Conditional Probability in a Uniform Sample Space
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Conditional Probability in a Non-Uniform Space
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- Tutorial 1
- Tutorial 2
- Exercise 1 - Part a
- Exercise 1 - Parts b-c
- Exercise 1 - Parts d-e
- Exercise 2 - Part a
- Exercise 2 - Parts b-c
- Exercise 2 - Part d
- Exercise 3 - Part a
- Exercise 3 - Parts b-c
- Exercise 3 - Part d
- Exercise 4 - Parts a-b
- Exercise 4 - Parts c-e
- Exercise 4 - Part f
- Exercise 5 - Part a
- Exercise 5 - Part b
- Exercise 5 - Part c

Tree Diagrams the Bayes Formula and the Complete Probability Formula
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- Tutorial 1
- Tutorial 2
- Tutorial 3
- Tutorial 4
- Tutorial 5
- Exercise 1 - Part a
- Exercise 1 - Part b
- Exercise 2 - Part a
- Exercise 2 - Part b
- Exercise 2 - Part c
- Exercise 2 - Part d
- Exercise 3 - Part a
- Exercise 3 - Part b
- Exercise 4 - Part a
- Exercise 4 - Part b
- Exercise 4 - Part c
- Exercise 5 - Part a
- Exercise 5 - Part b
- Exercise 5 - Part c
- Exercise 6 - Part a
- Exercise 6 - Part b
- Exercise 7 - Part a
- Exercise 7 - Parts b-c
- Exercise 8 - Part a
- Exercise 8 - Part b
- Exercise 8 - Part c
- Exercise 8 - Part d
- Exercise 9
- Exercise 10 - Part a
- Exercise 10 - Part b

Dependent and Independent Events
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Summary Questions
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- Introduction
- Exercise 1 - Part a
- Exercise 1 - Parts b-c
- Exercise 1 - Part d
- Exercise 2 - Part a
- Exercise 2 - Parts b-d
- Exercise 3 - Part a
- Exercise 3 - Parts b-d
- Exercise 4 - Part a
- Exercise 4 - Parts b-c
- Exercise 4 - Part d
- Exercise 5
- Exercise 6 - Part a
- Exercise 6 - Parts b-c
- Exercise 7 - Part a
- Exercise 7 - Parts b-c
- Exercise 7 - Part d
- Exercise 8

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[{"Name":"Conditional Probability in a Uniform Sample Space","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"1m 38s","ChapterTopicVideoID":12391,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:02.040 ","End":"00:05.670","Text":"conditional probability in a uniform sample space."},{"Start":"00:05.670 ","End":"00:09.090","Text":"Now, we sometimes need to calculate the probability of an event"},{"Start":"00:09.090 ","End":"00:12.960","Text":"occurring when we have information about a different event."},{"Start":"00:12.960 ","End":"00:17.580","Text":"Conditional probability is the chances that a given event will"},{"Start":"00:17.580 ","End":"00:22.410","Text":"occur when it\u0027s known that a different event did or did not occur."},{"Start":"00:22.410 ","End":"00:25.560","Text":"Now, the probability of A,"},{"Start":"00:25.560 ","End":"00:27.600","Text":"given that B has already occurred,"},{"Start":"00:27.600 ","End":"00:29.130","Text":"well, we write it like this,"},{"Start":"00:29.130 ","End":"00:32.895","Text":"the probability of A given B."},{"Start":"00:32.895 ","End":"00:34.695","Text":"Now how do we calculate that?"},{"Start":"00:34.695 ","End":"00:37.630","Text":"Well, when the sample space is uniform,"},{"Start":"00:37.630 ","End":"00:40.880","Text":"then the probability of A given B, well,"},{"Start":"00:40.880 ","End":"00:46.235","Text":"that equals to the size of the intersect"},{"Start":"00:46.235 ","End":"00:52.475","Text":"of the 2 events divided by the size of the event that\u0027s given to us."},{"Start":"00:52.475 ","End":"00:57.244","Text":"Now, what do we mean when we say a sample space is uniform?"},{"Start":"00:57.244 ","End":"00:59.600","Text":"When we roll the die, well,"},{"Start":"00:59.600 ","End":"01:05.270","Text":"the result form a sample space that\u0027s uniform. Why is that?"},{"Start":"01:05.270 ","End":"01:09.930","Text":"Well, we have the same probability for each and every result."},{"Start":"01:09.930 ","End":"01:15.760","Text":"The probability of getting a 1 is the same as the probability of getting a 5."},{"Start":"01:15.760 ","End":"01:19.955","Text":"That\u0027s what we mean when we say that the sample space is uniform."},{"Start":"01:19.955 ","End":"01:22.639","Text":"Again, when the sample space is uniform,"},{"Start":"01:22.639 ","End":"01:25.603","Text":"then the probability of A given B, well,"},{"Start":"01:25.603 ","End":"01:30.290","Text":"that equals to the size of the intersect of the 2 events"},{"Start":"01:30.290 ","End":"01:35.420","Text":"divided by the size of the event that\u0027s given to us."},{"Start":"01:35.420 ","End":"01:38.340","Text":"Let\u0027s take a look at an example now."}],"ID":12870},{"Watched":false,"Name":"Tutorial 2","Duration":"2m 54s","ChapterTopicVideoID":12392,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this example, we throw a die."},{"Start":"00:02.370 ","End":"00:07.290","Text":"Now we define the event A as getting an even result in the end,"},{"Start":"00:07.290 ","End":"00:10.770","Text":"the event B as getting a result that\u0027s larger than 3"},{"Start":"00:10.770 ","End":"00:15.405","Text":"and we\u0027re asked to calculate the probability of A given B."},{"Start":"00:15.405 ","End":"00:21.295","Text":"Well, by definition, the probability of A given B,"},{"Start":"00:21.295 ","End":"00:31.330","Text":"well that equals to the size of the intersect of A and B divided by the size of B."},{"Start":"00:31.330 ","End":"00:38.075","Text":"The size of the intersect of A and B divided by the size of the event that\u0027s given."},{"Start":"00:38.075 ","End":"00:41.135","Text":"Okay, let\u0027s just see what we have here."},{"Start":"00:41.135 ","End":"00:42.670","Text":"We\u0027re throwing a die,"},{"Start":"00:42.670 ","End":"00:47.780","Text":"and there are 6 possible results when you throw a die from 1-6."},{"Start":"00:47.780 ","End":"00:51.695","Text":"Now, A is getting an even result,"},{"Start":"00:51.695 ","End":"00:55.845","Text":"so what are the possible results for A?"},{"Start":"00:55.845 ","End":"01:00.180","Text":"That\u0027ll be 2, 4, and 6."},{"Start":"01:00.180 ","End":"01:02.870","Text":"What about the possible results for B?"},{"Start":"01:02.870 ","End":"01:06.725","Text":"Well, B says that it\u0027s a result that\u0027s larger than 3,"},{"Start":"01:06.725 ","End":"01:08.360","Text":"so it has to be 4,"},{"Start":"01:08.360 ","End":"01:11.575","Text":"5, and 6."},{"Start":"01:11.575 ","End":"01:15.590","Text":"Now, what about A intersect B?"},{"Start":"01:15.590 ","End":"01:20.365","Text":"Well, that\u0027s the results that are common to both A and B."},{"Start":"01:20.365 ","End":"01:25.670","Text":"The results that are common to A and B are 4 and 6."},{"Start":"01:26.510 ","End":"01:30.590","Text":"Now let\u0027s take a look at the size of A."},{"Start":"01:30.590 ","End":"01:32.395","Text":"Well, the size of A,"},{"Start":"01:32.395 ","End":"01:34.500","Text":"well that equals to 3."},{"Start":"01:34.500 ","End":"01:38.310","Text":"We have 3 results. The size of B?"},{"Start":"01:38.310 ","End":"01:41.255","Text":"Well, I have 3 results here."},{"Start":"01:41.255 ","End":"01:45.160","Text":"Now, what about the size of A intersect B?"},{"Start":"01:45.160 ","End":"01:47.505","Text":"Well, that\u0027s 2."},{"Start":"01:47.505 ","End":"01:50.055","Text":"I only have 2 results here."},{"Start":"01:50.055 ","End":"01:53.990","Text":"Let\u0027s now take a look at this."},{"Start":"01:53.990 ","End":"01:58.160","Text":"Let\u0027s calculate the probability of A given B."},{"Start":"01:58.160 ","End":"02:02.210","Text":"Well, that\u0027s the size of A intersect B divided by the size of B."},{"Start":"02:02.210 ","End":"02:04.100","Text":"Well, the size of A intersect B,"},{"Start":"02:04.100 ","End":"02:08.115","Text":"that\u0027s 2 divided by the size of B."},{"Start":"02:08.115 ","End":"02:10.335","Text":"Well, the size of B is 3."},{"Start":"02:10.335 ","End":"02:13.490","Text":"The probability of A given B,"},{"Start":"02:13.490 ","End":"02:15.995","Text":"the probability of getting an even result,"},{"Start":"02:15.995 ","End":"02:19.310","Text":"given that the result is greater than 3?"},{"Start":"02:19.310 ","End":"02:20.870","Text":"Well, that\u0027s 2/3."},{"Start":"02:20.870 ","End":"02:23.000","Text":"Now, let\u0027s see why is that?"},{"Start":"02:23.000 ","End":"02:26.540","Text":"Well, we\u0027re given that the result is larger than 3,"},{"Start":"02:26.540 ","End":"02:34.775","Text":"so we have 3 possible results that we have to choose from and getting an even result."},{"Start":"02:34.775 ","End":"02:36.896","Text":"Well, out of the 3 results,"},{"Start":"02:36.896 ","End":"02:38.980","Text":"well, 2 of them are even."},{"Start":"02:38.980 ","End":"02:41.600","Text":"That\u0027s these guys right here,"},{"Start":"02:41.600 ","End":"02:44.270","Text":"that\u0027s the intersect between A and B."},{"Start":"02:44.270 ","End":"02:49.459","Text":"We have here a probability of 2/3 the size"},{"Start":"02:49.459 ","End":"02:55.350","Text":"of the intersect divided by the size of the event that\u0027s given."}],"ID":12871},{"Watched":false,"Name":"Exercise 1","Duration":"3m 12s","ChapterTopicVideoID":12393,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this question, we\u0027re told that an even number is chosen randomly between 0 and 9, and we\u0027re asked "},{"Start":"00:04.185 ","End":"00:09.220","Text":"what are the chances that the number is larger than 6?"},{"Start":"00:09.350 ","End":"00:13.860","Text":"Here we\u0027re dealing with a conditional probability."},{"Start":"00:13.860 ","End":"00:15.900","Text":"Why is that? Well, first of all,"},{"Start":"00:15.900 ","End":"00:18.030","Text":"we have 2 events."},{"Start":"00:18.030 ","End":"00:23.335","Text":"We have an event that an even number is chosen randomly"},{"Start":"00:23.335 ","End":"00:28.965","Text":"and the next event is that the number is larger than 6."},{"Start":"00:28.965 ","End":"00:33.735","Text":"Let\u0027s, first of all, define the 2 events."},{"Start":"00:33.735 ","End":"00:36.255","Text":"What\u0027s given to us?"},{"Start":"00:36.255 ","End":"00:42.345","Text":"Well, we\u0027re given that we have an even number."},{"Start":"00:42.345 ","End":"00:46.060","Text":"We\u0027re told an even number is chosen randomly."},{"Start":"00:46.060 ","End":"00:51.320","Text":"B here, that\u0027s the event that\u0027s given."},{"Start":"00:51.320 ","End":"00:53.780","Text":"Well, what numbers do we have here?"},{"Start":"00:53.780 ","End":"00:55.235","Text":"What are the even numbers?"},{"Start":"00:55.235 ","End":"00:57.335","Text":"Well, that\u0027s 0, 2,"},{"Start":"00:57.335 ","End":"01:00.690","Text":"4, 6, and 8."},{"Start":"01:00.690 ","End":"01:06.105","Text":"We\u0027re asked about the probability of this."},{"Start":"01:06.105 ","End":"01:10.385","Text":"What are the probabilities that the number is larger than 6?"},{"Start":"01:10.385 ","End":"01:12.725","Text":"Well, what is the event?"},{"Start":"01:12.725 ","End":"01:16.210","Text":"What are the numbers that are larger than 6?"},{"Start":"01:16.210 ","End":"01:17.525","Text":"Well, that\u0027ll be 7,"},{"Start":"01:17.525 ","End":"01:20.220","Text":"8, and 9."},{"Start":"01:20.220 ","End":"01:24.215","Text":"Now, since this is a conditional probability,"},{"Start":"01:24.215 ","End":"01:29.950","Text":"then we\u0027re looking for the probability of A given B."},{"Start":"01:29.950 ","End":"01:36.710","Text":"What\u0027s the probability of getting a number that\u0027s larger than 6,"},{"Start":"01:36.710 ","End":"01:40.630","Text":"given that we have even numbers?"},{"Start":"01:40.630 ","End":"01:44.990","Text":"Well, we know the definition of a conditional probability,"},{"Start":"01:44.990 ","End":"01:50.380","Text":"that\u0027s the size of A intersect B divided by the size of B."},{"Start":"01:50.380 ","End":"01:53.225","Text":"The size of the event that\u0027s given to us."},{"Start":"01:53.225 ","End":"01:56.925","Text":"Well, what\u0027s the size of B?"},{"Start":"01:56.925 ","End":"02:00.045","Text":"We have here 1, 2, 3, 4, 5."},{"Start":"02:00.045 ","End":"02:01.500","Text":"That\u0027s the size of B."},{"Start":"02:01.500 ","End":"02:03.705","Text":"We have 5 numbers."},{"Start":"02:03.705 ","End":"02:07.640","Text":"What about A intersect B?"},{"Start":"02:07.640 ","End":"02:11.255","Text":"Well, what are the numbers in A intersect B?"},{"Start":"02:11.255 ","End":"02:14.300","Text":"What are the common numbers between A and B?"},{"Start":"02:14.300 ","End":"02:18.195","Text":"Well, the only common number is 8."},{"Start":"02:18.195 ","End":"02:21.915","Text":"What\u0027s the size of A intersect B?"},{"Start":"02:21.915 ","End":"02:24.180","Text":"That equals to 1."},{"Start":"02:24.180 ","End":"02:26.910","Text":"Here, let\u0027s just plug in the numbers."},{"Start":"02:26.910 ","End":"02:28.680","Text":"The size of A intersect B,"},{"Start":"02:28.680 ","End":"02:30.383","Text":"well that\u0027s 1,"},{"Start":"02:30.383 ","End":"02:34.060","Text":"divided by the size of B, that\u0027s 5."},{"Start":"02:34.060 ","End":"02:37.925","Text":"That equals to 0.2 or 20%."},{"Start":"02:37.925 ","End":"02:43.205","Text":"The probability of getting a number that\u0027s larger than 6,"},{"Start":"02:43.205 ","End":"02:46.925","Text":"given that we choose an even number,"},{"Start":"02:46.925 ","End":"02:48.800","Text":"well, that\u0027s 1 out of 5,"},{"Start":"02:48.800 ","End":"02:50.420","Text":"and it makes sense."},{"Start":"02:50.420 ","End":"02:54.755","Text":"We\u0027re told that we have only even numbers to choose from,"},{"Start":"02:54.755 ","End":"02:56.885","Text":"so we have 5 even numbers."},{"Start":"02:56.885 ","End":"03:02.235","Text":"How many numbers are larger than 6 out of the 5 numbers here?"},{"Start":"03:02.235 ","End":"03:05.145","Text":"Well, only 1 is larger than 6, and that\u0027s the 8."},{"Start":"03:05.145 ","End":"03:08.070","Text":"We have 1 out of 5."},{"Start":"03:08.070 ","End":"03:13.030","Text":"That\u0027s the probability of A given B."}],"ID":12872},{"Watched":false,"Name":"Exercise 2","Duration":"2m 15s","ChapterTopicVideoID":12394,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this question, we were told that Joe throws a die."},{"Start":"00:02.640 ","End":"00:05.610","Text":"We\u0027re asked, what are the chances that the result was"},{"Start":"00:05.610 ","End":"00:10.210","Text":"4 if it\u0027s known that the result was an even number?"},{"Start":"00:10.220 ","End":"00:13.140","Text":"Here we know that we\u0027re dealing with"},{"Start":"00:13.140 ","End":"00:16.350","Text":"a conditional probability because we\u0027re given something."},{"Start":"00:16.350 ","End":"00:18.090","Text":"What are we given?"},{"Start":"00:18.090 ","End":"00:20.310","Text":"We\u0027re given that when throwing a die,"},{"Start":"00:20.310 ","End":"00:22.620","Text":"the result was an even number."},{"Start":"00:22.620 ","End":"00:27.615","Text":"When we throw a die, we get the results from 1-6."},{"Start":"00:27.615 ","End":"00:30.945","Text":"B here, that\u0027s the event that\u0027s given."},{"Start":"00:30.945 ","End":"00:32.625","Text":"Now, when throwing a die,"},{"Start":"00:32.625 ","End":"00:37.185","Text":"we\u0027re told that the result was even so B contains the values 2,"},{"Start":"00:37.185 ","End":"00:40.110","Text":"4, and 6 only."},{"Start":"00:40.110 ","End":"00:43.945","Text":"We\u0027re asked about A, what\u0027s A?"},{"Start":"00:43.945 ","End":"00:48.180","Text":"A is the event of getting a 4."},{"Start":"00:48.920 ","End":"00:56.465","Text":"Now, these are the events and we\u0027re asked what\u0027s the probability of getting a 4,"},{"Start":"00:56.465 ","End":"01:00.320","Text":"that\u0027s A, given that when we threw the die,"},{"Start":"01:00.320 ","End":"01:02.710","Text":"we got an even number?"},{"Start":"01:02.710 ","End":"01:05.700","Text":"This is what we\u0027re looking for."},{"Start":"01:05.700 ","End":"01:13.970","Text":"This by definition is the size of A intersect B divided by the size of B,"},{"Start":"01:13.970 ","End":"01:15.410","Text":"the event that\u0027s given."},{"Start":"01:15.410 ","End":"01:17.120","Text":"Let\u0027s calculate that."},{"Start":"01:17.120 ","End":"01:19.795","Text":"Well, the size of B,"},{"Start":"01:19.795 ","End":"01:21.975","Text":"that equals to 1, 2, 3."},{"Start":"01:21.975 ","End":"01:24.270","Text":"We have 3 elements here,"},{"Start":"01:24.270 ","End":"01:26.430","Text":"a 2, a 4, and a 6."},{"Start":"01:26.430 ","End":"01:30.005","Text":"Now, what about A intersect B?"},{"Start":"01:30.005 ","End":"01:31.460","Text":"What are the values here?"},{"Start":"01:31.460 ","End":"01:34.160","Text":"What\u0027s common to both A and B?"},{"Start":"01:34.160 ","End":"01:37.885","Text":"Well, the value 4."},{"Start":"01:37.885 ","End":"01:41.715","Text":"That means that the size of A intersect B,"},{"Start":"01:41.715 ","End":"01:43.860","Text":"that equals to 1."