start":"04:40.310 ","End":"04:45.025","Text":"So this area A intersect C was counted twice."},{"Start":"04:45.025 ","End":"04:53.435","Text":"The same thing for B and C. We have to take away the probability of B intersect C. Again,"},{"Start":"04:53.435 ","End":"05:00.620","Text":"we counted B and then we counted C. This area right here was counted twice,"},{"Start":"05:00.620 ","End":"05:02.420","Text":"so we have to take that away."},{"Start":"05:02.420 ","End":"05:08.090","Text":"But when we took away these areas in red, well,"},{"Start":"05:08.090 ","End":"05:16.725","Text":"we also took away the intersect of all 3 areas 3 times."},{"Start":"05:16.725 ","End":"05:19.160","Text":"Once for A intersect B,"},{"Start":"05:19.160 ","End":"05:21.095","Text":"1 for A intersect C,"},{"Start":"05:21.095 ","End":"05:29.360","Text":"and 1 for B intersect C. That means that we have to add this section back on again."},{"Start":"05:29.360 ","End":"05:32.900","Text":"That\u0027ll be plus the probability of"},{"Start":"05:32.900 ","End":"05:40.940","Text":"A intersect B intersect C. But this is exactly what we\u0027re trying to calculate."},{"Start":"05:40.940 ","End":"05:45.050","Text":"Let\u0027s now plug in the numbers because we have all of"},{"Start":"05:45.050 ","End":"05:49.805","Text":"the probabilities here. Let\u0027s get started."},{"Start":"05:49.805 ","End":"05:57.950","Text":"The probability of A union B union C. We\u0027ve calculated that right here, that\u0027s 0.87."},{"Start":"05:57.950 ","End":"06:00.425","Text":"That\u0027ll be 0.87,"},{"Start":"06:00.425 ","End":"06:05.180","Text":"and that equals to the probability of A that\u0027s 0.4,"},{"Start":"06:05.180 ","End":"06:07.615","Text":"plus the probability of B,"},{"Start":"06:07.615 ","End":"06:14.490","Text":"that\u0027s 0.52, plus the probability of C that\u0027s 0.2."},{"Start":"06:14.490 ","End":"06:19.115","Text":"Now, we have to take away the probability of the intersects."},{"Start":"06:19.115 ","End":"06:21.740","Text":"The probability of A intersect B,"},{"Start":"06:21.740 ","End":"06:26.390","Text":"that\u0027s 0.15, that\u0027ll be minus 0.15,"},{"Start":"06:26.390 ","End":"06:30.170","Text":"minus the probability of A intersect C,"},{"Start":"06:30.170 ","End":"06:37.985","Text":"well that\u0027s 0.07 minus the intersect of B and C,"},{"Start":"06:37.985 ","End":"06:45.200","Text":"that\u0027s 0.08, and we have to add this thing back on again,"},{"Start":"06:45.200 ","End":"06:54.560","Text":"plus the probability of A intersect B intersect C. This is what we\u0027re looking for."},{"Start":"06:54.560 ","End":"06:59.070","Text":"Now, when we do all the calculations."},{"Start":"06:59.070 ","End":"07:01.800","Text":"After switching everything around,"},{"Start":"07:01.800 ","End":"07:07.955","Text":"the probability of A intersect B intersect C. Well,"},{"Start":"07:07.955 ","End":"07:14.075","Text":"that comes out to 0.05 and this is exactly what we\u0027re looking for."},{"Start":"07:14.075 ","End":"07:21.520","Text":"The probability of a person having all 3 cards, that\u0027s 5 percent"}],"ID":12772},{"Watched":false,"Name":"Exercise 19 - Part b","Duration":"1m 23s","ChapterTopicVideoID":12290,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12290.jpeg","UploadDate":"2019-01-31T23:27:50.7600000","DurationForVideoObject":"PT1M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.225","Text":"In section B we\u0027re asked,"},{"Start":"00:02.225 ","End":"00:04.720","Text":"what proportion of people have both a Master card"},{"Start":"00:04.720 ","End":"00:08.020","Text":"and a Visa card but not an American Express cards?"