},{"Start":"01:43.860 ","End":"01:46.940","Text":"Getting back to our conditional probability,"},{"Start":"01:46.940 ","End":"01:49.400","Text":"the size of A intersect B,"},{"Start":"01:49.400 ","End":"01:53.000","Text":"that\u0027s 1, divided by the size of B,"},{"Start":"01:53.000 ","End":"01:55.010","Text":"that\u0027s 3, and again,"},{"Start":"01:55.010 ","End":"02:03.090","Text":"that makes perfect sense because we\u0027re told that we were rolling an even number."},{"Start":"02:03.090 ","End":"02:05.300","Text":"Out of the 3 even numbers,"},{"Start":"02:05.300 ","End":"02:07.940","Text":"what\u0027s the probability of getting a 4?"},{"Start":"02:07.940 ","End":"02:11.015","Text":"Well, that\u0027s 1 out of the 3, that\u0027s a third."},{"Start":"02:11.015 ","End":"02:15.930","Text":"So that\u0027s the probability of A given B."}],"ID":12873},{"Watched":false,"Name":"Exercise 3","Duration":"2m 50s","ChapterTopicVideoID":12395,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.715","Text":"In this question, we\u0027re rolling a pair of dice."},{"Start":"00:02.715 ","End":"00:04.890","Text":"Now we define 2 events."},{"Start":"00:04.890 ","End":"00:10.400","Text":"Event A is the sum of the results of the 2 throws is equal to 7,"},{"Start":"00:10.400 ","End":"00:14.760","Text":"and B, the product of the result of the 2 throws is equal to 12."},{"Start":"00:14.760 ","End":"00:18.885","Text":"We\u0027re asked to calculate the probability of A given B."},{"Start":"00:18.885 ","End":"00:23.760","Text":"The probability that the sum of the results of the 2 throws is 7,"},{"Start":"00:23.760 ","End":"00:28.185","Text":"given that the product of the results of the 2 throws is equal to 12."},{"Start":"00:28.185 ","End":"00:37.975","Text":"Now, by definition, that equals to the size of A intersect B divided by the size of B."},{"Start":"00:37.975 ","End":"00:48.375","Text":"First of all, let\u0027s see what are the pairs of numbers that are relevant for A and for B?"},{"Start":"00:48.375 ","End":"00:54.740","Text":"For A, we\u0027re looking for pairs of numbers that their sum equals to 7."},{"Start":"00:54.740 ","End":"00:58.530","Text":"So we have 1 and 6,"},{"Start":"00:58.530 ","End":"01:00.250","Text":"we have 2 and 5,"},{"Start":"01:00.250 ","End":"01:02.210","Text":"3 and 4,"},{"Start":"01:02.210 ","End":"01:04.990","Text":"4 and 3,"},{"Start":"01:04.990 ","End":"01:08.415","Text":"5 and 2, and 6 and 1."},{"Start":"01:08.415 ","End":"01:11.210","Text":"Now what about for B?"},{"Start":"01:11.210 ","End":"01:16.610","Text":"Well, we\u0027re looking for pairs of numbers that their product equals to 12."},{"Start":"01:16.610 ","End":"01:19.085","Text":"Well, that\u0027ll be 2 and 6,"},{"Start":"01:19.085 ","End":"01:22.010","Text":"6 and 2, 3 and 4,"},{"Start":"01:22.010 ","End":"01:24.810","Text":"and 4 and 3."},{"Start":"01:25.250 ","End":"01:30.235","Text":"What about A intersect B?"},{"Start":"01:30.235 ","End":"01:35.675","Text":"Well, what are the pairs of numbers that are common both to A and to B?"},{"Start":"01:35.675 ","End":"01:39.650","Text":"Well, these are the pairs of numbers right here, 3 and 4,"},{"Start":"01:39.650 ","End":"01:43.595","Text":"and 4 and 3, and we have here 3 and 4, 4 and 3."},{"Start":"01:43.595 ","End":"01:45.640","Text":"These are the common numbers here."},{"Start":"01:45.640 ","End":"01:50.290","Text":"There\u0027ll be 3 and 4, 4 and 3."},{"Start":"01:50.990 ","End":"01:54.690","Text":"What is the size of B?"},{"Start":"01:54.690 ","End":"01:57.315","Text":"Well, that\u0027s 1, 2, 3, 4."},{"Start":"01:57.315 ","End":"02:00.735","Text":"What\u0027s the size of A intersect B?"},{"Start":"02:00.735 ","End":"02:02.850","Text":"Well that\u0027s 1 and 2."},{"Start":"02:02.850 ","End":"02:04.940","Text":"Let\u0027s plug in these numbers right here."},{"Start":"02:04.940 ","End":"02:06.980","Text":"The size of A intersect B,"},{"Start":"02:06.980 ","End":"02:09.305","Text":"that\u0027s 2 the size of B,"},{"Start":"02:09.305 ","End":"02:10.670","Text":"That equals to 4,"},{"Start":"02:10.670 ","End":"02:14.665","Text":"so we have a probability of a 1/2."},{"Start":"02:14.665 ","End":"02:19.670","Text":"The probability that the sum of the results of the 2 throws is 7,"},{"Start":"02:19.670 ","End":"02:23.555","Text":"given that the product of the results of the 2 throws is 12,"},{"Start":"02:23.555 ","End":"02:24.937","Text":"well, that equals to a 1/2,"},{"Start":"02:24.937 ","End":"02:27.920","Text":"and it makes sense because given that we\u0027re here,"},{"Start":"02:27.920 ","End":"02:31.610","Text":"that the product of the results of the 2 throws is 12."},{"Start":"02:31.610 ","End":"02:39.210","Text":"Well, what\u0027s the probability of getting a sum that is 7?"},{"Start":"02:39.210 ","End":"02:42.780","Text":"Well, we have 2 events here 3 and 4,"},{"Start":"02:42.780 ","End":"02:44.280","Text":"4 and 3,"},{"Start":"02:44.280 ","End":"02:50.620","Text":"so there\u0027ll be 2 out of the 4. So that equals to a 1/2."}],"ID":12874},{"Watched":false,"Name":"Exercise 4","Duration":"2m 19s","ChapterTopicVideoID":12396,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this question, a coin was tossed twice."},{"Start":"00:02.730 ","End":"00:07.470","Text":"Now, it was known that there was 1 head at most and we\u0027re asked,"},{"Start":"00:07.470 ","End":"00:10.350","Text":"what are the chances that there were 2 heads?"},{"Start":"00:10.350 ","End":"00:13.320","Text":"First of all, we know that this is"},{"Start":"00:13.320 ","End":"00:16.650","Text":"a conditional probability because we\u0027re given something,"},{"Start":"00:16.650 ","End":"00:19.395","Text":"it\u0027s known something, we know something."},{"Start":"00:19.395 ","End":"00:24.495","Text":"What do we know? That there was at most 1 head."},{"Start":"00:24.495 ","End":"00:28.650","Text":"Let\u0027s define that as event B,"},{"Start":"00:28.650 ","End":"00:31.005","Text":"that there was at most 1 head."},{"Start":"00:31.005 ","End":"00:33.150","Text":"Now, when there was at most 1 head,"},{"Start":"00:33.150 ","End":"00:38.600","Text":"we could have gotten tails and tails because a coin was tossed twice."},{"Start":"00:38.600 ","End":"00:43.110","Text":"It could be heads and tails and it could be tails and heads."},{"Start":"00:44.450 ","End":"00:50.085","Text":"Those are the results of getting at most 1 head."},{"Start":"00:50.085 ","End":"00:52.485","Text":"Now, what about A?"},{"Start":"00:52.485 ","End":"00:59.260","Text":"Well, A says that there were 2 heads; heads and heads."},{"Start":"01:00.050 ","End":"01:04.430","Text":"We\u0027re looking for the probability of A getting"},{"Start":"01:04.430 ","End":"01:09.530","Text":"2 heads given that there was at most 1 head."},{"Start":"01:09.530 ","End":"01:20.240","Text":"Now, that equals to the size of A intersect B divided by the size of B."},{"Start":"01:20.240 ","End":"01:25.080","Text":"Well, the size of B here that equals to 3;"},{"Start":"01:25.080 ","End":"01:27.075","Text":"1, 2, and 3."},{"Start":"01:27.075 ","End":"01:30.960","Text":"What about A intersect B?"},{"Start":"01:30.960 ","End":"01:33.180","Text":"What A intersect B,"},{"Start":"01:33.180 ","End":"01:36.105","Text":"well, this is an empty set."},{"Start":"01:36.105 ","End":"01:41.300","Text":"It\u0027s usually written like this with the Greek letter Phi or we"},{"Start":"01:41.300 ","End":"01:47.460","Text":"can write it as blank squiggly brackets."},{"Start":"01:47.460 ","End":"01:52.665","Text":"The size of A intersect B,"},{"Start":"01:52.665 ","End":"01:57.460","Text":"the size of an empty set that equals to 0."},{"Start":"01:57.460 ","End":"01:59.930","Text":"This thing, let\u0027s plug in the numbers,"},{"Start":"01:59.930 ","End":"02:02.990","Text":"that\u0027s 0 divided by 3 and that equals to 0."},{"Start":"02:02.990 ","End":"02:11.315","Text":"Again, that makes sense because if we know that we had at most 1 head,"},{"Start":"02:11.315 ","End":"02:16.265","Text":"well, the chances of getting 2 heads is null, is 0."},{"Start":"02:16.265 ","End":"02:20.160","Text":"This then is the probability."}],"ID":12875},{"Watched":false,"Name":"Exercise 5","Duration":"3m 31s","ChapterTopicVideoID":12397,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this question, we have a person that throws a pair of"},{"Start":"00:03.300 ","End":"00:06.810","Text":"dice and gets the same results in both dice."},{"Start":"00:06.810 ","End":"00:11.790","Text":"We\u0027re asked what is the chances that at least 1 of the results was a 5?"},{"Start":"00:11.790 ","End":"00:15.330","Text":"Here we have a conditional probability. Why is that?"},{"Start":"00:15.330 ","End":"00:16.995","Text":"Because we\u0027re given something,"},{"Start":"00:16.995 ","End":"00:18.060","Text":"we\u0027re told something."},{"Start":"00:18.060 ","End":"00:19.500","Text":"What\u0027s that?"},{"Start":"00:19.500 ","End":"00:23.145","Text":"That the person gets the same results in both dice,"},{"Start":"00:23.145 ","End":"00:28.845","Text":"and we\u0027re asked about the probability that 1 of the results was a 5."},{"Start":"00:28.845 ","End":"00:31.425","Text":"Let\u0027s define B,"},{"Start":"00:31.425 ","End":"00:39.755","Text":"the event that\u0027s given as that the same results are given on both dice."},{"Start":"00:39.755 ","End":"00:46.115","Text":"Now, what\u0027s that? What are the possible results of getting the same result in both dice?"},{"Start":"00:46.115 ","End":"00:49.260","Text":"Well, there could be a 1 and a 1, 2 and a 2,"},{"Start":"00:49.260 ","End":"00:52.215","Text":"3,3, 4,4,"},{"Start":"00:52.215 ","End":"00:55.515","Text":"5,5, and 6,6."},{"Start":"00:55.515 ","End":"01:00.020","Text":"These are the possible sets of numbers for event B."},{"Start":"01:00.020 ","End":"01:02.240","Text":"Now, what about event A?"},{"Start":"01:02.240 ","End":"01:10.069","Text":"Well, that event A says that we have at least a 5 in 1 of the 2 results."},{"Start":"01:10.069 ","End":"01:13.355","Text":"That could be a 1 and a 5,"},{"Start":"01:13.355 ","End":"01:15.080","Text":"2 and a 5,"},{"Start":"01:15.080 ","End":"01:16.645","Text":"3 and a 5,"},{"Start":"01:16.645 ","End":"01:20.625","Text":"4 and a 5, 5 and 5, 6 and 5."},{"Start":"01:20.625 ","End":"01:24.255","Text":"But we can also get a 5 and a 1,"},{"Start":"01:24.255 ","End":"01:25.815","Text":"5 and 2,"},{"Start":"01:25.815 ","End":"01:27.645","Text":"5 and 3,"},{"Start":"01:27.645 ","End":"01:29.625","Text":"5 and 4,"},{"Start":"01:29.625 ","End":"01:31.620","Text":"5 and 5 we have."},{"Start":"01:31.620 ","End":"01:33.585","Text":"We don\u0027t need to write it again,"},{"Start":"01:33.585 ","End":"01:37.365","Text":"and we have 5 and 6."},{"Start":"01:37.365 ","End":"01:41.205","Text":"These are the pairs of numbers for A."},{"Start":"01:41.205 ","End":"01:46.230","Text":"Now what about A intersect B?"},{"Start":"01:46.230 ","End":"01:50.475","Text":"Now, what\u0027s common to both A and B?"},{"Start":"01:50.475 ","End":"01:54.100","Text":"Well, 5 and 5 here, that\u0027s common."},{"Start":"01:55.730 ","End":"02:04.100","Text":"Again, now that we have these 2 events defined and we have the intersect defined,"},{"Start":"02:04.100 ","End":"02:05.690","Text":"well, what are we looking for?"},{"Start":"02:05.690 ","End":"02:09.995","Text":"We\u0027re looking for the chances that at least 1 of the results was 5"},{"Start":"02:09.995 ","End":"02:16.470","Text":"given that we got the same results on both dice,"},{"Start":"02:16.470 ","End":"02:19.625","Text":"so we\u0027re looking for the probability of A,"},{"Start":"02:19.625 ","End":"02:26.660","Text":"all these things, given that we\u0027re here, given B."},{"Start":"02:26.660 ","End":"02:36.245","Text":"Now, that equals to the size of A intersect B divided by the size of B."},{"Start":"02:36.245 ","End":"02:39.515","Text":"Well, how many pairs of numbers do I have here?"},{"Start":"02:39.515 ","End":"02:40.880","Text":"1, 2, 3, 4,"},{"Start":"02:40.880 ","End":"02:42.455","Text":"5, and 6."},{"Start":"02:42.455 ","End":"02:45.860","Text":"What about the size of A intersect B?"},{"Start":"02:45.860 ","End":"02:47.300","Text":"Well, that\u0027s 1."},{"Start":"02:47.300 ","End":"02:49.280","Text":"We have 1 pair here."},{"Start":"02:49.280 ","End":"02:51.200","Text":"Let\u0027s just plug in the numbers,"},{"Start":"02:51.200 ","End":"02:53.135","Text":"that\u0027s 1 divided by 6,"},{"Start":"02:53.135 ","End":"02:56.350","Text":"the size of the intersect of A and B,"},{"Start":"02:56.350 ","End":"03:00.840","Text":"that\u0027s 1 divided by the size of B, well that\u0027s 6."},{"Start":"03:00.840 ","End":"03:04.010","Text":"It makes sense because again,"},{"Start":"03:04.010 ","End":"03:06.710","Text":"we\u0027re given that when you roll the dice,"},{"Start":"03:06.710 ","End":"03:10.835","Text":"he had the same results on both dice,"},{"Start":"03:10.835 ","End":"03:12.320","Text":"and out of that,"},{"Start":"03:12.320 ","End":"03:17.060","Text":"we\u0027re asking about the probability that at least 1 of the results was a 5."},{"Start":"03:17.060 ","End":"03:20.515","Text":"Well, the only thing here is a 5 and a 5."},{"Start":"03:20.515 ","End":"03:29.235","Text":"This is the only pair of numbers out of 6 possible numbers,"},{"Start":"03:29.235 ","End":"03:31.900","Text":"so that\u0027s 1 out of 6."}],"ID":12876},{"Watched":false,"Name":"Exercise 6","Duration":"3m 43s","ChapterTopicVideoID":12398,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In this question, a person throws a pair of dice and gets at least 1, 4."},{"Start":"00:04.830 ","End":"00:08.970","Text":"We\u0027re asked what are the chances that 1 of the results was a 5."},{"Start":"00:08.970 ","End":"00:12.540","Text":"Here we have a conditional probability. Why is that?"},{"Start":"00:12.540 ","End":"00:16.200","Text":"Because we\u0027re told that when a person throws a pair of dice,"},{"Start":"00:16.200 ","End":"00:18.420","Text":"he gets at least 1, 4."},{"Start":"00:18.420 ","End":"00:23.880","Text":"We\u0027re asked about another event that\u0027s conditional on this 1, that says,"},{"Start":"00:23.880 ","End":"00:30.105","Text":"\"What are the chances that 1 of the results was a 5?\""},{"Start":"00:30.105 ","End":"00:33.840","Text":"Let\u0027s just define the 2 events here,"},{"Start":"00:33.840 ","End":"00:36.575","Text":"let\u0027s define the event that was given"},{"Start":"00:36.575 ","End":"00:42.530","Text":"B that 1 of the results when throwing a pair of dice is a 4,"},{"Start":"00:42.530 ","End":"00:47.465","Text":"what are the possible pairs of results here?"},{"Start":"00:47.465 ","End":"00:51.325","Text":"Well, that could be a 1 and a 4, 2 and a 4,"},{"Start":"00:51.325 ","End":"00:52.845","Text":"3 and a 4,"},{"Start":"00:52.845 ","End":"00:55.815","Text":"4 and a 4, 5 and a 4,"},{"Start":"00:55.815 ","End":"00:57.315","Text":"6 and a 4,"},{"Start":"00:57.315 ","End":"01:00.530","Text":"but you could also get a 4 and a 1, 4,"},{"Start":"01:00.530 ","End":"01:02.880","Text":"2, 4, 3,"},{"Start":"01:02.880 ","End":"01:05.090","Text":"now 4, 4 we\u0027ve already written down,"},{"Start":"01:05.090 ","End":"01:07.505","Text":"so we don\u0027t have to write that down again."},{"Start":"01:07.505 ","End":"01:12.430","Text":"We have a 4 and a 5 and 4 and a 6."},{"Start":"01:12.430 ","End":"01:15.920","Text":"These are the all the possible pairs of"},{"Start":"01:15.920 ","End":"01:21.335","Text":"numbers when a person throws the dice and gets at least 1, 4."},{"Start":"01:21.335 ","End":"01:24.500","Text":"Now what about the other event?"},{"Start":"01:24.500 ","End":"01:26.870","Text":"Well, the other event, we\u0027ll call that A,"},{"Start":"01:26.870 ","End":"01:32.455","Text":"and that\u0027s the results of getting at least 1, 5."},{"Start":"01:32.455 ","End":"01:34.770","Text":"Well, let\u0027s write that down."},{"Start":"01:34.770 ","End":"01:36.855","Text":"He can get a 1 and a 5,"},{"Start":"01:36.855 ","End":"01:38.160","Text":"a 2 and a 5,"},{"Start":"01:38.160 ","End":"01:39.600","Text":"3 and a 5,"},{"Start":"01:39.600 ","End":"01:41.130","Text":"4 and a 5,"},{"Start":"01:41.130 ","End":"01:42.645","Text":"5 and a 5,"},{"Start":"01:42.645 ","End":"01:44.310","Text":"and a 6 and a 5."},{"Start":"01:44.310 ","End":"01:47.655","Text":"But you can also get 5 and 1,"},{"Start":"01:47.655 ","End":"01:49.800","Text":"5 and 2,"},{"Start":"01:49.800 ","End":"01:51.510","Text":"5 and 3,"},{"Start":"01:51.510 ","End":"01:53.295","Text":"5 and 4,"},{"Start":"01:53.295 ","End":"01:55.800","Text":"5 and 5, we don\u0027t need to write that down,"},{"Start":"01:55.800 ","End":"02:01.780","Text":"we\u0027ve already written that down and we have a 5 and a 6."},{"Start":"02:01.780 ","End":"02:09.230","Text":"These are the possible pairs of numbers where when throwing a pair of die,"},{"Start":"02:09.230 ","End":"02:11.705","Text":"1 of the values here is a 5."},{"Start":"02:11.705 ","End":"02:15.800","Text":"Now, what about A intersect B?"