},{"Start":"00:08.020 ","End":"00:09.385","Text":"What\u0027s the event?"},{"Start":"00:09.385 ","End":"00:13.625","Text":"The event is A and B,"},{"Start":"00:13.625 ","End":"00:18.790","Text":"and not C and we\u0027re looking for this probability right here."},{"Start":"00:18.790 ","End":"00:23.785","Text":"What\u0027s this event on the Venn diagram right here?"},{"Start":"00:23.785 ","End":"00:27.860","Text":"That\u0027s this area right here."},{"Start":"00:28.370 ","End":"00:31.005","Text":"That\u0027s A and B,"},{"Start":"00:31.005 ","End":"00:32.140","Text":"that\u0027s this area,"},{"Start":"00:32.140 ","End":"00:35.935","Text":"but not C. In the last section,"},{"Start":"00:35.935 ","End":"00:41.390","Text":"we\u0027ve calculated this area right here, that\u0027s 0.05."},{"Start":"00:41.840 ","End":"00:46.160","Text":"That\u0027s the probability of a person"},{"Start":"00:46.160 ","End":"00:51.485","Text":"having an American Express and a Visa and a Master card."},{"Start":"00:51.485 ","End":"00:56.830","Text":"We were given that the probability of A intersect B,"},{"Start":"00:56.830 ","End":"00:58.305","Text":"all of this area,"},{"Start":"00:58.305 ","End":"01:00.795","Text":"that equals to 0.15,"},{"Start":"01:00.795 ","End":"01:09.980","Text":"so the area in red would be 0.15 minus 0.05 and that equals to 0.1."},{"Start":"01:09.980 ","End":"01:11.960","Text":"So this area right here,"},{"Start":"01:11.960 ","End":"01:14.315","Text":"that equals to 0.1."},{"Start":"01:14.315 ","End":"01:22.890","Text":"So the probability of A intersect B intersect not C that equals to 0.1."}],"ID":12773},{"Watched":false,"Name":"Exercise 19 - Part c","Duration":"5m 22s","ChapterTopicVideoID":12291,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12291.jpeg","UploadDate":"2019-01-31T23:28:12.3870000","DurationForVideoObject":"PT5M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"In this section we\u0027re asked,"},{"Start":"00:01.830 ","End":"00:04.710","Text":"what proportion of people have only 1 of the 3 cards?"},{"Start":"00:04.710 ","End":"00:07.860","Text":"How did we define this event right here?"},{"Start":"00:07.860 ","End":"00:12.525","Text":"Having only 1 of the 3 cards means that I have A,"},{"Start":"00:12.525 ","End":"00:14.310","Text":"and not B,"},{"Start":"00:14.310 ","End":"00:22.965","Text":"and not C. Or I have not A and B and not C,"},{"Start":"00:22.965 ","End":"00:30.840","Text":"or I have not A and not B and C. This is the event."},{"Start":"00:30.840 ","End":"00:34.900","Text":"How does this event look on a Venn diagram?"},{"Start":"00:35.030 ","End":"00:42.185","Text":"Here we go, this is how this event looks on the Venn diagram. Why is that?"},{"Start":"00:42.185 ","End":"00:48.980","Text":"A and not B and not C. I took all of A and I took away"},{"Start":"00:48.980 ","End":"00:56.015","Text":"the intersect with B and the intersect with C. That\u0027s this shaded area right here."},{"Start":"00:56.015 ","End":"00:59.760","Text":"What about not A and B and not C?"