},{"Start":"02:15.800 ","End":"02:20.810","Text":"Well, what are the common pairs of numbers to both A and B?"},{"Start":"02:20.810 ","End":"02:24.830","Text":"Well, that would be a 4 and a 5,"},{"Start":"02:24.830 ","End":"02:27.200","Text":"and a 5 and a 4,"},{"Start":"02:27.200 ","End":"02:29.630","Text":"these pairs of numbers right here,"},{"Start":"02:29.630 ","End":"02:31.040","Text":"4 and a 5,"},{"Start":"02:31.040 ","End":"02:33.095","Text":"and a 5 and a 4."},{"Start":"02:33.095 ","End":"02:34.940","Text":"Let\u0027s just write that down,"},{"Start":"02:34.940 ","End":"02:38.515","Text":"that\u0027s 4 and 5, and 5 and a 4."},{"Start":"02:38.515 ","End":"02:39.770","Text":"Now, what do we asked?"},{"Start":"02:39.770 ","End":"02:44.440","Text":"Were asked what are the chances that 1 of the results was a 5?"},{"Start":"02:44.440 ","End":"02:47.945","Text":"We\u0027re looking for the probability of A,"},{"Start":"02:47.945 ","End":"02:49.685","Text":"1 of these guys,"},{"Start":"02:49.685 ","End":"02:54.770","Text":"given that he threw 1 of these guys, given B."},{"Start":"02:54.770 ","End":"02:58.249","Text":"Now, that equals according to the definition,"},{"Start":"02:58.249 ","End":"03:03.980","Text":"to the size of A intersect B divided by the size of B."},{"Start":"03:03.980 ","End":"03:07.700","Text":"Well, what\u0027s the size of B here?"},{"Start":"03:07.700 ","End":"03:10.430","Text":"Well, how many pairs of numbers do I have here?"},{"Start":"03:10.430 ","End":"03:12.410","Text":"We\u0027ll have 1, 2, 3, 4, 5,"},{"Start":"03:12.410 ","End":"03:16.775","Text":"6, 7, 8, 9, 10, 11."},{"Start":"03:16.775 ","End":"03:19.340","Text":"We have 11 pairs of numbers."},{"Start":"03:19.340 ","End":"03:22.355","Text":"We don\u0027t want to account 4, 4 twice."},{"Start":"03:22.355 ","End":"03:26.030","Text":"What about the size of A intersect B?"},{"Start":"03:26.030 ","End":"03:30.160","Text":"Well, we only have 2 pairs of numbers here."},{"Start":"03:30.160 ","End":"03:32.980","Text":"When we plug that in, we have 2,"},{"Start":"03:32.980 ","End":"03:36.905","Text":"the size of a intersect B divided by the size of B,"},{"Start":"03:36.905 ","End":"03:38.945","Text":"2 divided by 11."},{"Start":"03:38.945 ","End":"03:43.710","Text":"That\u0027s the probability of A intersect B."}],"ID":12877},{"Watched":false,"Name":"Exercise 7","Duration":"2m 9s","ChapterTopicVideoID":12399,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"In this question, we have a family with 2 children."},{"Start":"00:02.865 ","End":"00:06.900","Text":"Now, it\u0027s known that 1 of the children is a boy and we\u0027re asked,"},{"Start":"00:06.900 ","End":"00:09.420","Text":"what are the chances that both children are boys?"},{"Start":"00:09.420 ","End":"00:13.155","Text":"Here again, we have a conditional probability."},{"Start":"00:13.155 ","End":"00:17.775","Text":"Why is that? Because we\u0027re given something, we know something."},{"Start":"00:17.775 ","End":"00:19.950","Text":"That is that out of the 2 children,"},{"Start":"00:19.950 ","End":"00:21.420","Text":"1 of them is a boy,"},{"Start":"00:21.420 ","End":"00:26.370","Text":"and we\u0027re asked about another event that both children are boys."},{"Start":"00:26.370 ","End":"00:28.575","Text":"Let\u0027s just define the events here."},{"Start":"00:28.575 ","End":"00:30.135","Text":"Let\u0027s define B,"},{"Start":"00:30.135 ","End":"00:31.710","Text":"the event that\u0027s given,"},{"Start":"00:31.710 ","End":"00:35.145","Text":"that at least 1 of the children is a boy."},{"Start":"00:35.145 ","End":"00:36.960","Text":"Now, what could that be?"},{"Start":"00:36.960 ","End":"00:38.730","Text":"Well, we could have a boy and a boy,"},{"Start":"00:38.730 ","End":"00:45.600","Text":"we can have a boy and a girl and we can have a girl and a boy."},{"Start":"00:45.600 ","End":"00:49.210","Text":"Now, what about A,"},{"Start":"00:49.210 ","End":"00:52.880","Text":"what are the chances that both children are boys?"},{"Start":"00:52.880 ","End":"00:56.195","Text":"That means that we have a boy and a boy,"},{"Start":"00:56.195 ","End":"00:59.135","Text":"that\u0027s the event that both children are boys."},{"Start":"00:59.135 ","End":"01:03.435","Text":"Now what about A intersect B?"},{"Start":"01:03.435 ","End":"01:05.355","Text":"What\u0027s that?"},{"Start":"01:05.355 ","End":"01:07.965","Text":"What\u0027s common to A, and to B?"},{"Start":"01:07.965 ","End":"01:11.385","Text":"Well, 2 boys, a boy and a boy."},{"Start":"01:11.385 ","End":"01:21.440","Text":"Excellent. Now we\u0027re looking for the probability of A given B. What does that mean?"},{"Start":"01:21.440 ","End":"01:31.410","Text":"What is the chances that both children are boys given that at least 1 child is a boy."},{"Start":"01:31.570 ","End":"01:33.890","Text":"What does that equal to?"},{"Start":"01:33.890 ","End":"01:35.885","Text":"Well, according to the definition,"},{"Start":"01:35.885 ","End":"01:44.735","Text":"that equals to the size of A intersect B divided by the size of B."},{"Start":"01:44.735 ","End":"01:49.950","Text":"We know that the size of B that equals to 1, 2, 3,"},{"Start":"01:49.950 ","End":"01:53.415","Text":"and the size of A intersect B,"},{"Start":"01:53.415 ","End":"01:54.750","Text":"that\u0027s right here,"},{"Start":"01:54.750 ","End":"01:56.430","Text":"that equals to 1."},{"Start":"01:56.430 ","End":"01:58.560","Text":"That equals to 1/3."},{"Start":"01:58.560 ","End":"02:03.330","Text":"That\u0027s the probability of getting"},{"Start":"02:03.330 ","End":"02:10.180","Text":"2 boys in the family when we know that at least 1 of the children is a boy."}],"ID":12878},{"Watched":false,"Name":"Exercise 8","Duration":"3m 39s","ChapterTopicVideoID":12400,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"In this question, we have a family with 3 children."},{"Start":"00:03.330 ","End":"00:06.690","Text":"Now, given that the middle child is a boy,"},{"Start":"00:06.690 ","End":"00:11.460","Text":"what are the chances that there are girls in the family?"},{"Start":"00:11.460 ","End":"00:15.120","Text":"Again, we have a conditional probability. Why is that?"},{"Start":"00:15.120 ","End":"00:17.730","Text":"Because we\u0027re given something, we know something,"},{"Start":"00:17.730 ","End":"00:20.780","Text":"we know that the middle child is a boy and we\u0027re asked"},{"Start":"00:20.780 ","End":"00:24.665","Text":"about the event that there are girls in the family."},{"Start":"00:24.665 ","End":"00:30.530","Text":"Let\u0027s just see what are the possible combinations of"},{"Start":"00:30.530 ","End":"00:36.620","Text":"children\u0027s boys and girls, in both events."},{"Start":"00:36.620 ","End":"00:42.240","Text":"Where A is the event that we want to talk about."},{"Start":"00:42.240 ","End":"00:44.295","Text":"What\u0027s the probability of A here?"},{"Start":"00:44.295 ","End":"00:45.845","Text":"Well, that would be,"},{"Start":"00:45.845 ","End":"00:47.750","Text":"where we have girls in the family."},{"Start":"00:47.750 ","End":"00:49.795","Text":"It could be girl, girl,"},{"Start":"00:49.795 ","End":"00:52.625","Text":"girl, we have 3 girls in the family."},{"Start":"00:52.625 ","End":"00:55.925","Text":"We have girl, girl, and boy."},{"Start":"00:55.925 ","End":"00:59.960","Text":"We have a girl, boy, girl."},{"Start":"00:59.960 ","End":"01:02.090","Text":"We have a boy,"},{"Start":"01:02.090 ","End":"01:04.670","Text":"girl and a girl."},{"Start":"01:04.670 ","End":"01:09.150","Text":"We have a boy, boy and girl."},{"Start":"01:09.150 ","End":"01:13.365","Text":"We have a boy, girl and boy."},{"Start":"01:13.365 ","End":"01:18.790","Text":"We have a girl, boy and boy."},{"Start":"01:19.400 ","End":"01:26.330","Text":"These are all the possible combinations of having a girl in the family."},{"Start":"01:26.330 ","End":"01:28.605","Text":"Now, what about B,"},{"Start":"01:28.605 ","End":"01:30.315","Text":"the event that\u0027s given?"},{"Start":"01:30.315 ","End":"01:34.235","Text":"Well, we\u0027re given that the middle child is a boy."},{"Start":"01:34.235 ","End":"01:37.250","Text":"What are the possible combinations here?"},{"Start":"01:37.250 ","End":"01:41.685","Text":"Well, we can have a girl, boy, girl."},{"Start":"01:41.685 ","End":"01:46.175","Text":"We can have a girl, boy, boy."},{"Start":"01:46.175 ","End":"01:49.280","Text":"We can have a boy, boy,"},{"Start":"01:49.280 ","End":"01:54.980","Text":"girl, or a boy, boy and boy."},{"Start":"01:54.980 ","End":"01:58.430","Text":"These are all the possible combination of children,"},{"Start":"01:58.430 ","End":"02:04.100","Text":"boys and girls in the family where the boy is the middle child."},{"Start":"02:04.100 ","End":"02:08.840","Text":"Now, what about A intersect B?"},{"Start":"02:08.840 ","End":"02:13.880","Text":"Well, what\u0027s common to both A and B?"},{"Start":"02:13.880 ","End":"02:19.110","Text":"Well, here we have girl, boy, boy."},{"Start":"02:19.110 ","End":"02:22.595","Text":"We have girl, boy,"},{"Start":"02:22.595 ","End":"02:28.070","Text":"girl, and we have boy, boy, girl."},{"Start":"02:28.070 ","End":"02:35.900","Text":"These are the 3 combinations of girls and boys that are common to both A and B."},{"Start":"02:35.900 ","End":"02:42.965","Text":"Now we\u0027re asked for the probability of A given B."},{"Start":"02:42.965 ","End":"02:47.825","Text":"What is the probability that there are girls in the family,"},{"Start":"02:47.825 ","End":"02:53.825","Text":"given that we have a boy as a middle child?"},{"Start":"02:53.825 ","End":"03:00.013","Text":"Well, by definition, that equals to the size of A intersect B,"},{"Start":"03:00.013 ","End":"03:02.495","Text":"divided by the size of B."},{"Start":"03:02.495 ","End":"03:05.045","Text":"Well, what\u0027s the size of B here?"},{"Start":"03:05.045 ","End":"03:06.732","Text":"The size of B,"},{"Start":"03:06.732 ","End":"03:10.050","Text":"well that equals to 1, 2, 3, 4."},{"Start":"03:10.050 ","End":"03:13.805","Text":"What about the size of A intersect B?"},{"Start":"03:13.805 ","End":"03:16.220","Text":"Well, that\u0027s 1, 2 and 3."},{"Start":"03:16.220 ","End":"03:18.155","Text":"Let\u0027s just plug that in."},{"Start":"03:18.155 ","End":"03:20.180","Text":"The size of A intersect B,"},{"Start":"03:20.180 ","End":"03:23.645","Text":"that\u0027ll be 3, the size of B that\u0027s 4."},{"Start":"03:23.645 ","End":"03:30.095","Text":"The probability that there are girls in the family,"},{"Start":"03:30.095 ","End":"03:33.335","Text":"given that the middle child is a boy."},{"Start":"03:33.335 ","End":"03:39.990","Text":"Well, that\u0027s 0.75 or 3/4 or 75 percent."}],"ID":12879},{"Watched":false,"Name":"Exercise 9","Duration":"2m 54s","ChapterTopicVideoID":12401,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Now, the following questions require some knowledge on combinatorics."},{"Start":"00:04.680 ","End":"00:08.700","Text":"For this question, we have 4 children that are selected"},{"Start":"00:08.700 ","End":"00:13.350","Text":"from a class containing 6 boys and 7 girls."},{"Start":"00:13.350 ","End":"00:18.020","Text":"Now, if it\u0027s known that 2 boys and 2 girls were selected,"},{"Start":"00:18.020 ","End":"00:19.715","Text":"what are the chances that Mike,"},{"Start":"00:19.715 ","End":"00:22.190","Text":"one of the boys, was not selected?"},{"Start":"00:22.190 ","End":"00:27.785","Text":"We have here a question in conditional probability."},{"Start":"00:27.785 ","End":"00:30.475","Text":"Why is that? Because we\u0027re given something."},{"Start":"00:30.475 ","End":"00:34.605","Text":"It\u0027s known that 2 boys and 2 girls were selected."},{"Start":"00:34.605 ","End":"00:37.755","Text":"Let\u0027s define the Event B,"},{"Start":"00:37.755 ","End":"00:40.380","Text":"that 2 boys and 2 girls were selected."},{"Start":"00:40.380 ","End":"00:44.205","Text":"We have 2 boys plus 2 girls."},{"Start":"00:44.205 ","End":"00:46.380","Text":"Now, what\u0027s A?"},{"Start":"00:46.380 ","End":"00:51.875","Text":"We\u0027re looking for the probability that Mike was not selected."},{"Start":"00:51.875 ","End":"00:57.375","Text":"Mike, one of the boys, was not selected."},{"Start":"00:57.375 ","End":"01:01.085","Text":"We\u0027re looking for the probability of A,"},{"Start":"01:01.085 ","End":"01:07.370","Text":"that Mike was not selected given that we selected 2 boys and 2 girls."},{"Start":"01:07.370 ","End":"01:13.985","Text":"Now, that equals to the size of A intersect B divided by the size of B."},{"Start":"01:13.985 ","End":"01:15.880","Text":"Now, let\u0027s think about it."},{"Start":"01:15.880 ","End":"01:18.885","Text":"What is the size of B?"},{"Start":"01:18.885 ","End":"01:27.210","Text":"What is the size of the event that 2 boys are selected out of 6 boys,"},{"Start":"01:27.210 ","End":"01:29.115","Text":"and out of the 7 girls,"},{"Start":"01:29.115 ","End":"01:30.960","Text":"2 girls are selected?"},{"Start":"01:30.960 ","End":"01:35.010","Text":"Well, that\u0027s 7 out of the 7 girls,"},{"Start":"01:35.010 ","End":"01:36.390","Text":"we\u0027re looking for 2 girls."},{"Start":"01:36.390 ","End":"01:38.175","Text":"We\u0027re selecting 2 girls."},{"Start":"01:38.175 ","End":"01:39.990","Text":"Out of the 6 boys,"},{"Start":"01:39.990 ","End":"01:43.350","Text":"we\u0027re selecting 2 boys."},{"Start":"01:43.350 ","End":"01:48.255","Text":"This is non-sequential sampling without returns."},{"Start":"01:48.255 ","End":"01:51.180","Text":"This would be the size of the Event B."},{"Start":"01:51.180 ","End":"01:57.915","Text":"What about the size of A intersect B? What does that mean?"},{"Start":"01:57.915 ","End":"02:01.050","Text":"That means that Mike was not selected."},{"Start":"02:01.050 ","End":"02:04.980","Text":"Some other 2 boys were selected, but not Mike."},{"Start":"02:04.980 ","End":"02:10.695","Text":"We\u0027re selecting 2 boys but now not out of 6 boys,"},{"Start":"02:10.695 ","End":"02:12.435","Text":"out of 5 boys."},{"Start":"02:12.435 ","End":"02:14.430","Text":"Because we have 6 boys,"},{"Start":"02:14.430 ","End":"02:16.110","Text":"Mike was not selected,"},{"Start":"02:16.110 ","End":"02:18.165","Text":"so we have 5 boys."},{"Start":"02:18.165 ","End":"02:22.660","Text":"We have to select 2 out of these 5."},{"Start":"02:22.660 ","End":"02:24.905","Text":"Now, what about the girls?"},{"Start":"02:24.905 ","End":"02:27.440","Text":"Well, there\u0027s no constraint on the girls,"},{"Start":"02:27.440 ","End":"02:31.050","Text":"so we have to select 2 girls out of the 7."},{"Start":"02:31.050 ","End":"02:33.455","Text":"That\u0027s the probability."},{"Start":"02:33.455 ","End":"02:38.700","Text":"Now, this thing comes out to the following."},{"Start":"02:38.700 ","End":"02:40.830","Text":"Well, 7/2,"},{"Start":"02:40.830 ","End":"02:42.525","Text":"just cancels each other out."},{"Start":"02:42.525 ","End":"02:47.325","Text":"5/2, that equals to 10; and 6/2,"},{"Start":"02:47.325 ","End":"02:48.915","Text":"that equals to 15,"},{"Start":"02:48.915 ","End":"02:54.700","Text":"so the probability here would be equal to 2/3."}],"ID":12880},{"Watched":false,"Name":"Exercise 10","Duration":"3m 20s","ChapterTopicVideoID":12402,"CourseChapterTopicPlaylistID":245031,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"In this question, we have 5 friends that went to the movies and sat"},{"Start":"00:03.420 ","End":"00:07.720","Text":"next to each other randomly in seats number 5-9."},{"Start":"00:07.720 ","End":"00:11.700","Text":"Now, if it\u0027s known that Joe and Mike sat next to each other,"},{"Start":"00:11.700 ","End":"00:17.340","Text":"what\u0027s the probability that they\u0027re sitting in seats number 6 and 7."},{"Start":"00:17.340 ","End":"00:22.365","Text":"Here are seats number 5-9."},{"Start":"00:22.365 ","End":"00:26.490","Text":"Now, we have here a conditional probability. Now, why is that?"},{"Start":"00:26.490 ","End":"00:29.565","Text":"Because we\u0027re given something, it\u0027s known."},{"Start":"00:29.565 ","End":"00:32.685","Text":"What\u0027s known? That Joe and Mike sat next to each other."},{"Start":"00:32.685 ","End":"00:34.350","Text":"Let\u0027s define event B,"},{"Start":"00:34.350 ","End":"00:39.210","Text":"what\u0027s known as Joe and Mike sitting"},{"Start":"00:39.210 ","End":"00:45.460","Text":"next to each other, sitting together."},{"Start":"00:46.670 ","End":"00:53.180","Text":"Now, what\u0027s A? A is that they\u0027re sitting in seats number 6 and 7,"},{"Start":"00:53.180 ","End":"00:55.385","Text":"that they\u0027re sitting right here."},{"Start":"00:55.385 ","End":"01:05.370","Text":"Joe and Mike are sitting in seats number 6 and 7."},{"Start":"01:05.370 ","End":"01:07.610","Text":"Let\u0027s get to work here."},{"Start":"01:07.610 ","End":"01:13.850","Text":"We\u0027re looking for the probability of Joe and Mike sitting in seats number 6 and 7,"},{"Start":"01:13.850 ","End":"01:17.460","Text":"given that they\u0027re sitting together."