},{"Start":"00:59.760 ","End":"01:05.280","Text":"I took B and then I took away the intersect with A,"},{"Start":"01:05.280 ","End":"01:08.945","Text":"the intersect with C. That\u0027s the shaded area right here."},{"Start":"01:08.945 ","End":"01:13.100","Text":"This thing, not A and not B and C. Well I took"},{"Start":"01:13.100 ","End":"01:17.735","Text":"C and then I took away the intersects with A and with B."},{"Start":"01:17.735 ","End":"01:21.480","Text":"What\u0027s this probability right here?"},{"Start":"01:21.480 ","End":"01:28.240","Text":"As we can see, these events right here,"},{"Start":"01:28.240 ","End":"01:32.525","Text":"shaded in yellow, they\u0027re mutually exclusive."},{"Start":"01:32.525 ","End":"01:33.860","Text":"I\u0027m not saying that A, B,"},{"Start":"01:33.860 ","End":"01:35.690","Text":"and C are mutually exclusive,"},{"Start":"01:35.690 ","End":"01:40.850","Text":"but I\u0027m saying that these elements right here, they\u0027re mutually exclusive."},{"Start":"01:40.850 ","End":"01:45.360","Text":"As such, I could add these probabilities up"},{"Start":"01:45.360 ","End":"01:50.555","Text":"this thing would be equal to the probability of A and not B,"},{"Start":"01:50.555 ","End":"01:52.255","Text":"and not C,"},{"Start":"01:52.255 ","End":"01:55.965","Text":"plus the probability of not A,"},{"Start":"01:55.965 ","End":"01:58.440","Text":"and B and not C,"},{"Start":"01:58.440 ","End":"02:01.650","Text":"plus the probability of not A,"},{"Start":"02:01.650 ","End":"02:07.760","Text":"and not B and C. What are these probabilities?"},{"Start":"02:07.760 ","End":"02:10.130","Text":"How can we calculate them?"},{"Start":"02:10.130 ","End":"02:12.320","Text":"From the last section,"},{"Start":"02:12.320 ","End":"02:17.570","Text":"we know that the intersect of all 3 events that equals to 0.05."},{"Start":"02:17.570 ","End":"02:20.390","Text":"That\u0027s the probability of the intersect of the 3 events."},{"Start":"02:20.390 ","End":"02:22.250","Text":"Let\u0027s write that down here."},{"Start":"02:22.250 ","End":"02:25.460","Text":"That\u0027ll be 0.05."},{"Start":"02:25.460 ","End":"02:30.500","Text":"What about this area right here."},{"Start":"02:30.500 ","End":"02:33.610","Text":"The intersect between A and B,"},{"Start":"02:33.610 ","End":"02:36.410","Text":"well that equals to 0.15,"},{"Start":"02:36.410 ","End":"02:40.865","Text":"but that\u0027s all of this area right here."},{"Start":"02:40.865 ","End":"02:45.500","Text":"What\u0027s the probability of this area right here."},{"Start":"02:45.500 ","End":"02:53.125","Text":"Well that be 0.15 minus 0.05, that\u0027s 0.1."},{"Start":"02:53.125 ","End":"02:57.580","Text":"What about this area right here?"},{"Start":"02:57.860 ","End":"03:03.330","Text":"That\u0027s the intersect between B and C and all that area,"},{"Start":"03:03.330 ","End":"03:07.415","Text":"the probability of all that area is 0.08."},{"Start":"03:07.415 ","End":"03:12.140","Text":"We have to take away 0.05 from there."},{"Start":"03:12.140 ","End":"03:17.015","Text":"This area right here, that\u0027s 0.03."},{"Start":"03:17.015 ","End":"03:20.635","Text":"What about this area right here?"},{"Start":"03:20.635 ","End":"03:26.850","Text":"That\u0027s the intersect between A and C. The intersect between A and C is 0.07."},{"Start":"03:26.850 ","End":"03:34.465","Text":"We have to take away 0.05 here, so that\u0027s 0.02."