},{"Start":"01:17.460 ","End":"01:26.020","Text":"That equals to the size of A intersect B divided by the size of event B."},{"Start":"01:26.020 ","End":"01:28.160","Text":"Now, let\u0027s calculate, first of all,"},{"Start":"01:28.160 ","End":"01:34.920","Text":"the size of event B that Joe and Mike are sitting together. What do we have here?"},{"Start":"01:34.920 ","End":"01:36.525","Text":"We have 5 friends,"},{"Start":"01:36.525 ","End":"01:40.620","Text":"1, 2, 3, 4."},{"Start":"01:40.620 ","End":"01:43.500","Text":"But we have here Joe and Mike,"},{"Start":"01:43.500 ","End":"01:46.250","Text":"that\u0027s 2 people,"},{"Start":"01:46.250 ","End":"01:48.110","Text":"but because they\u0027re sitting together,"},{"Start":"01:48.110 ","End":"01:51.215","Text":"we\u0027ll combine them into 1 object,"},{"Start":"01:51.215 ","End":"01:53.855","Text":"and we have 3 other people."},{"Start":"01:53.855 ","End":"01:56.105","Text":"We have 4 objects."},{"Start":"01:56.105 ","End":"02:03.400","Text":"In how many ways can we order for objects?"},{"Start":"02:03.400 ","End":"02:06.337","Text":"Well, 4 factorial,"},{"Start":"02:06.337 ","End":"02:09.125","Text":"but we have the internal ordering here."},{"Start":"02:09.125 ","End":"02:11.360","Text":"It could be Joe and Mike or Mike and Joe."},{"Start":"02:11.360 ","End":"02:14.315","Text":"We have to multiply that by 2 factorial."},{"Start":"02:14.315 ","End":"02:17.600","Text":"There we go. That\u0027s the size of event B,"},{"Start":"02:17.600 ","End":"02:20.435","Text":"where Joe and Mike sit together."},{"Start":"02:20.435 ","End":"02:26.915","Text":"What\u0027s the probability that they\u0027re sitting in seats number 6 and 7?"},{"Start":"02:26.915 ","End":"02:29.655","Text":"If they\u0027re sitting right here,"},{"Start":"02:29.655 ","End":"02:32.040","Text":"Joe can be sitting here and Mike here or"},{"Start":"02:32.040 ","End":"02:35.490","Text":"the other way around where Mike sits here and Joe sits here."},{"Start":"02:35.490 ","End":"02:42.040","Text":"We have 2 factorial ways of having Joe and Mike sitting in seats 6 and 7."},{"Start":"02:42.040 ","End":"02:46.925","Text":"We have 3 other friends that are sitting in seats number 5,"},{"Start":"02:46.925 ","End":"02:49.520","Text":"8, and 9. We order that."},{"Start":"02:49.520 ","End":"02:52.910","Text":"We have 3 factorial combinations of"},{"Start":"02:52.910 ","End":"02:57.220","Text":"having the other 3 people sitting in the other 3 seats."},{"Start":"02:57.220 ","End":"02:59.040","Text":"This is the probability."},{"Start":"02:59.040 ","End":"03:02.030","Text":"Now, 2 factorials cancel each other out,"},{"Start":"03:02.030 ","End":"03:05.675","Text":"so we\u0027re left with 3 factorial divided by 4 factorial,"},{"Start":"03:05.675 ","End":"03:12.575","Text":"and that equals to 1 divided by 4 or 25 percent."},{"Start":"03:12.575 ","End":"03:18.560","Text":"So that\u0027s the probability of Joe and Mike sitting in seats number 6 and 7,"},{"Start":"03:18.560 ","End":"03:21.600","Text":"given that they\u0027re sitting together."}],"ID":12881}],"Thumbnail":null,"ID":245031},{"Name":"Conditional Probability in a Non-Uniform Space","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"1m 18s","ChapterTopicVideoID":12403,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12403.jpeg","UploadDate":"2018-07-22T17:16:25.9530000","DurationForVideoObject":"PT1M18S","Description":null,"MetaTitle":"Tutorial 1: Video + Workbook | Proprep","MetaDescription":"Conditional Probability - Conditional Probability in a Non-Uniform Space. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/probability/conditional-probability/conditional-probability-in-a-non_uniform-space/vid12882","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this chapter, we\u0027ll be talking about conditional probability in a non-uniform space."},{"Start":"00:05.190 ","End":"00:10.575","Text":"Now, we\u0027re looking at the chances that event A will occur,"},{"Start":"00:10.575 ","End":"00:14.325","Text":"given that event B has already occurred."},{"Start":"00:14.325 ","End":"00:17.550","Text":"Now, what is that probability?"},{"Start":"00:17.550 ","End":"00:20.250","Text":"Well, this is a conditional probability."},{"Start":"00:20.250 ","End":"00:27.300","Text":"We\u0027re looking for the probability of A conditioned on the fact that event B has occurred."},{"Start":"00:27.300 ","End":"00:28.755","Text":"Now, we write it like this."},{"Start":"00:28.755 ","End":"00:31.164","Text":"That\u0027s the probability of A,"},{"Start":"00:31.164 ","End":"00:34.305","Text":"the straight line here is given, B, right?"},{"Start":"00:34.305 ","End":"00:36.715","Text":"How do we calculate that?"},{"Start":"00:36.715 ","End":"00:38.900","Text":"Well, that equals to the following,"},{"Start":"00:38.900 ","End":"00:40.130","Text":"that\u0027s the ratio right here."},{"Start":"00:40.130 ","End":"00:44.870","Text":"The denominator, we have the probability of the event that has occurred,"},{"Start":"00:44.870 ","End":"00:46.610","Text":"the probability of B."},{"Start":"00:46.610 ","End":"00:47.975","Text":"That\u0027s thing right here."},{"Start":"00:47.975 ","End":"00:52.725","Text":"In the numerator, we have the probability of the intersect;"},{"Start":"00:52.725 ","End":"00:56.555","Text":"the event that we\u0027re looking to calculate its probability"},{"Start":"00:56.555 ","End":"01:00.745","Text":"and the event that has already occurred."},{"Start":"01:00.745 ","End":"01:02.539","Text":"Again, in the numerator,"},{"Start":"01:02.539 ","End":"01:06.050","Text":"that\u0027s the chances of the intersection of the 2 events: the 1 whose"},{"Start":"01:06.050 ","End":"01:10.100","Text":"probability we want to know and the 1 we know has occurred."},{"Start":"01:10.100 ","End":"01:11.525","Text":"In the denominator,"},{"Start":"01:11.525 ","End":"01:16.100","Text":"that\u0027s the chances of the event we know has occurred."},{"Start":"01:16.100 ","End":"01:19.680","Text":"Let\u0027s just take a look at an example here."}],"ID":12882},{"Watched":false,"Name":"Tutorial 2","Duration":"3m 38s","ChapterTopicVideoID":12404,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this example, we\u0027re sampling families that have 2 cars."},{"Start":"00:04.185 ","End":"00:07.110","Text":"Now, we\u0027re told that in 30 percent of these families,"},{"Start":"00:07.110 ","End":"00:11.070","Text":"the make of the older car is European and in 60 percent of these families,"},{"Start":"00:11.070 ","End":"00:13.170","Text":"make of the newer car is European,"},{"Start":"00:13.170 ","End":"00:15.870","Text":"and in 15 percent of these families,"},{"Start":"00:15.870 ","End":"00:18.090","Text":"the make them both cars is European."},{"Start":"00:18.090 ","End":"00:21.960","Text":"We\u0027re asked, if a family\u0027s older car is European,"},{"Start":"00:21.960 ","End":"00:26.235","Text":"what\u0027s the probability that the newer 1 is also European?"},{"Start":"00:26.235 ","End":"00:28.880","Text":"How do we approach this question?"},{"Start":"00:28.880 ","End":"00:31.385","Text":"Well, the first thing we need to know is"},{"Start":"00:31.385 ","End":"00:34.925","Text":"if we\u0027re dealing with a conditional probability or not."},{"Start":"00:34.925 ","End":"00:39.800","Text":"How do we know that we are dealing with a conditional probability?"},{"Start":"00:39.800 ","End":"00:41.810","Text":"Well, we have 2 events,"},{"Start":"00:41.810 ","End":"00:43.295","Text":"and what are the events?"},{"Start":"00:43.295 ","End":"00:46.850","Text":"That the family has an older car that\u0027s"},{"Start":"00:46.850 ","End":"00:52.310","Text":"European and a family that has a newer car that\u0027s European."},{"Start":"00:52.310 ","End":"00:54.620","Text":"Now, what are we given?"},{"Start":"00:54.620 ","End":"00:58.655","Text":"We\u0027re given that the family has an older car."},{"Start":"00:58.655 ","End":"01:02.510","Text":"We\u0027re told if a family\u0027s older car is European,"},{"Start":"01:02.510 ","End":"01:04.610","Text":"that\u0027s given to us, that\u0027s known,"},{"Start":"01:04.610 ","End":"01:06.140","Text":"we\u0027ll call that B,"},{"Start":"01:06.140 ","End":"01:09.040","Text":"and we\u0027re asked about another event,"},{"Start":"01:09.040 ","End":"01:12.730","Text":"the probability that the newer 1 is also European."},{"Start":"01:12.730 ","End":"01:15.045","Text":"We\u0027ll call that A."},{"Start":"01:15.045 ","End":"01:23.575","Text":"We\u0027re looking for the probability of A given that B occurred,"},{"Start":"01:23.575 ","End":"01:26.260","Text":"that the families older cars European."},{"Start":"01:26.260 ","End":"01:29.030","Text":"Let\u0027s just write everything down here."},{"Start":"01:29.090 ","End":"01:37.465","Text":"These are the events: A is that the new cars European and B that the old cars European,"},{"Start":"01:37.465 ","End":"01:40.960","Text":"and we said that we\u0027re dealing with a conditional probability."},{"Start":"01:40.960 ","End":"01:46.060","Text":"We\u0027re looking to calculate the probability of A given B."},{"Start":"01:46.060 ","End":"01:48.670","Text":"The probability that the family has"},{"Start":"01:48.670 ","End":"01:52.700","Text":"a newer car given that they already have an older car."},{"Start":"01:52.700 ","End":"01:55.970","Text":"By definition, that equals to the probability of"},{"Start":"01:55.970 ","End":"02:02.235","Text":"A intersect B divided by the probability of B."},{"Start":"02:02.235 ","End":"02:05.090","Text":"Let\u0027s calculate these probabilities."},{"Start":"02:05.090 ","End":"02:08.435","Text":"Let\u0027s see if they\u0027re given to us in any way."},{"Start":"02:08.435 ","End":"02:12.120","Text":"Well, what\u0027s the probability of A,"},{"Start":"02:12.120 ","End":"02:14.225","Text":"that the family has a new car?"},{"Start":"02:14.225 ","End":"02:16.310","Text":"That\u0027s given to us right here."},{"Start":"02:16.310 ","End":"02:18.500","Text":"In 60 percent of these families,"},{"Start":"02:18.500 ","End":"02:23.975","Text":"the make of the newer car is European so the probability of A is 0.6."},{"Start":"02:23.975 ","End":"02:26.890","Text":"What\u0027s the probability of B,"},{"Start":"02:26.890 ","End":"02:29.120","Text":"that the old cars European?"},{"Start":"02:29.120 ","End":"02:31.475","Text":"In 30 percent of these families,"},{"Start":"02:31.475 ","End":"02:34.380","Text":"the make of the older car is Europeans,"},{"Start":"02:34.380 ","End":"02:37.140","Text":"so B equals to 0.3."},{"Start":"02:37.140 ","End":"02:42.380","Text":"We\u0027re also told about the probability of A intersect B"},{"Start":"02:42.380 ","End":"02:47.716","Text":"that the new car and the old car is European, that\u0027s right here."},{"Start":"02:47.716 ","End":"02:52.704","Text":"15 percent, 0.15 or 15 percent of these families,"},{"Start":"02:52.704 ","End":"02:55.575","Text":"the make both cars is European."},{"Start":"02:55.575 ","End":"02:58.790","Text":"We have all the probabilities that we need here,"},{"Start":"02:58.790 ","End":"03:01.680","Text":"all we need to do is plug everything in."},{"Start":"03:01.730 ","End":"03:06.470","Text":"We have here the probability of A given B is"},{"Start":"03:06.470 ","End":"03:10.640","Text":"equal to the probability of A intersect B divided by the probability of B."},{"Start":"03:10.640 ","End":"03:12.320","Text":"Now, let\u0027s plug in the numbers."},{"Start":"03:12.320 ","End":"03:15.140","Text":"The probability of A intersect B, that\u0027s right here,"},{"Start":"03:15.140 ","End":"03:22.370","Text":"that\u0027s 0.15 divided by the probability of B, that\u0027s 0.3."},{"Start":"03:22.370 ","End":"03:26.550","Text":"Now, that comes out to 0.5."},{"Start":"03:26.550 ","End":"03:31.910","Text":"The probability of a family having a second car or newer car"},{"Start":"03:31.910 ","End":"03:39.670","Text":"that\u0027s European given that their older car is European is equal to 1/2."}],"ID":12883},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"5m 39s","ChapterTopicVideoID":12405,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.200","Text":"This question we\u0027re told that a student takes 2 exams in"},{"Start":"00:03.200 ","End":"00:06.615","Text":"a semester: an economics exam and a statistics exam."},{"Start":"00:06.615 ","End":"00:09.570","Text":"Now, we define the following events: A is passing"},{"Start":"00:09.570 ","End":"00:12.900","Text":"the statistics exam and B is passing the economics exam."},{"Start":"00:12.900 ","End":"00:16.800","Text":"Now, the chances of passing the economics exam are 0.8,"},{"Start":"00:16.800 ","End":"00:20.730","Text":"so the probability of B that equals to 0.8."},{"Start":"00:20.730 ","End":"00:24.180","Text":"The chances of passing the statistics exam is 0.9,"},{"Start":"00:24.180 ","End":"00:28.650","Text":"so the probability of A that equals to 0.9."},{"Start":"00:28.650 ","End":"00:34.050","Text":"The chances of passing both exams are 0.75,"},{"Start":"00:34.050 ","End":"00:37.350","Text":"so the probability of A intersect B,"},{"Start":"00:37.350 ","End":"00:41.620","Text":"A and B, that equals to 0.75."},{"Start":"00:41.620 ","End":"00:44.990","Text":"Now, we\u0027re asked to calculate the chances of the following events."},{"Start":"00:44.990 ","End":"00:46.735","Text":"The first 1 is this,"},{"Start":"00:46.735 ","End":"00:49.640","Text":"the student passes the statistics exam."},{"Start":"00:49.640 ","End":"00:52.010","Text":"That\u0027s what we\u0027re told, and we\u0027re asked what\u0027s"},{"Start":"00:52.010 ","End":"00:54.920","Text":"the probability that he pass the economics exam?"},{"Start":"00:54.920 ","End":"00:58.040","Text":"Now, this is a conditional probability. Why is that?"},{"Start":"00:58.040 ","End":"01:04.570","Text":"Because we\u0027re asked about the probability of event B,"},{"Start":"01:04.570 ","End":"01:07.085","Text":"passing the economics exam,"},{"Start":"01:07.085 ","End":"01:12.560","Text":"given that he passed a statistics exam, given A."},{"Start":"01:12.560 ","End":"01:18.190","Text":"We\u0027re looking for the probability of B given A."},{"Start":"01:18.190 ","End":"01:22.280","Text":"Now, before we calculate this,"},{"Start":"01:22.280 ","End":"01:28.430","Text":"let\u0027s just see if we can arrange the data in a way that\u0027ll be comfortable for us."},{"Start":"01:28.430 ","End":"01:34.505","Text":"The way that we do that is by using our probability matrix."},{"Start":"01:34.505 ","End":"01:35.870","Text":"Now what do I mean by that?"},{"Start":"01:35.870 ","End":"01:38.300","Text":"Well, we have the 2 events A and B,"},{"Start":"01:38.300 ","End":"01:41.510","Text":"passing the statistics exam and passing the economics exam."},{"Start":"01:41.510 ","End":"01:43.520","Text":"Now here we have the matrix."},{"Start":"01:43.520 ","End":"01:46.010","Text":"We have the probability of A,"},{"Start":"01:46.010 ","End":"01:49.880","Text":"which is the probability of passing the statistics exam."},{"Start":"01:49.880 ","End":"01:52.970","Text":"Here we have the probability of B,"},{"Start":"01:52.970 ","End":"01:56.285","Text":"which is the probability of passing the economics exam."},{"Start":"01:56.285 ","End":"02:00.050","Text":"Now, if A is passing the statistics exam,"},{"Start":"02:00.050 ","End":"02:04.390","Text":"then A bar is failing the statistics exam."},{"Start":"02:04.390 ","End":"02:07.595","Text":"If B is passing the economics exam,"},{"Start":"02:07.595 ","End":"02:11.375","Text":"then B bar is failing the economics exam."},{"Start":"02:11.375 ","End":"02:15.275","Text":"Let\u0027s now plug in the numbers here."},{"Start":"02:15.275 ","End":"02:20.705","Text":"Well, we know, first of all that all the probabilities here have to add up to 1."},{"Start":"02:20.705 ","End":"02:24.590","Text":"So we\u0027ll put a 1 right here in this corner."},{"Start":"02:24.590 ","End":"02:27.950","Text":"Now, we\u0027re looking for the probability of A,"},{"Start":"02:27.950 ","End":"02:31.280","Text":"the probability of passing the statistics exam."},{"Start":"02:31.280 ","End":"02:33.620","Text":"We\u0027re given that, that\u0027s 0.9."},{"Start":"02:33.620 ","End":"02:37.840","Text":"So we\u0027ll write that down here, that\u0027s 0.9,"},{"Start":"02:37.840 ","End":"02:40.980","Text":"and the probability of passing the economics exam,"},{"Start":"02:40.980 ","End":"02:45.130","Text":"that\u0027s 0.8, so we\u0027ll write that here, 0.8."},{"Start":"02:45.380 ","End":"02:49.370","Text":"Now, in the middle here,"},{"Start":"02:49.370 ","End":"02:54.395","Text":"we have the probabilities of the intersect of the events."},{"Start":"02:54.395 ","End":"02:59.810","Text":"If we have A intersect B and we\u0027re looking for the probability, that\u0027s right here."},{"Start":"02:59.810 ","End":"03:03.035","Text":"This is A, this is B,"},{"Start":"03:03.035 ","End":"03:07.730","Text":"and the probability here will then be 0.75."