},{"Start":"03:34.465 ","End":"03:37.265","Text":"Now that we have this,"},{"Start":"03:37.265 ","End":"03:43.105","Text":"what are the probabilities of the shaded areas in yellow?"},{"Start":"03:43.105 ","End":"03:46.505","Text":"Let\u0027s take a look at this area right here."},{"Start":"03:46.505 ","End":"03:49.550","Text":"The shaded area in yellow here,"},{"Start":"03:49.550 ","End":"03:56.364","Text":"that\u0027s all the probability of A minus the probabilities of the intersect."},{"Start":"03:56.364 ","End":"04:01.065","Text":"That\u0027ll be 0.4 minus 0.1,"},{"Start":"04:01.065 ","End":"04:05.055","Text":"minus 0.05, minus 0.02."},{"Start":"04:05.055 ","End":"04:08.350","Text":"That equals to 0.23."},{"Start":"04:08.750 ","End":"04:12.135","Text":"What about the shaded area here?"},{"Start":"04:12.135 ","End":"04:17.835","Text":"That\u0027s the probability of B that\u0027s 0.52 minus the intersects."},{"Start":"04:17.835 ","End":"04:22.535","Text":"That\u0027ll be 0.52 minus 0.1, minus 0.05,"},{"Start":"04:22.535 ","End":"04:27.463","Text":"minus 0.03, and that equals to 0.34."},{"Start":"04:27.463 ","End":"04:31.670","Text":"What about this shaded area right here?"},{"Start":"04:31.670 ","End":"04:37.190","Text":"That\u0027s the probability of C 0.2 minus the intersect."},{"Start":"04:37.190 ","End":"04:43.145","Text":"That\u0027ll be minus 0.202 plus minus 0.05,"},{"Start":"04:43.145 ","End":"04:48.415","Text":"minus 0.03 and that leaves me with 0.1 here."},{"Start":"04:48.415 ","End":"04:52.490","Text":"Let\u0027s now add up all these probabilities."},{"Start":"04:52.490 ","End":"04:56.050","Text":"The probability of A and not B and not C,"},{"Start":"04:56.050 ","End":"04:58.250","Text":"that equals to 0.23,"},{"Start":"04:58.250 ","End":"05:02.735","Text":"plus the probability of not A and B and not C,"},{"Start":"05:02.735 ","End":"05:09.135","Text":"that\u0027s 0.34, plus the probability of not A and not B and C,"},{"Start":"05:09.135 ","End":"05:11.370","Text":"that equals to 0.1."},{"Start":"05:11.370 ","End":"05:14.490","Text":"All these comes to 0.67."},{"Start":"05:14.490 ","End":"05:22.720","Text":"So 67 percent of the people have only 1 of the 3 cards."}],"ID":12774},{"Watched":false,"Name":"Exercise 20","Duration":"57s","ChapterTopicVideoID":12292,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12292.jpeg","UploadDate":"2019-01-31T23:28:14.9500000","DurationForVideoObject":"PT57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this question, we\u0027re asked to prove this expression right here."},{"Start":"00:04.620 ","End":"00:06.690","Text":"Now, this is a mathematical proof,"},{"Start":"00:06.690 ","End":"00:08.595","Text":"so let\u0027s get to work."},{"Start":"00:08.595 ","End":"00:14.580","Text":"Now, the probability of not A and not B, well,"},{"Start":"00:14.580 ","End":"00:16.344","Text":"according to De Morgan,"},{"Start":"00:16.344 ","End":"00:21.665","Text":"that equals to 1 minus the probability of A union B."},{"Start":"00:21.665 ","End":"00:26.045","Text":"Now, what\u0027s the probability of A union B?"},{"Start":"00:26.045 ","End":"00:31.455","Text":"Well, we know that that equals to the probability of A"},{"Start":"00:31.455 ","End":"00:38.555","Text":"plus the probability of B minus the probability of A intersect B."},{"Start":"00:38.555 ","End":"00:45.440","Text":"Let\u0027s just open the brackets here and that equals to 1 minus the probability of A"},{"Start":"00:45.440 ","End":"00:52.395","Text":"minus the probability of B plus the probability of A intersect B."