},{"Start":"03:07.730 ","End":"03:09.350","Text":"That\u0027s given to us."},{"Start":"03:09.350 ","End":"03:12.740","Text":"Now, since we have all this,"},{"Start":"03:12.740 ","End":"03:17.810","Text":"let\u0027s just complete the probability matrix."},{"Start":"03:17.810 ","End":"03:22.400","Text":"We know that the sum of the probabilities have to add up to 1."},{"Start":"03:22.400 ","End":"03:25.590","Text":"The probability of passing, let\u0027s say,"},{"Start":"03:25.590 ","End":"03:29.000","Text":"the economics exam plus the probability"},{"Start":"03:29.000 ","End":"03:32.672","Text":"of failing the economics exam that has to equal to 1,"},{"Start":"03:32.672 ","End":"03:36.005","Text":"so 0.8 plus what equals to 1?"},{"Start":"03:36.005 ","End":"03:41.305","Text":"That\u0027s 0.2. The same for the statistics exam,"},{"Start":"03:41.305 ","End":"03:43.760","Text":"0.9 is the probability of passing"},{"Start":"03:43.760 ","End":"03:49.280","Text":"the statistics exam plus the probability of failing the statistics exam equals 1,"},{"Start":"03:49.280 ","End":"03:51.925","Text":"so that\u0027ll be 0.1."},{"Start":"03:51.925 ","End":"03:56.885","Text":"Let\u0027s take a look at the probabilities"},{"Start":"03:56.885 ","End":"04:03.075","Text":"inside the matrix right here, the intersect probabilities."},{"Start":"04:03.075 ","End":"04:09.629","Text":"0.75 plus what equals to 0.9? That\u0027s 0.15."},{"Start":"04:09.629 ","End":"04:13.620","Text":"0.75 plus what equals to 0.8?"},{"Start":"04:13.620 ","End":"04:19.110","Text":"That\u0027s 0.05, and 0.15 plus what equals 0.2?"},{"Start":"04:19.110 ","End":"04:21.555","Text":"That\u0027s 0.05 again."},{"Start":"04:21.555 ","End":"04:28.620","Text":"Now let\u0027s just make sure 0.05 plus 0.05 that equals to 0.1, we\u0027re okay."},{"Start":"04:28.620 ","End":"04:33.770","Text":"Now we can begin to calculate this probability."},{"Start":"04:33.770 ","End":"04:39.930","Text":"We\u0027re looking for the probability of B given A,"},{"Start":"04:39.930 ","End":"04:42.440","Text":"the student passed the statistics exam,"},{"Start":"04:42.440 ","End":"04:47.540","Text":"we\u0027re given A, what\u0027s the probability that he passed the economics exam?"},{"Start":"04:47.540 ","End":"04:52.730","Text":"Well, that equals by definition to the probability of A intersect B,"},{"Start":"04:52.730 ","End":"04:54.050","Text":"that\u0027s in the numerator,"},{"Start":"04:54.050 ","End":"04:56.990","Text":"and then the denominator we\u0027re given"},{"Start":"04:56.990 ","End":"05:00.335","Text":"or we have to write the probability of what\u0027s given,"},{"Start":"05:00.335 ","End":"05:03.505","Text":"probability now of A."},{"Start":"05:03.505 ","End":"05:06.169","Text":"Well, we have all this information."},{"Start":"05:06.169 ","End":"05:09.245","Text":"The probability of A that\u0027s right here,"},{"Start":"05:09.245 ","End":"05:16.535","Text":"that\u0027s 0.9, and what about the probability of A intersect B?"},{"Start":"05:16.535 ","End":"05:23.420","Text":"That\u0027s right here in the intersection of A and B, and that\u0027s 0.75."},{"Start":"05:23.420 ","End":"05:29.730","Text":"This comes out to 0.833. What\u0027s that?"},{"Start":"05:29.730 ","End":"05:34.610","Text":"Well, this is the probability of the student passing"},{"Start":"05:34.610 ","End":"05:40.590","Text":"the economics exam given that he\u0027s already passed the statistics exam."}],"ID":12884},{"Watched":false,"Name":"Exercise 1 - Parts b-c","Duration":"3m 4s","ChapterTopicVideoID":12406,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.997","Text":"In this section, we\u0027re given that the student passed economics exam now."},{"Start":"00:03.997 ","End":"00:08.175","Text":"We\u0027re asked what\u0027s the probability that he passed the statistics exam?"},{"Start":"00:08.175 ","End":"00:13.330","Text":"We\u0027re looking at the probability of what?"},{"Start":"00:13.730 ","End":"00:17.985","Text":"We have a conditional probability because we\u0027re given something,"},{"Start":"00:17.985 ","End":"00:20.390","Text":"the student passed the economics exam, right?"},{"Start":"00:20.390 ","End":"00:26.115","Text":"So we\u0027re given B and we\u0027re asked about the probability of A."},{"Start":"00:26.115 ","End":"00:29.955","Text":"The student passed the economics exam, that\u0027s B,"},{"Start":"00:29.955 ","End":"00:35.160","Text":"and we\u0027re asked about the probability that they passed the statistics exam, that\u0027s A."},{"Start":"00:35.160 ","End":"00:37.910","Text":"Again, in the denominator,"},{"Start":"00:37.910 ","End":"00:40.555","Text":"we have the probability of what\u0027s given,"},{"Start":"00:40.555 ","End":"00:42.050","Text":"and in the numerator,"},{"Start":"00:42.050 ","End":"00:45.745","Text":"we have the probability of the intersect."},{"Start":"00:45.745 ","End":"00:51.590","Text":"We have the probability matrix here and we have all the information that we need."},{"Start":"00:51.590 ","End":"00:53.285","Text":"The probability of B,"},{"Start":"00:53.285 ","End":"00:55.520","Text":"what\u0027s given, that\u0027s right here,"},{"Start":"00:55.520 ","End":"01:00.360","Text":"that\u0027s 0.8, and the probability of A intersect B,"},{"Start":"01:00.360 ","End":"01:04.305","Text":"that\u0027s, right here, that\u0027s 0.75."},{"Start":"01:04.305 ","End":"01:12.575","Text":"Now, that comes out to 0.9375. What does this mean?"},{"Start":"01:12.575 ","End":"01:17.750","Text":"This means that the probability of passing the statistics exam"},{"Start":"01:17.750 ","End":"01:25.100","Text":"given that he\u0027s already passed the economics exam is 0.9375."},{"Start":"01:25.100 ","End":"01:28.920","Text":"In this section, the student passed the economics exam,"},{"Start":"01:28.920 ","End":"01:34.145","Text":"we\u0027re asked, what\u0027s the probability that he failed the statistics exam?"},{"Start":"01:34.145 ","End":"01:42.080","Text":"We\u0027re looking for the probability that the student failed the statistics exam."},{"Start":"01:42.080 ","End":"01:45.170","Text":"Now passing the statistics examine is A,"},{"Start":"01:45.170 ","End":"01:49.565","Text":"so we\u0027re looking for the probability of A bar and what\u0027s given to us."},{"Start":"01:49.565 ","End":"01:52.610","Text":"Well, the student passed the economics exam,"},{"Start":"01:52.610 ","End":"01:54.280","Text":"so this is B."},{"Start":"01:54.280 ","End":"01:59.430","Text":"We\u0027re looking for the probability of A bar given B."},{"Start":"01:59.430 ","End":"02:04.790","Text":"Let\u0027s go to our probability matrix and see what we have here."},{"Start":"02:04.790 ","End":"02:12.335","Text":"This is the probability of A bar intersect B divided by the probability of B."},{"Start":"02:12.335 ","End":"02:17.115","Text":"Now, probability of B is the probability of what\u0027s given,"},{"Start":"02:17.115 ","End":"02:19.735","Text":"that the student passed the economics exam,"},{"Start":"02:19.735 ","End":"02:22.130","Text":"and that equals to 0.8."},{"Start":"02:22.130 ","End":"02:26.030","Text":"What about the probability of A bar intersect B?"},{"Start":"02:26.030 ","End":"02:29.540","Text":"Well, here\u0027s the probability of B right here in this slide,"},{"Start":"02:29.540 ","End":"02:33.065","Text":"and here\u0027s the probability of A bar right here."},{"Start":"02:33.065 ","End":"02:37.430","Text":"The intersect of B and A bar is right here"},{"Start":"02:37.430 ","End":"02:44.510","Text":"in this [inaudible] and the probability here would be 0.05."},{"Start":"02:44.510 ","End":"02:52.235","Text":"Now, that comes out to 0.0625."},{"Start":"02:52.235 ","End":"02:55.700","Text":"The probability of failing"},{"Start":"02:55.700 ","End":"03:04.480","Text":"the statistics exam given that he passed the economics exam is 0.0625."}],"ID":12885},{"Watched":false,"Name":"Exercise 1 - Parts d-e","Duration":"6m 34s","ChapterTopicVideoID":12407,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"In this section, we\u0027re told that the student failed the statistics exam and we are asked,"},{"Start":"00:04.635 ","End":"00:08.955","Text":"what\u0027s the probability that he also failed the economics exam?"},{"Start":"00:08.955 ","End":"00:11.790","Text":"Failing the statistics exam, well,"},{"Start":"00:11.790 ","End":"00:17.835","Text":"that\u0027s A bar and failing the economics exam, that\u0027s B bar."},{"Start":"00:17.835 ","End":"00:21.975","Text":"So we\u0027re asked about this probability."},{"Start":"00:21.975 ","End":"00:25.680","Text":"The probability that he failed the economics exam,"},{"Start":"00:25.680 ","End":"00:32.460","Text":"that\u0027s B bar, given that he failed the statistics exam, given A bar."},{"Start":"00:32.460 ","End":"00:35.575","Text":"Well, that equals to the probability of"},{"Start":"00:35.575 ","End":"00:43.480","Text":"A bar intersect B bar divided by the probability of A bar."},{"Start":"00:44.240 ","End":"00:47.525","Text":"Let\u0027s go to our probability matrix."},{"Start":"00:47.525 ","End":"00:51.020","Text":"Let\u0027s first of all calculate the probability of what\u0027s given,"},{"Start":"00:51.020 ","End":"00:55.655","Text":"the probability that he failed the statistics exam, A bar."},{"Start":"00:55.655 ","End":"00:58.460","Text":"Well, the probability of A bar right here,"},{"Start":"00:58.460 ","End":"01:02.515","Text":"this probability right here, that\u0027s 0.1."},{"Start":"01:02.515 ","End":"01:05.600","Text":"Now, what about the intersecting probability?"},{"Start":"01:05.600 ","End":"01:07.680","Text":"A bar intersect B bar."},{"Start":"01:07.680 ","End":"01:09.485","Text":"Here\u0027s A bar right here,"},{"Start":"01:09.485 ","End":"01:11.920","Text":"and here\u0027s B bar right here."},{"Start":"01:11.920 ","End":"01:14.270","Text":"So the intersect is this cell right here,"},{"Start":"01:14.270 ","End":"01:17.570","Text":"and the probability is 0.05."},{"Start":"01:17.570 ","End":"01:21.560","Text":"Now, that comes out to 0.5,"},{"Start":"01:21.560 ","End":"01:26.629","Text":"so the probability that the student failed"},{"Start":"01:26.629 ","End":"01:29.270","Text":"the economics exam given that he\u0027s"},{"Start":"01:29.270 ","End":"01:33.110","Text":"already failed the statistics exam, well, that\u0027s a 1/2."},{"Start":"01:33.110 ","End":"01:37.768","Text":"In this section, we\u0027re told that the student passed at least 1 exam,"},{"Start":"01:37.768 ","End":"01:41.885","Text":"and we\u0027re asked what\u0027s the probability that he passed both exams?"},{"Start":"01:41.885 ","End":"01:44.090","Text":"This is a little bit more complicated,"},{"Start":"01:44.090 ","End":"01:45.830","Text":"so let\u0027s get to it."},{"Start":"01:45.830 ","End":"01:52.890","Text":"Let\u0027s define the event that the student passed at least 1 exam."},{"Start":"01:53.570 ","End":"01:59.565","Text":"Now, that equals to A union B."},{"Start":"01:59.565 ","End":"02:02.540","Text":"He can either pass the statistics exam,"},{"Start":"02:02.540 ","End":"02:05.555","Text":"or the economics exam, or both of them."},{"Start":"02:05.555 ","End":"02:09.600","Text":"That means that he passed at least 1 exam."},{"Start":"02:09.600 ","End":"02:14.555","Text":"What about the event of passing both exams?"},{"Start":"02:14.555 ","End":"02:20.900","Text":"Well, passing both exam means that he passed A and he passed B,"},{"Start":"02:20.900 ","End":"02:24.605","Text":"he passed the statistics exam and the economics exam."},{"Start":"02:24.605 ","End":"02:26.570","Text":"In this conditional probability,"},{"Start":"02:26.570 ","End":"02:27.665","Text":"what are we asked?"},{"Start":"02:27.665 ","End":"02:30.375","Text":"We\u0027re asked for this."},{"Start":"02:30.375 ","End":"02:34.540","Text":"Well, what\u0027s given to us that the student passed at least 1 exam?"},{"Start":"02:34.540 ","End":"02:37.115","Text":"Here we write what\u0027s given,"},{"Start":"02:37.115 ","End":"02:40.190","Text":"and we\u0027re given that he passed at least 1 exam,"},{"Start":"02:40.190 ","End":"02:43.760","Text":"so that means that we\u0027re looking at A union B,"},{"Start":"02:43.760 ","End":"02:48.125","Text":"and we\u0027re looking for the probability that he passed both exams."},{"Start":"02:48.125 ","End":"02:52.080","Text":"Both exams is A intersect B."},{"Start":"02:53.000 ","End":"02:58.310","Text":"This then is the conditional probability that we have to calculate."},{"Start":"02:58.310 ","End":"03:01.430","Text":"Now, that equals to the following. In the denominator,"},{"Start":"03:01.430 ","End":"03:04.090","Text":"we have the probability of what\u0027s given,"},{"Start":"03:04.090 ","End":"03:06.060","Text":"A union B,"},{"Start":"03:06.060 ","End":"03:09.500","Text":"and in the numerator we have the intersects."},{"Start":"03:09.500 ","End":"03:14.225","Text":"That\u0027s the probability of A intersect B,"},{"Start":"03:14.225 ","End":"03:19.890","Text":"intersect A union B."},{"Start":"03:20.900 ","End":"03:24.995","Text":"This looks complicated, but it really isn\u0027t."},{"Start":"03:24.995 ","End":"03:28.085","Text":"Let\u0027s just calculate the probabilities right"},{"Start":"03:28.085 ","End":"03:31.850","Text":"here for the denominator and the numerator separately,"},{"Start":"03:31.850 ","End":"03:35.455","Text":"and then we\u0027ll calculate the total probability here."},{"Start":"03:35.455 ","End":"03:38.310","Text":"The probability of the denominator,"},{"Start":"03:38.310 ","End":"03:41.730","Text":"well, that\u0027s the probability of A union B."},{"Start":"03:41.730 ","End":"03:47.300","Text":"Now, let\u0027s use the rule that this thing equals to the probability of A plus"},{"Start":"03:47.300 ","End":"03:53.635","Text":"the probability of B minus the probability of A intersect B."},{"Start":"03:53.635 ","End":"03:58.790","Text":"Now, let\u0027s go to our probability matrix and see what these probabilities are."},{"Start":"03:58.790 ","End":"04:02.825","Text":"Well, the probability of A that equals to 0.9,"},{"Start":"04:02.825 ","End":"04:10.010","Text":"and the probability of B that equals to 0.8 minus the probability of A intersect B,"},{"Start":"04:10.010 ","End":"04:17.250","Text":"well, that equals to 0.75 and this thing comes out to 0.95."},{"Start":"04:17.250 ","End":"04:19.028","Text":"So we have the denominator."},{"Start":"04:19.028 ","End":"04:21.210","Text":"Now, what about the numerator?"},{"Start":"04:21.210 ","End":"04:24.425","Text":"This looks complicated and frightening,"},{"Start":"04:24.425 ","End":"04:31.280","Text":"but let\u0027s look at these events right here using the Venn diagram."},{"Start":"04:33.110 ","End":"04:39.030","Text":"The first thing that we want to do is to color in A intersect B."},{"Start":"04:39.030 ","End":"04:41.660","Text":"We have here A, we have here B."},{"Start":"04:41.660 ","End":"04:47.150","Text":"So A intersect B would be this area right here."},{"Start":"04:47.150 ","End":"04:50.060","Text":"What about A union B?"},{"Start":"04:50.060 ","End":"04:55.745","Text":"Well, A union B would be all this area of A and B."},{"Start":"04:55.745 ","End":"04:59.565","Text":"That\u0027s this area in yellow right here."},{"Start":"04:59.565 ","End":"05:03.380","Text":"Again, A union B is the area in yellow,"},{"Start":"05:03.380 ","End":"05:05.240","Text":"A intersect B, well,"},{"Start":"05:05.240 ","End":"05:09.295","Text":"that\u0027s the area in red."},{"Start":"05:09.295 ","End":"05:13.155","Text":"Now, what\u0027s the intersect of both these areas?"},{"Start":"05:13.155 ","End":"05:19.115","Text":"Well, the intersect of both these areas would be A intersect B and that makes sense."},{"Start":"05:19.115 ","End":"05:27.630","Text":"Because if the student passed at least 1 of the exams,"},{"Start":"05:27.630 ","End":"05:34.085","Text":"and we want to intersect that with the student passing both of the exams."},{"Start":"05:34.085 ","End":"05:38.165","Text":"Well, that means that the student actually passed both of the exams."},{"Start":"05:38.165 ","End":"05:40.910","Text":"This thing right here,"},{"Start":"05:40.910 ","End":"05:52.620","Text":"that equals to the probability of A intersect B divided by the probability of A union B."},{"Start":"05:52.620 ","End":"05:55.770","Text":"Well, what does that equal to?"},{"Start":"05:55.770 ","End":"06:04.350","Text":"The probability of A intersect B divided by the probability of A union B."},{"Start":"06:04.350 ","End":"06:06.860","Text":"The probability of A union B,"},{"Start":"06:06.860 ","End":"06:09.616","Text":"we\u0027ve calculated that, that\u0027s 0.95,"},{"Start":"06:09.616 ","End":"06:12.740","Text":"and the probability of A intersect B,"},{"Start":"06:12.740 ","End":"06:16.100","Text":"well, that\u0027s right here, that\u0027s 0.75."},{"Start":"06:16.100 ","End":"06:21.160","Text":"Now, this comes out to 0.789."},{"Start":"06:22.580 ","End":"06:28.250","Text":"This then would be the probability that the student"},{"Start":"06:28.250 ","End":"06:34.500","Text":"passed both exams given that he passed at least 1 of them."}],"ID":12886},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"3m 40s","ChapterTopicVideoID":12408,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"In this question, we\u0027re told that there are"},{"Start":"00:01.890 ","End":"00:05.965","Text":"2 cellular telephone companies in a country: Soft and Bell."},{"Start":"00:05.965 ","End":"00:09.090","Text":"Now 30 percent of the adults in the country are registered with"},{"Start":"00:09.090 ","End":"00:12.750","Text":"Bell and 60 percent of them are registered with Soft."},{"Start":"00:12.750 ","End":"00:16.350","Text":"15 percent of the adults in the country have no cellular telephone."},{"Start":"00:16.350 ","End":"00:21.825","Text":"We\u0027re asked, what\u0027s the proportion of adults that are registered with both companies?"},{"Start":"00:21.825 ","End":"00:27.300","Text":"The first thing that we want to do is define our events."},{"Start":"00:27.300 ","End":"00:32.900","Text":"Let\u0027s define A as adults who are registered at"},{"Start":"00:32.900 ","End":"00:37.100","Text":"the Bell telephone company and B is the event"},{"Start":"00:37.100 ","End":"00:41.545","Text":"that the adults are registered at the Soft telephone company."},{"Start":"00:41.545 ","End":"00:44.375","Text":"We\u0027re given the following information."},{"Start":"00:44.375 ","End":"00:46.460","Text":"The probability of A,"},{"Start":"00:46.460 ","End":"00:51.430","Text":"the probability of people or adults that are registered with Bell,"},{"Start":"00:51.430 ","End":"00:55.010","Text":"that equals to 0.3 or 30 percent."},{"Start":"00:55.010 ","End":"00:58.040","Text":"The probability of people that are registered at"},{"Start":"00:58.040 ","End":"01:02.640","Text":"the Soft telephone company, that\u0027s 60 percent."},{"Start":"01:02.640 ","End":"01:05.670","Text":"The probability now here,"},{"Start":"01:05.670 ","End":"01:10.250","Text":"60 percent of the adults in the country have no cellular telephone."},{"Start":"01:10.250 ","End":"01:16.380","Text":"We\u0027re looking at the probability of not A and not B,"},{"Start":"01:16.380 ","End":"01:19.420","Text":"and that equals to 0.15."},{"Start":"01:19.970 ","End":"01:24.755","Text":"Again, as with most of these types of questions,"},{"Start":"01:24.755 ","End":"01:29.665","Text":"the best thing to do is to create a probability matrix."},{"Start":"01:29.665 ","End":"01:31.635","Text":"Here we have our matrix."},{"Start":"01:31.635 ","End":"01:33.480","Text":"This is A, this is B,"},{"Start":"01:33.480 ","End":"01:35.570","Text":"this is not A and not B."},{"Start":"01:35.570 ","End":"01:39.350","Text":"Now we know that the sum of all the probabilities have to add up to 1,"},{"Start":"01:39.350 ","End":"01:40.940","Text":"so we\u0027ll put a 1 here."},{"Start":"01:40.940 ","End":"01:44.060","Text":"The probability of A,"},{"Start":"01:44.060 ","End":"01:48.560","Text":"number of adults or the proportion of adults that are"},{"Start":"01:48.560 ","End":"01:53.550","Text":"registered at the Bell telephone company, that\u0027s 0.3."},{"Start":"01:53.890 ","End":"02:00.005","Text":"The probability of B, that equals to 0.6."},{"Start":"02:00.005 ","End":"02:06.630","Text":"Now, the probability of not A intersect not B, that\u0027s right here."},{"Start":"02:06.630 ","End":"02:08.520","Text":"This is not B this is not A,"},{"Start":"02:08.520 ","End":"02:10.470","Text":"so the intersect is this cell right here."},{"Start":"02:10.470 ","End":"02:13.390","Text":"Well, that equals to 0.15."},{"Start":"02:14.780 ","End":"02:17.130","Text":"Now that we have this,"},{"Start":"02:17.130 ","End":"02:21.794","Text":"let\u0027s complete the probability matrix."},{"Start":"02:21.794 ","End":"02:25.230","Text":"Well, 0.6 plus 0.4,"},{"Start":"02:25.230 ","End":"02:26.550","Text":"that equals to 1."},{"Start":"02:26.550 ","End":"02:29.910","Text":"0.3 plus 0.7,"},{"Start":"02:29.910 ","End":"02:31.785","Text":"that equals to 1."},{"Start":"02:31.785 ","End":"02:38.220","Text":"0.55 plus 0.15, that equals to 0.7."},{"Start":"02:38.220 ","End":"02:42.960","Text":"0.05 plus 0.55,"},{"Start":"02:42.960 ","End":"02:45.240","Text":"that equals to 0.6."},{"Start":"02:45.240 ","End":"02:52.770","Text":"0.25 plus 0.15, that equals to 0.4."},{"Start":"02:52.770 ","End":"02:54.905","Text":"Now that we have that,"},{"Start":"02:54.905 ","End":"02:58.825","Text":"we can start answering the questions."},{"Start":"02:58.825 ","End":"03:01.640","Text":"In the first section, we\u0027re asked,"},{"Start":"03:01.640 ","End":"03:06.800","Text":"what is the proportion of adults that are registered with both companies?"},{"Start":"03:06.800 ","End":"03:12.510","Text":"We\u0027re looking here for the probability of A and B."},{"Start":"03:12.510 ","End":"03:15.470","Text":"We see that this is not a conditional probability,"},{"Start":"03:15.470 ","End":"03:17.990","Text":"it\u0027s a straight probability question."},{"Start":"03:17.990 ","End":"03:21.665","Text":"Well, we have our probability matrix here."},{"Start":"03:21.665 ","End":"03:24.985","Text":"The probability of A intersect B,"},{"Start":"03:24.985 ","End":"03:29.117","Text":"adults that are registered at Bell and at Soft,"},{"Start":"03:29.117 ","End":"03:31.520","Text":"that\u0027s this intersection right here,"},{"Start":"03:31.520 ","End":"03:32.810","Text":"this is A, this is B,"},{"Start":"03:32.810 ","End":"03:34.640","Text":"so the intersection is right here."},{"Start":"03:34.640 ","End":"03:41.200","Text":"That equals to 0.05 or 5 percent."}],"ID":12887},{"Watched":false,"Name":"Exercise 2 - Parts b-c","Duration":"3m 4s","ChapterTopicVideoID":12410,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.900","Text":"In this section, we\u0027re told that a person is selected and he\u0027s registered with Soft,"},{"Start":"00:04.900 ","End":"00:09.479","Text":"and we\u0027re asked what\u0027s the probability that he\u0027s also registered with Bell."},{"Start":"00:09.479 ","End":"00:12.540","Text":"This is a conditional probability. Why is that?"},{"Start":"00:12.540 ","End":"00:14.625","Text":"Because we\u0027re given something,"},{"Start":"00:14.625 ","End":"00:18.105","Text":"the selected person is registered with Soft."},{"Start":"00:18.105 ","End":"00:23.130","Text":"We\u0027re looking at the probability, and what\u0027s given?"},{"Start":"00:23.130 ","End":"00:24.720","Text":"That he\u0027s registered with Soft,"},{"Start":"00:24.720 ","End":"00:26.700","Text":"that means B here."},{"Start":"00:26.700 ","End":"00:29.010","Text":"We\u0027re asked about the probability"},{"Start":"00:29.010 ","End":"00:33.660","Text":"of this person also being registered with Bell, with A."},{"Start":"00:33.660 ","End":"00:36.050","Text":"This is the probability that we\u0027re looking at,"},{"Start":"00:36.050 ","End":"00:38.060","Text":"the probability of A given B,"},{"Start":"00:38.060 ","End":"00:41.420","Text":"the probability that a person is"},{"Start":"00:41.420 ","End":"00:46.040","Text":"registered with Bell given that he\u0027s registered with Soft."},{"Start":"00:46.040 ","End":"00:51.755","Text":"In the denominator, we have the probability of what\u0027s given,"},{"Start":"00:51.755 ","End":"00:53.203","Text":"that\u0027s the probability of B,"},{"Start":"00:53.203 ","End":"00:54.635","Text":"and in the numerator,"},{"Start":"00:54.635 ","End":"00:57.110","Text":"we have the probability of the intersect,"},{"Start":"00:57.110 ","End":"00:59.625","Text":"that\u0027s A intersect B."},{"Start":"00:59.625 ","End":"01:02.660","Text":"Well, what\u0027s the probability of B?"},{"Start":"01:02.660 ","End":"01:04.895","Text":"Let\u0027s go to our probability table,"},{"Start":"01:04.895 ","End":"01:07.190","Text":"and this is equal to 0.6."},{"Start":"01:07.190 ","End":"01:09.530","Text":"The probability of A intersect B,"},{"Start":"01:09.530 ","End":"01:11.360","Text":"here\u0027s A, here\u0027s B,"},{"Start":"01:11.360 ","End":"01:15.095","Text":"and that\u0027s the intersect, that\u0027s 0.05."},{"Start":"01:15.095 ","End":"01:20.730","Text":"Now, this comes out to 0.0833."},{"Start":"01:20.730 ","End":"01:24.170","Text":"So this is a probability that a person is"},{"Start":"01:24.170 ","End":"01:29.010","Text":"registered at Bell given that he\u0027s registered with Soft."},{"Start":"01:29.010 ","End":"01:32.940","Text":"In this section, we\u0027re asked if a person is not registered with Bell,"},{"Start":"01:32.940 ","End":"01:36.350","Text":"what\u0027s the probability that he\u0027s registered with Soft?"},{"Start":"01:36.350 ","End":"01:39.290","Text":"Again, this is a conditional probability because"},{"Start":"01:39.290 ","End":"01:42.550","Text":"we\u0027re given that this person is not registered with Bell,"},{"Start":"01:42.550 ","End":"01:46.720","Text":"and we\u0027re asked about the probability that he\u0027s registered with Soft."},{"Start":"01:46.720 ","End":"01:48.530","Text":"Let\u0027s just write the probability."},{"Start":"01:48.530 ","End":"01:50.330","Text":"This is the probability."},{"Start":"01:50.330 ","End":"01:51.875","Text":"Now, what\u0027s given?"},{"Start":"01:51.875 ","End":"01:54.200","Text":"That he\u0027s not registered with Bell,"},{"Start":"01:54.200 ","End":"01:56.225","Text":"so that means not A."},{"Start":"01:56.225 ","End":"01:58.490","Text":"What do we want to find out?"},{"Start":"01:58.490 ","End":"02:01.175","Text":"The probability that he\u0027s registered with Soft,"},{"Start":"02:01.175 ","End":"02:06.930","Text":"so that\u0027s the probability of B given not A."},{"Start":"02:06.930 ","End":"02:09.184","Text":"Well, that equals,"},{"Start":"02:09.184 ","End":"02:12.875","Text":"in the denominator we have the probability of what\u0027s given,"},{"Start":"02:12.875 ","End":"02:14.680","Text":"probability of not A,"},{"Start":"02:14.680 ","End":"02:18.460","Text":"and in the numerator we have the probability of the intersect,"},{"Start":"02:18.460 ","End":"02:21.865","Text":"not A and B."},{"Start":"02:21.865 ","End":"02:24.920","Text":"Let\u0027s go to our probability table."},{"Start":"02:24.920 ","End":"02:26.840","Text":"The probability of not A,"},{"Start":"02:26.840 ","End":"02:28.790","Text":"here\u0027s the not A column,"},{"Start":"02:28.790 ","End":"02:31.910","Text":"so the probability of not A is 0.7."},{"Start":"02:31.910 ","End":"02:35.070","Text":"In the denominator, we have 0.7."},{"Start":"02:35.070 ","End":"02:36.450","Text":"In the numerator,"},{"Start":"02:36.450 ","End":"02:39.390","Text":"naught A and B, that\u0027s the intersect."},{"Start":"02:39.390 ","End":"02:43.440","Text":"Here\u0027s not A and here is B right here,"},{"Start":"02:43.440 ","End":"02:45.375","Text":"so the Intersect is this cell."},{"Start":"02:45.375 ","End":"02:47.295","Text":"That\u0027s 0.55."},{"Start":"02:47.295 ","End":"02:49.680","Text":"That\u0027s the probability of the intersect."},{"Start":"02:49.680 ","End":"02:54.375","Text":"Now, this comes out to 0.78."},{"Start":"02:54.375 ","End":"02:58.850","Text":"This is the probability that a person is registered with"},{"Start":"02:58.850 ","End":"03:04.860","Text":"Soft if it\u0027s known that he\u0027s not registered with Bell."}],"ID":12888},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"5m 1s","ChapterTopicVideoID":12409,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this section we\u0027re asked if a person is registered with only 1 company,"},{"Start":"00:03.930 ","End":"00:07.080","Text":"what\u0027s the probability that he\u0027s registered with Soft?"},{"Start":"00:07.080 ","End":"00:10.830","Text":"Well, again, this is a conditional probability question. Why is that?"},{"Start":"00:10.830 ","End":"00:12.870","Text":"Because we\u0027re given something."},{"Start":"00:12.870 ","End":"00:16.755","Text":"That the person is registered with only 1 company."},{"Start":"00:16.755 ","End":"00:20.910","Text":"Let\u0027s write down this probability. Here we go."},{"Start":"00:20.910 ","End":"00:24.435","Text":"This is a probability that a person is registered in Soft,"},{"Start":"00:24.435 ","End":"00:27.465","Text":"given that he is registered in 1 company."},{"Start":"00:27.465 ","End":"00:31.965","Text":"Now let\u0027s translate that into the events here."},{"Start":"00:31.965 ","End":"00:35.600","Text":"Well, that equals to the probability."},{"Start":"00:35.600 ","End":"00:37.700","Text":"Now if a person is registered at Soft,"},{"Start":"00:37.700 ","End":"00:43.400","Text":"that\u0027s B, given that that person is registered at 1 company."},{"Start":"00:43.400 ","End":"00:45.470","Text":"Now if he is registered in 1 company,"},{"Start":"00:45.470 ","End":"00:53.255","Text":"then that person is registered either at Bell and not at Soft or at Soft and not at Bell."},{"Start":"00:53.255 ","End":"00:54.890","Text":"Let\u0027s write that down."},{"Start":"00:54.890 ","End":"01:04.870","Text":"The person could be at A and not B or at not A and B."},{"Start":"01:04.880 ","End":"01:08.160","Text":"That\u0027s the probability."},{"Start":"01:08.160 ","End":"01:11.990","Text":"That\u0027s the events that were defined here by Soft,"},{"Start":"01:11.990 ","End":"01:13.895","Text":"that\u0027s B, and 1 company,"},{"Start":"01:13.895 ","End":"01:17.525","Text":"that\u0027s A and not B or not A and B."},{"Start":"01:17.525 ","End":"01:20.045","Text":"This is a conditional probability."},{"Start":"01:20.045 ","End":"01:21.920","Text":"How do we solve that?"},{"Start":"01:21.920 ","End":"01:26.510","Text":"In the denominator we have the probability of what\u0027s given."},{"Start":"01:26.510 ","End":"01:36.705","Text":"That\u0027s the probability of A and not B or not A and B."},{"Start":"01:36.705 ","End":"01:40.280","Text":"In the numerator, where we have the intersect,"},{"Start":"01:40.280 ","End":"01:44.740","Text":"so we have here the probability of B."},{"Start":"01:44.740 ","End":"01:47.100","Text":"Now what do we have here?"},{"Start":"01:47.100 ","End":"01:57.120","Text":"We have A and not B or not A and B."},{"Start":"01:57.120 ","End":"02:07.135","Text":"Now, this then is the probability that we have to calculate in the numerator."},{"Start":"02:07.135 ","End":"02:12.705","Text":"Let\u0027s just take a look at each 1 of the probabilities independently,"},{"Start":"02:12.705 ","End":"02:14.670","Text":"the numerator and the denominator."},{"Start":"02:14.670 ","End":"02:19.910","Text":"We\u0027ll calculate them and then we\u0027ll calculate the whole conditional probability."},{"Start":"02:19.910 ","End":"02:24.750","Text":"The best way to do that is to use the Venn diagram."},{"Start":"02:26.570 ","End":"02:29.775","Text":"Here\u0027s a Venn diagram."},{"Start":"02:29.775 ","End":"02:33.275","Text":"What we want to do right now is to calculate"},{"Start":"02:33.275 ","End":"02:37.870","Text":"the probability of the denominator and see what are the events in the denominator."},{"Start":"02:37.870 ","End":"02:40.305","Text":"Well, that\u0027s A and not B."},{"Start":"02:40.305 ","End":"02:44.950","Text":"A and not B is this area right here."},{"Start":"02:46.190 ","End":"02:53.220","Text":"That\u0027s this area in yellow or not A and B."},{"Start":"02:53.220 ","End":"02:54.405","Text":"Well, not A and B,"},{"Start":"02:54.405 ","End":"02:57.700","Text":"that\u0027s this area right here."},{"Start":"02:57.950 ","End":"03:01.620","Text":"Again, that\u0027s also in yellow."},{"Start":"03:01.620 ","End":"03:06.185","Text":"What would be the probability here?"},{"Start":"03:06.185 ","End":"03:08.790","Text":"Well, A and not B,"},{"Start":"03:08.790 ","End":"03:10.760","Text":"let\u0027s look at our probability table."},{"Start":"03:10.760 ","End":"03:14.940","Text":"Here\u0027s A and here\u0027s not B, that\u0027s 0.25."},{"Start":"03:19.190 ","End":"03:22.750","Text":"Here\u0027s not A and here\u0027s B,"},{"Start":"03:22.750 ","End":"03:27.820","Text":"so this thing right here would be 0.55."},{"Start":"03:28.220 ","End":"03:38.090","Text":"We know that in the denominator we have 0.25 plus 0.55."},{"Start":"03:38.090 ","End":"03:41.255","Text":"Now let\u0027s take a look at the numerator."},{"Start":"03:41.255 ","End":"03:48.880","Text":"The numerator is this area in yellow intersect B."},{"Start":"03:49.130 ","End":"03:53.375","Text":"Let\u0027s intersect this area in yellow with B."},{"Start":"03:53.375 ","End":"03:57.060","Text":"That\u0027ll be this area right here."},{"Start":"03:58.760 ","End":"04:02.535","Text":"That\u0027s the area then in green."},{"Start":"04:02.535 ","End":"04:05.685","Text":"But what is this area right here?"},{"Start":"04:05.685 ","End":"04:09.810","Text":"This is actually B and not A."},{"Start":"04:09.810 ","End":"04:14.225","Text":"This thing right here,"},{"Start":"04:14.225 ","End":"04:16.715","Text":"the numerator right here,"},{"Start":"04:16.715 ","End":"04:22.055","Text":"that\u0027ll be the probability of B and not A."