},{"Start":"00:52.395 ","End":"00:56.730","Text":"This is exactly what we were asked to prove here."}],"ID":12775},{"Watched":false,"Name":"Exercise 21","Duration":"2m 9s","ChapterTopicVideoID":12293,"CourseChapterTopicPlaylistID":264985,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12293.jpeg","UploadDate":"2019-01-31T23:28:22.6170000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this question, we\u0027re given A and B that are events in the sample space."},{"Start":"00:04.980 ","End":"00:11.085","Text":"We\u0027re asked, is it correct to say that the chances of exactly 1 event occurring are"},{"Start":"00:11.085 ","End":"00:14.070","Text":"the probability of A plus the probability of B"},{"Start":"00:14.070 ","End":"00:18.660","Text":"minus 2 times the probability of the intercept?"},{"Start":"00:18.660 ","End":"00:25.245","Text":"First of all, let\u0027s take a look at a Venn diagram with 2 events in a sample space."},{"Start":"00:25.245 ","End":"00:28.080","Text":"Here\u0027s my Venn diagram."},{"Start":"00:28.080 ","End":"00:30.090","Text":"I have Event A,"},{"Start":"00:30.090 ","End":"00:32.085","Text":"that\u0027s right here,"},{"Start":"00:32.085 ","End":"00:38.490","Text":"and I have Event B, that\u0027s right here."},{"Start":"00:38.490 ","End":"00:42.380","Text":"Also, I have the intersect right here,"},{"Start":"00:42.380 ","End":"00:45.810","Text":"which is this area right here."},{"Start":"00:47.630 ","End":"00:53.450","Text":"If I\u0027m looking at the chances of exactly 1 event occurring,"},{"Start":"00:53.450 ","End":"00:56.330","Text":"how would that look on a Venn diagram?"},{"Start":"00:56.330 ","End":"00:59.195","Text":"Again, here\u0027s my Venn diagram."},{"Start":"00:59.195 ","End":"01:01.700","Text":"If exactly 1 event occurs,"},{"Start":"01:01.700 ","End":"01:05.490","Text":"then A should occur and"},{"Start":"01:05.490 ","End":"01:12.795","Text":"not B or B will occur and not A."},{"Start":"01:12.795 ","End":"01:16.265","Text":"Let\u0027s just see what the probabilities are."},{"Start":"01:16.265 ","End":"01:22.210","Text":"Here, that\u0027ll be the probability of A plus the probability of B."},{"Start":"01:22.210 ","End":"01:25.835","Text":"Here, if exactly 1 event occurs,"},{"Start":"01:25.835 ","End":"01:32.450","Text":"then I don\u0027t want to calculate the probability of the intersect right here."},{"Start":"01:32.450 ","End":"01:36.725","Text":"If I add the probability of A plus the probability of B,"},{"Start":"01:36.725 ","End":"01:40.580","Text":"then I\u0027ve added the intersect twice."},{"Start":"01:40.580 ","End":"01:45.710","Text":"So I have to take 2 times the probability of"},{"Start":"01:45.710 ","End":"01:52.130","Text":"the Intersect away in order to get only the shaded areas here."},{"Start":"01:52.130 ","End":"01:56.750","Text":"Yes, it\u0027s correct to say that the chances"},{"Start":"01:56.750 ","End":"02:00.920","Text":"of exactly 1 event occurring is this expression right here,"},{"Start":"02:00.920 ","End":"02:03.935","Text":"the probability of A plus the probability of B"},{"Start":"02:03.935 ","End":"02:09.510","Text":"minus 2 times the probability of the intersect."}],"ID":12776}],"Thumbnail":null,"ID":264985}]

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