},{"Start":"04:22.055 ","End":"04:25.585","Text":"What\u0027s the probability of B and not A?"},{"Start":"04:25.585 ","End":"04:27.600","Text":"Here\u0027s B, here\u0027s not A,"},{"Start":"04:27.600 ","End":"04:28.995","Text":"so that\u0027s right here."},{"Start":"04:28.995 ","End":"04:31.620","Text":"That\u0027s 0.55."},{"Start":"04:31.620 ","End":"04:35.510","Text":"Here we go. We\u0027ve calculated the probabilities for"},{"Start":"04:35.510 ","End":"04:40.250","Text":"the numerator and the denominator and that equals to"},{"Start":"04:40.250 ","End":"04:50.380","Text":"0.55 divided by 0.8 and that comes out to 0.6875."},{"Start":"04:51.100 ","End":"04:56.330","Text":"This then is the probability that a person"},{"Start":"04:56.330 ","End":"05:01.830","Text":"is registered with Soft if he\u0027s registered with only 1 company."}],"ID":12889},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"3m 13s","ChapterTopicVideoID":12411,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.936","Text":"In this question, we have 2 parking lots at a college"},{"Start":"00:02.936 ","End":"00:05.790","Text":": a small parking lot and a large parking lot."},{"Start":"00:05.790 ","End":"00:07.485","Text":"Now, at 8:00 AM,"},{"Start":"00:07.485 ","End":"00:09.930","Text":"there\u0027s a 60 percent chance that there\u0027s"},{"Start":"00:09.930 ","End":"00:13.545","Text":"an available parking place in the large parking lot,"},{"Start":"00:13.545 ","End":"00:18.255","Text":"a 30 percent chance that there\u0027s an available parking place in the small parking lot,"},{"Start":"00:18.255 ","End":"00:23.940","Text":"and a 20 percent chance that there\u0027s an available parking space in both parking lots."},{"Start":"00:23.940 ","End":"00:27.660","Text":"We\u0027re asked what\u0027s the probability that there is a parking place"},{"Start":"00:27.660 ","End":"00:33.570","Text":"available at 8:00 AM only in the college\u0027s large parking lot?"},{"Start":"00:33.570 ","End":"00:35.520","Text":"Before we answer that,"},{"Start":"00:35.520 ","End":"00:40.140","Text":"let\u0027s just write down the information that we have."},{"Start":"00:40.140 ","End":"00:43.625","Text":"So we define 2 events, A and B."},{"Start":"00:43.625 ","End":"00:47.490","Text":"A is a parking that\u0027s available in the large parking lot,"},{"Start":"00:47.490 ","End":"00:50.705","Text":"and B, parking that\u0027s available on the small parking lot."},{"Start":"00:50.705 ","End":"00:54.200","Text":"Now we\u0027re given that the probability of A,"},{"Start":"00:54.200 ","End":"00:56.330","Text":"that equals to 0.6,"},{"Start":"00:56.330 ","End":"00:58.694","Text":"and the probability of B,"},{"Start":"00:58.694 ","End":"01:01.460","Text":"that equals to 0.3,"},{"Start":"01:01.460 ","End":"01:10.020","Text":"and the probability of A intersect B to this available parking at both parking lots,"},{"Start":"01:10.020 ","End":"01:13.065","Text":"that equals to 0.2."},{"Start":"01:13.065 ","End":"01:17.210","Text":"Let\u0027s just write this out in such a way that will be"},{"Start":"01:17.210 ","End":"01:22.175","Text":"comfortable for us using the probability matrix."},{"Start":"01:22.175 ","End":"01:24.485","Text":"Here\u0027s our matrix."},{"Start":"01:24.485 ","End":"01:27.260","Text":"Now, the probability of A,"},{"Start":"01:27.260 ","End":"01:31.730","Text":"that\u0027s 0.6, that will be right here, 0.6."},{"Start":"01:31.730 ","End":"01:35.720","Text":"The probability of B, that\u0027s 0.3."},{"Start":"01:35.720 ","End":"01:37.295","Text":"We\u0027ll write that here."},{"Start":"01:37.295 ","End":"01:40.070","Text":"The probability of A intersect B,"},{"Start":"01:40.070 ","End":"01:43.240","Text":"that\u0027s this cell right here, that\u0027s 0.2."},{"Start":"01:43.240 ","End":"01:46.415","Text":"We know that the sum of the probabilities,"},{"Start":"01:46.415 ","End":"01:48.020","Text":"that equals to 1."},{"Start":"01:48.020 ","End":"01:54.290","Text":"Let\u0027s just now complete the probabilities here,"},{"Start":"01:54.290 ","End":"01:57.530","Text":"0.3 plus 0.7, that\u0027s 1."},{"Start":"01:57.530 ","End":"02:01.235","Text":"0.6 plus 0.4, that\u0027s 1."},{"Start":"02:01.235 ","End":"02:05.265","Text":"0.2 plus 0.1, that\u0027s 0.3."},{"Start":"02:05.265 ","End":"02:09.330","Text":"0.2 plus 0.4, that\u0027s 0.6,"},{"Start":"02:09.330 ","End":"02:12.480","Text":"and 0.1 plus 0.3,"},{"Start":"02:12.480 ","End":"02:14.970","Text":"that equals to 0.4."},{"Start":"02:14.970 ","End":"02:18.395","Text":"0.4 plus 0.3, that equals to 0.7."},{"Start":"02:18.395 ","End":"02:20.915","Text":"We\u0027re good. Now,"},{"Start":"02:20.915 ","End":"02:22.565","Text":"let\u0027s go to the question."},{"Start":"02:22.565 ","End":"02:27.050","Text":"We\u0027re asked, what\u0027s the probability that there\u0027s a parking place available"},{"Start":"02:27.050 ","End":"02:31.160","Text":"at 8:00 AM only in the college\u0027s large parking lot?"},{"Start":"02:31.160 ","End":"02:40.640","Text":"We\u0027re looking for the probability of A intersect not B."},{"Start":"02:40.640 ","End":"02:46.550","Text":"Why is that? Well, we\u0027re looking at the large parking lot only."},{"Start":"02:46.550 ","End":"02:48.320","Text":"If it\u0027s only,"},{"Start":"02:48.320 ","End":"02:51.000","Text":"then it\u0027s A and not B."},{"Start":"02:51.000 ","End":"02:55.305","Text":"It\u0027s in the large parking lot and not the small parking lot."},{"Start":"02:55.305 ","End":"02:58.005","Text":"That\u0027s A and not B."},{"Start":"02:58.005 ","End":"03:01.700","Text":"What\u0027s this probability right here?"},{"Start":"03:01.700 ","End":"03:04.865","Text":"Let\u0027s take a look at a probability matrix,"},{"Start":"03:04.865 ","End":"03:08.840","Text":"the intersect between A and not B,"},{"Start":"03:08.840 ","End":"03:10.490","Text":"that\u0027s this cell right here."},{"Start":"03:10.490 ","End":"03:14.460","Text":"That equals to 0.4."}],"ID":12890},{"Watched":false,"Name":"Exercise 3 - Parts b-c","Duration":"2m 46s","ChapterTopicVideoID":12413,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"In this section, it\u0027s known that there\u0027s"},{"Start":"00:02.250 ","End":"00:05.625","Text":"a parking place available in the small parking lot at 8:00 AM."},{"Start":"00:05.625 ","End":"00:08.370","Text":"We\u0027re asked, what\u0027s the probability that there will be"},{"Start":"00:08.370 ","End":"00:11.220","Text":"a parking space available in the large parking lot?"},{"Start":"00:11.220 ","End":"00:15.690","Text":"Well, this is a conditional probability because we\u0027re given something."},{"Start":"00:15.690 ","End":"00:18.060","Text":"Something is known, and what\u0027s that?"},{"Start":"00:18.060 ","End":"00:21.150","Text":"That there\u0027s a parking place available in the small parking lot."},{"Start":"00:21.150 ","End":"00:23.325","Text":"So when we write the probability,"},{"Start":"00:23.325 ","End":"00:24.945","Text":"what\u0027s given to us?"},{"Start":"00:24.945 ","End":"00:27.910","Text":"Well, we\u0027re given B,"},{"Start":"00:27.910 ","End":"00:31.400","Text":"that there\u0027s a parking place available in the small parking lot."},{"Start":"00:31.400 ","End":"00:34.100","Text":"We\u0027re asked, what\u0027s the probability that there will be"},{"Start":"00:34.100 ","End":"00:37.470","Text":"a parking space available in the large parking lot? That\u0027s A."},{"Start":"00:37.470 ","End":"00:41.795","Text":"So this is a conditional probability that we\u0027re looking for."},{"Start":"00:41.795 ","End":"00:47.450","Text":"That equals to the probability of A intersect B, that\u0027s the numerator."},{"Start":"00:47.450 ","End":"00:50.210","Text":"The intersect of the 2 events divided by"},{"Start":"00:50.210 ","End":"00:54.410","Text":"the probability of the events that\u0027s given or known."},{"Start":"00:54.410 ","End":"00:58.590","Text":"The probability of A intersect B,"},{"Start":"00:58.590 ","End":"01:00.470","Text":"let\u0027s look at a probability table,"},{"Start":"01:00.470 ","End":"01:07.835","Text":"that\u0027s 0.2 divided by the probability of B, that\u0027s 0.3."},{"Start":"01:07.835 ","End":"01:15.380","Text":"That equals to 2/3 or 0.667."},{"Start":"01:15.380 ","End":"01:17.990","Text":"In this section we\u0027re asked if there\u0027s no parking place"},{"Start":"01:17.990 ","End":"01:20.810","Text":"available in the large parking lot at 8:00 AM,"},{"Start":"01:20.810 ","End":"01:22.790","Text":"what\u0027s the probability that there will be"},{"Start":"01:22.790 ","End":"01:26.120","Text":"a parking place available in the small parking lot?"},{"Start":"01:26.120 ","End":"01:29.675","Text":"We\u0027re looking at a conditional probability, right?"},{"Start":"01:29.675 ","End":"01:35.360","Text":"We\u0027re given that there\u0027s no parking space available in the large parking lot."},{"Start":"01:35.360 ","End":"01:37.760","Text":"So we\u0027re looking for the probability."},{"Start":"01:37.760 ","End":"01:39.245","Text":"Now, what\u0027s given?"},{"Start":"01:39.245 ","End":"01:41.625","Text":"Not A."},{"Start":"01:41.625 ","End":"01:46.846","Text":"A means that there is parking available on the large parking lot,"},{"Start":"01:46.846 ","End":"01:50.750","Text":"and here, we\u0027re saying that there\u0027s no parking place."},{"Start":"01:50.750 ","End":"01:53.870","Text":"That\u0027s what\u0027s given. What are we asked?"},{"Start":"01:53.870 ","End":"01:56.300","Text":"What\u0027s the probability that there will be"},{"Start":"01:56.300 ","End":"01:58.940","Text":"a parking place available in the small parking lot?"},{"Start":"01:58.940 ","End":"02:00.755","Text":"So we\u0027re looking at B here."},{"Start":"02:00.755 ","End":"02:03.050","Text":"So this is a conditional probability,"},{"Start":"02:03.050 ","End":"02:06.710","Text":"the probability of B given not A."},{"Start":"02:06.710 ","End":"02:09.380","Text":"Well, again, in the denominator,"},{"Start":"02:09.380 ","End":"02:12.110","Text":"we have the probability of what\u0027s given,"},{"Start":"02:12.110 ","End":"02:13.460","Text":"and in the numerator,"},{"Start":"02:13.460 ","End":"02:18.805","Text":"we have the probability of the intersect B and not A."},{"Start":"02:18.805 ","End":"02:21.890","Text":"Let\u0027s see what we have here."},{"Start":"02:21.890 ","End":"02:24.035","Text":"The probability of not A, well,"},{"Start":"02:24.035 ","End":"02:27.170","Text":"let\u0027s look at our probability table."},{"Start":"02:27.170 ","End":"02:29.740","Text":"Well, that\u0027s 0.4,"},{"Start":"02:29.740 ","End":"02:35.476","Text":"and the probability of B and not A, that\u0027s the intersect."},{"Start":"02:35.476 ","End":"02:38.390","Text":"Here\u0027s not A and here\u0027s B,"},{"Start":"02:38.390 ","End":"02:40.790","Text":"so that\u0027ll be equal to 0.1."},{"Start":"02:40.790 ","End":"02:46.920","Text":"So that equals to 1/4, or 0.25."}],"ID":12891},{"Watched":false,"Name":"Exercise 3 - Part d","Duration":"3m 57s","ChapterTopicVideoID":12412,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.335","Text":"In this section, we\u0027re asked,"},{"Start":"00:01.335 ","End":"00:06.620","Text":"given that at least 1 of the parking lots has an available parking place at 8:00 AM,"},{"Start":"00:06.620 ","End":"00:08.520","Text":"what\u0027s the probability that there will be"},{"Start":"00:08.520 ","End":"00:12.325","Text":"a parking place available in the large parking lot?"},{"Start":"00:12.325 ","End":"00:14.970","Text":"Here, we have 2 events."},{"Start":"00:14.970 ","End":"00:19.230","Text":"The first event is that there will be a parking place"},{"Start":"00:19.230 ","End":"00:24.570","Text":"available in the large parking lot and that\u0027s event A."},{"Start":"00:24.570 ","End":"00:26.800","Text":"Now, what about the second event here?"},{"Start":"00:26.800 ","End":"00:32.930","Text":"Given that at least 1 of the parking lots has available parking place."},{"Start":"00:32.930 ","End":"00:36.420","Text":"When we\u0027re talking about at least 1 of the parking lots,"},{"Start":"00:36.420 ","End":"00:38.520","Text":"it could be A, it could be B,"},{"Start":"00:38.520 ","End":"00:40.095","Text":"or it could be both of them."},{"Start":"00:40.095 ","End":"00:44.655","Text":"We\u0027re looking at A union B right here."},{"Start":"00:44.655 ","End":"00:47.960","Text":"The conditional probability, and it\u0027s"},{"Start":"00:47.960 ","End":"00:52.850","Text":"a conditional probability that we\u0027re looking for because we\u0027re given A union B,"},{"Start":"00:52.850 ","End":"00:57.485","Text":"we\u0027re given that at least 1 of the parking lots has an available parking place."},{"Start":"00:57.485 ","End":"00:59.155","Text":"How do we write that?"},{"Start":"00:59.155 ","End":"01:02.265","Text":"That\u0027s the probability. Now here,"},{"Start":"01:02.265 ","End":"01:03.885","Text":"we write what\u0027s given."},{"Start":"01:03.885 ","End":"01:05.777","Text":"That\u0027s A union B,"},{"Start":"01:05.777 ","End":"01:09.815","Text":"and we\u0027re looking for the probability of A."},{"Start":"01:09.815 ","End":"01:14.350","Text":"This is the conditional probability that we want to calculate."},{"Start":"01:14.350 ","End":"01:16.725","Text":"That equals to the following."},{"Start":"01:16.725 ","End":"01:20.395","Text":"In the denominator, we have the probability of what\u0027s given,"},{"Start":"01:20.395 ","End":"01:22.305","Text":"A union B,"},{"Start":"01:22.305 ","End":"01:24.014","Text":"and in the numerator,"},{"Start":"01:24.014 ","End":"01:25.350","Text":"we have the intersect."},{"Start":"01:25.350 ","End":"01:34.050","Text":"We have the probability of A intersect A union B."},{"Start":"01:35.270 ","End":"01:38.300","Text":"Let\u0027s look at each 1 of"},{"Start":"01:38.300 ","End":"01:42.785","Text":"the probabilities separately in the denominator and then in the numerator."},{"Start":"01:42.785 ","End":"01:45.880","Text":"Let\u0027s start with the denominator."},{"Start":"01:45.880 ","End":"01:49.530","Text":"Probability of A union B,"},{"Start":"01:49.530 ","End":"01:54.095","Text":"that equals to the probability of A plus the probability of"},{"Start":"01:54.095 ","End":"01:58.940","Text":"B minus the probability of A intersect B."},{"Start":"01:58.940 ","End":"02:03.450","Text":"The probability of A,"},{"Start":"02:03.450 ","End":"02:05.850","Text":"that equals to 0.6,"},{"Start":"02:05.850 ","End":"02:08.250","Text":"plus the probability of B,"},{"Start":"02:08.250 ","End":"02:11.020","Text":"that equals to 0.3,"},{"Start":"02:11.020 ","End":"02:14.480","Text":"minus the probability of A intersect B,"},{"Start":"02:14.480 ","End":"02:17.165","Text":"that\u0027s right here, that\u0027s 0.2,"},{"Start":"02:17.165 ","End":"02:23.695","Text":"and that comes out to 0.7."},{"Start":"02:23.695 ","End":"02:26.960","Text":"This is what we have in the denominator."},{"Start":"02:26.960 ","End":"02:28.985","Text":"What about the numerator?"},{"Start":"02:28.985 ","End":"02:32.330","Text":"A intersect A union B."},{"Start":"02:32.330 ","End":"02:35.255","Text":"Now, in order to calculate this probability,"},{"Start":"02:35.255 ","End":"02:38.359","Text":"let\u0027s use a Venn diagram."},{"Start":"02:38.359 ","End":"02:41.405","Text":"Here\u0027s our Venn diagram."},{"Start":"02:41.405 ","End":"02:42.950","Text":"Let\u0027s take a look at this."},{"Start":"02:42.950 ","End":"02:46.620","Text":"This is A. Let\u0027s color in A."},{"Start":"02:46.970 ","End":"02:52.320","Text":"This would be A right here, that\u0027s the blue."},{"Start":"02:52.320 ","End":"02:55.050","Text":"What about A union B?"},{"Start":"02:55.050 ","End":"02:56.475","Text":"That\u0027ll be in red."},{"Start":"02:56.475 ","End":"03:02.080","Text":"A union B, that\u0027s all this area right here."},{"Start":"03:05.030 ","End":"03:12.109","Text":"Now, what\u0027s the intersect of A and A union B?"},{"Start":"03:12.109 ","End":"03:17.615","Text":"That\u0027s the area where we have both blue and red."},{"Start":"03:17.615 ","End":"03:21.510","Text":"This is the area that\u0027s for A,"},{"Start":"03:21.510 ","End":"03:23.685","Text":"this is the event A."},{"Start":"03:23.685 ","End":"03:28.805","Text":"This thing right here would be equal to the probability of A"},{"Start":"03:28.805 ","End":"03:34.795","Text":"divided by the probability of A union B."},{"Start":"03:34.795 ","End":"03:38.010","Text":"Now, the probability of A union B,"},{"Start":"03:38.010 ","End":"03:41.450","Text":"that equals to 0.7."},{"Start":"03:41.450 ","End":"03:43.535","Text":"We\u0027ve calculated that right here."},{"Start":"03:43.535 ","End":"03:45.970","Text":"Now what\u0027s the probability of A?"},{"Start":"03:45.970 ","End":"03:52.085","Text":"We\u0027re going back here to the probability matrix, that\u0027s 0.6."},{"Start":"03:52.085 ","End":"03:58.380","Text":"This whole probability goes down to 6 divided by 7."}],"ID":12892},{"Watched":false,"Name":"Exercise 4 - Parts a-b","Duration":"2m 59s","ChapterTopicVideoID":12415,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this question, we have a group of people that\u0027s comprised"},{"Start":"00:03.450 ","End":"00:07.500","Text":"of 200 company employees and 100 self-employed people."},{"Start":"00:07.500 ","End":"00:10.170","Text":"Now 20 of the company employees and 30 of"},{"Start":"00:10.170 ","End":"00:15.180","Text":"the self-employed people are university graduates."},{"Start":"00:15.180 ","End":"00:19.870","Text":"We\u0027re asked to build a joint frequency table for the data."},{"Start":"00:19.880 ","End":"00:23.400","Text":"Here\u0027s our joint frequency table."},{"Start":"00:23.400 ","End":"00:28.515","Text":"We have here the graduates column and the not graduates column."},{"Start":"00:28.515 ","End":"00:32.958","Text":"Person can be either a university graduate or not,"},{"Start":"00:32.958 ","End":"00:38.515","Text":"and a person can also be a company employee or self-employed."},{"Start":"00:38.515 ","End":"00:40.805","Text":"Now, what do we have here?"},{"Start":"00:40.805 ","End":"00:45.785","Text":"We have 200 people that are company employees."},{"Start":"00:45.785 ","End":"00:51.245","Text":"So here, there\u0027ll be 200 and 100 self-employed people."},{"Start":"00:51.245 ","End":"00:52.613","Text":"That means that self-employed,"},{"Start":"00:52.613 ","End":"00:54.710","Text":"there will be 100 here."},{"Start":"00:54.710 ","End":"00:59.000","Text":"Now, 20 of the company employees and"},{"Start":"00:59.000 ","End":"01:03.050","Text":"30 of the self-employed people are university graduates."},{"Start":"01:03.050 ","End":"01:06.440","Text":"So if a person is a graduate and a company employee,"},{"Start":"01:06.440 ","End":"01:08.315","Text":"well, how many do we have?"},{"Start":"01:08.315 ","End":"01:15.540","Text":"We have 20, and 30 self-employed are also graduates."},{"Start":"01:15.540 ","End":"01:21.110","Text":"Now, we can fill in the rest of the frequencies here."},{"Start":"01:21.110 ","End":"01:24.020","Text":"20 plus 30 is 50,"},{"Start":"01:24.020 ","End":"01:29.810","Text":"so we have a total of 50 graduates out of a total of 300 people."},{"Start":"01:29.810 ","End":"01:34.400","Text":"Now, we have here 20 plus what equals 200?"},{"Start":"01:34.400 ","End":"01:40.703","Text":"We have here 180 company employees that are not graduate,"},{"Start":"01:40.703 ","End":"01:47.680","Text":"and we have here 70 people that are self-employed which are not graduates."},{"Start":"01:47.680 ","End":"01:54.680","Text":"All and all, we have 250 non-graduates in the population."},{"Start":"01:54.680 ","End":"01:58.850","Text":"This then is the joint frequency table because it\u0027s"},{"Start":"01:58.850 ","End":"02:03.210","Text":"a frequency table that combines 2 parameters or 2 variables."},{"Start":"02:03.210 ","End":"02:07.870","Text":"The fact that people are either graduates or not graduates,"},{"Start":"02:07.870 ","End":"02:12.250","Text":"and whether they\u0027re company employees or self-employed."},{"Start":"02:12.250 ","End":"02:16.000","Text":"In this section, a person is selected randomly and we\u0027re asked,"},{"Start":"02:16.000 ","End":"02:18.865","Text":"what\u0027s the probability that he is a company employee?"},{"Start":"02:18.865 ","End":"02:28.520","Text":"We\u0027re looking for the probability that a person is a company employee."},{"Start":"02:28.530 ","End":"02:32.160","Text":"How many company employees do we have?"},{"Start":"02:32.160 ","End":"02:34.620","Text":"We have 200."},{"Start":"02:34.620 ","End":"02:38.260","Text":"How many people in total do we have in the group?"},{"Start":"02:38.260 ","End":"02:40.170","Text":"We have 300."},{"Start":"02:40.170 ","End":"02:48.065","Text":"The probability that a person is a company employee would be 200 divided by 300."},{"Start":"02:48.065 ","End":"02:54.353","Text":"Now, that equals to 2/3 or 0.667,"},{"Start":"02:54.353 ","End":"03:00.280","Text":"or if you will, 66.7 percent."}],"ID":12893},{"Watched":false,"Name":"Exercise 4 - Parts c-e","Duration":"3m 45s","ChapterTopicVideoID":12416,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"00:02.580 ","End":"00:05.940","Text":"a company employee is not a university graduate?"},{"Start":"00:05.940 ","End":"00:10.830","Text":"We have the probability of a person being"},{"Start":"00:10.830 ","End":"00:20.230","Text":"a company employee and he\u0027s not a graduate."},{"Start":"00:21.500 ","End":"00:25.770","Text":"A company employee, that\u0027s this line right here,"},{"Start":"00:25.770 ","End":"00:28.635","Text":"and not a graduate is this line right here."},{"Start":"00:28.635 ","End":"00:31.620","Text":"The intersection is this cell right here."},{"Start":"00:31.620 ","End":"00:35.460","Text":"We have 180, that\u0027s the frequency,"},{"Start":"00:35.460 ","End":"00:39.175","Text":"that\u0027s the number of people who are company employees and not graduate,"},{"Start":"00:39.175 ","End":"00:43.550","Text":"out of a total of 300 people."},{"Start":"00:43.550 ","End":"00:51.555","Text":"That comes out to 3 divided by 5 or 0.6."},{"Start":"00:51.555 ","End":"00:52.955","Text":"In this section we\u0027re asked,"},{"Start":"00:52.955 ","End":"00:55.130","Text":"what\u0027s the probability that a person is"},{"Start":"00:55.130 ","End":"00:59.030","Text":"either a company employee or a university graduate?"},{"Start":"00:59.030 ","End":"01:04.325","Text":"The probability of a person either being"},{"Start":"01:04.325 ","End":"01:13.752","Text":"a company employee or a graduate."},{"Start":"01:13.752 ","End":"01:18.230","Text":"Well, we know that that equals to the probability of"},{"Start":"01:18.230 ","End":"01:23.070","Text":"a person being a company employee plus"},{"Start":"01:23.070 ","End":"01:31.570","Text":"the probability that a person is a graduate minus the probability"},{"Start":"01:31.570 ","End":"01:42.340","Text":"of a person being a company employee and a graduate."},{"Start":"01:42.440 ","End":"01:47.725","Text":"What does that equal to? The probability of a person being a company employee,"},{"Start":"01:47.725 ","End":"01:55.630","Text":"that\u0027s 200 divided by 300 plus the probability that a person is a graduate,"},{"Start":"01:55.630 ","End":"02:00.070","Text":"that\u0027s 50 divided by 300 minus"},{"Start":"02:00.070 ","End":"02:06.945","Text":"the probability of a person being a company employee and a graduate."},{"Start":"02:06.945 ","End":"02:10.905","Text":"That\u0027s right here, that\u0027s 20 divided by 300."},{"Start":"02:10.905 ","End":"02:19.850","Text":"That comes out to 230 divided by 300 or 23 divided by 30."},{"Start":"02:19.850 ","End":"02:23.950","Text":"In this section, we\u0027re asked if a self-employed person is selected,"},{"Start":"02:23.950 ","End":"02:27.380","Text":"what\u0027s the probability that he\u0027s a university graduate?"},{"Start":"02:27.380 ","End":"02:31.820","Text":"Here we have a conditional probability question."},{"Start":"02:31.820 ","End":"02:36.820","Text":"Why is that? Because we\u0027re given that the person selected is self-employed."},{"Start":"02:36.820 ","End":"02:38.800","Text":"Let\u0027s write the probability."},{"Start":"02:38.800 ","End":"02:42.560","Text":"We\u0027re saying, what\u0027s the probability that a person is"},{"Start":"02:42.560 ","End":"02:47.430","Text":"a graduate given that he is self-employed?"},{"Start":"02:49.010 ","End":"02:54.155","Text":"In the denominator we have the probability of what\u0027s given,"},{"Start":"02:54.155 ","End":"02:58.260","Text":"that a person is self-employed."},{"Start":"02:58.260 ","End":"03:01.800","Text":"In the numerator, we have the intersect,"},{"Start":"03:01.800 ","End":"03:07.100","Text":"that\u0027s the probability of a person being a graduate and"},{"Start":"03:07.100 ","End":"03:13.200","Text":"that that person is self-employed."},{"Start":"03:13.200 ","End":"03:15.790","Text":"Well, let\u0027s take a look at our frequency table."},{"Start":"03:15.790 ","End":"03:19.265","Text":"The probability that a person is self-employed,"},{"Start":"03:19.265 ","End":"03:23.155","Text":"that\u0027s 100 divided by 300,"},{"Start":"03:23.155 ","End":"03:28.385","Text":"the probability that a person is a graduate and self-employed,"},{"Start":"03:28.385 ","End":"03:31.490","Text":"here\u0027s the joint frequency, that\u0027ll be 30,"},{"Start":"03:31.490 ","End":"03:35.409","Text":"and to make it a probability we have to divide that by 300."},{"Start":"03:35.409 ","End":"03:46.560","Text":"The 300 cancels each other out and we have 30/100 or 0.3."}],"ID":12894},{"Watched":false,"Name":"Exercise 4 - Part f","Duration":"1m 55s","ChapterTopicVideoID":12414,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.264","Text":"In this section, we\u0027re asked if a person selected is not a university graduate,"},{"Start":"00:05.264 ","End":"00:09.405","Text":"then what\u0027s the probability that he is a company employee?"},{"Start":"00:09.405 ","End":"00:12.930","Text":"Well, again, we have here a conditional probability, right?"},{"Start":"00:12.930 ","End":"00:14.580","Text":"What are we given?"},{"Start":"00:14.580 ","End":"00:18.570","Text":"We\u0027re given that the person is not a university graduate and we\u0027re"},{"Start":"00:18.570 ","End":"00:22.575","Text":"looking for the probability that he is a company employee."},{"Start":"00:22.575 ","End":"00:25.275","Text":"Let\u0027s write the probability down."},{"Start":"00:25.275 ","End":"00:29.055","Text":"That\u0027s the probability. Now, what\u0027s given?"},{"Start":"00:29.055 ","End":"00:33.555","Text":"We\u0027re given that the person is not a graduate,"},{"Start":"00:33.555 ","End":"00:42.160","Text":"and we\u0027re looking for the probability that that person is a company employee."},{"Start":"00:42.170 ","End":"00:44.300","Text":"In the denominator,"},{"Start":"00:44.300 ","End":"00:50.635","Text":"we have the probability of what\u0027s given, not a graduate."},{"Start":"00:50.635 ","End":"00:56.077","Text":"In the numerator, we have the intersect of that 2 events"},{"Start":"00:56.077 ","End":"01:02.305","Text":": the probability that the person is a company employee,"},{"Start":"01:02.305 ","End":"01:08.140","Text":"and that he\u0027s not a graduate."},{"Start":"01:08.960 ","End":"01:11.389","Text":"Let\u0027s look at the denominator,"},{"Start":"01:11.389 ","End":"01:14.270","Text":"the probability that a person is not a graduate."},{"Start":"01:14.270 ","End":"01:15.620","Text":"That\u0027s right here,"},{"Start":"01:15.620 ","End":"01:18.635","Text":"the relative frequency is 250."},{"Start":"01:18.635 ","End":"01:20.360","Text":"To make it a probability,"},{"Start":"01:20.360 ","End":"01:24.325","Text":"that\u0027ll be 250 divided by 300."},{"Start":"01:24.325 ","End":"01:26.970","Text":"What about the numerator?"},{"Start":"01:26.970 ","End":"01:29.720","Text":"A person is a company employee,"},{"Start":"01:29.720 ","End":"01:31.160","Text":"that\u0027s this line,"},{"Start":"01:31.160 ","End":"01:33.440","Text":"and not a graduate, that\u0027s this column."},{"Start":"01:33.440 ","End":"01:37.445","Text":"We have a relative frequency of 180."},{"Start":"01:37.445 ","End":"01:38.960","Text":"To make it a probability,"},{"Start":"01:38.960 ","End":"01:41.159","Text":"we need to divide that by 300,"},{"Start":"01:41.159 ","End":"01:43.475","Text":"the 300s cancel each other out,"},{"Start":"01:43.475 ","End":"01:49.450","Text":"and we have a 180 divided by 250."},{"Start":"01:49.450 ","End":"01:55.860","Text":"That comes out to approximately 0.72."}],"ID":12895},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"7m 47s","ChapterTopicVideoID":12417,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this question, we have a company that published the following data about people over"},{"Start":"00:04.110 ","End":"00:08.145","Text":"the age of 21: 40 percent of the people hold a Visa card,"},{"Start":"00:08.145 ","End":"00:10.275","Text":"52 percent hold a Diners card,"},{"Start":"00:10.275 ","End":"00:12.990","Text":"20 percent hold an American Express card,"},{"Start":"00:12.990 ","End":"00:17.030","Text":"15 percent hold both a Visa and a Diners card,"},{"Start":"00:17.030 ","End":"00:20.265","Text":"8 percent hold both a Diners and an American Express card,"},{"Start":"00:20.265 ","End":"00:24.225","Text":"and 7 percent hold both a Visa and an American Express card."},{"Start":"00:24.225 ","End":"00:27.915","Text":"Now, 5 percent hold all 3 of these cards."},{"Start":"00:27.915 ","End":"00:31.140","Text":"Now, before we go ahead and solve any questions,"},{"Start":"00:31.140 ","End":"00:35.445","Text":"let\u0027s just write down all the information that we know."},{"Start":"00:35.445 ","End":"00:38.625","Text":"Let\u0027s define 3 events here."},{"Start":"00:38.625 ","End":"00:42.810","Text":"Event A is a person holding a Visa card, event B,"},{"Start":"00:42.810 ","End":"00:44.360","Text":"a person holding a Diners card,"},{"Start":"00:44.360 ","End":"00:47.555","Text":"and event C, a person holding an American Express card."},{"Start":"00:47.555 ","End":"00:50.180","Text":"Now, what about the probabilities?"},{"Start":"00:50.180 ","End":"00:54.380","Text":"Well, the probability of holding a Visa card,"},{"Start":"00:54.380 ","End":"00:57.124","Text":"that\u0027s right here, that\u0027s 40 percent."},{"Start":"00:57.124 ","End":"01:00.260","Text":"The probability of holding a Diners Club,"},{"Start":"01:00.260 ","End":"01:03.340","Text":"that equals to 52 percent."},{"Start":"01:03.340 ","End":"01:12.495","Text":"The probability of holding an American Express card, that\u0027s 20 percent."},{"Start":"01:12.495 ","End":"01:19.350","Text":"Now, the probability of holding both a Visa and a Diners,"},{"Start":"01:19.350 ","End":"01:21.845","Text":"that means A and B."},{"Start":"01:21.845 ","End":"01:27.410","Text":"That equals to 0.15."},{"Start":"01:27.410 ","End":"01:35.180","Text":"What about the probability of holding a Diners Club and an American Express card?"},{"Start":"01:35.180 ","End":"01:38.780","Text":"That equals to 0.08 or 8 percent."},{"Start":"01:38.780 ","End":"01:44.880","Text":"The probability of holding a Visa and an American Express card,"},{"Start":"01:44.880 ","End":"01:49.410","Text":"that equals to 0.07 or 7 percent."},{"Start":"01:49.410 ","End":"01:53.670","Text":"Now, the probability of holding 3 cards,"},{"Start":"01:53.670 ","End":"01:59.720","Text":"both the Visa and a Diners and an American Express,"},{"Start":"01:59.720 ","End":"02:05.280","Text":"that equals to 0.05 or 5 percent."},{"Start":"02:05.280 ","End":"02:09.275","Text":"Until now, we\u0027ve been using the probability matrix"},{"Start":"02:09.275 ","End":"02:13.985","Text":"to solve all the conditional probability questions,"},{"Start":"02:13.985 ","End":"02:17.450","Text":"but a probability matrix uses only 2 variables,"},{"Start":"02:17.450 ","End":"02:19.730","Text":"and here we have 3 variables."},{"Start":"02:19.730 ","End":"02:24.780","Text":"The best way to go is to use a Venn diagram."},{"Start":"02:25.040 ","End":"02:27.500","Text":"Here\u0027s our Venn diagram."},{"Start":"02:27.500 ","End":"02:29.570","Text":"Here, we have event A, B,"},{"Start":"02:29.570 ","End":"02:32.345","Text":"and C. A is holding a Visa card,"},{"Start":"02:32.345 ","End":"02:33.530","Text":"B is a Diners card,"},{"Start":"02:33.530 ","End":"02:35.495","Text":"and C is an American Express card."},{"Start":"02:35.495 ","End":"02:39.980","Text":"Let\u0027s try and put in all these probabilities on"},{"Start":"02:39.980 ","End":"02:45.635","Text":"the Venn diagram and we\u0027ll start with the intersect of all 3 events."},{"Start":"02:45.635 ","End":"02:48.815","Text":"The probability of A and B and C,"},{"Start":"02:48.815 ","End":"02:54.320","Text":"that\u0027s right here in the middle right here and this is 0.05."},{"Start":"02:54.320 ","End":"02:57.020","Text":"Now, let\u0027s go to this probability."},{"Start":"02:57.020 ","End":"03:01.115","Text":"The probability of A and B is 0.15."},{"Start":"03:01.115 ","End":"03:04.985","Text":"Now A and B is this area right here,"},{"Start":"03:04.985 ","End":"03:09.830","Text":"but it already has this area which is 0.05."},{"Start":"03:09.830 ","End":"03:14.600","Text":"Right here in this area, that\u0027ll be 0.1."},{"Start":"03:14.600 ","End":"03:18.515","Text":"Because 0.1 plus 0.05,"},{"Start":"03:18.515 ","End":"03:24.840","Text":"that\u0027s 0.15, and this is the intersect of A and B."},{"Start":"03:24.840 ","End":"03:26.730","Text":"Now, what about B and C?"},{"Start":"03:26.730 ","End":"03:32.150","Text":"That\u0027s 0.08, that\u0027s this common area right here."},{"Start":"03:32.150 ","End":"03:35.180","Text":"But we already have 0.05 here in the middle,"},{"Start":"03:35.180 ","End":"03:39.550","Text":"so out here, that\u0027ll be 0.03."},{"Start":"03:39.550 ","End":"03:41.685","Text":"What about A and C?"},{"Start":"03:41.685 ","End":"03:45.740","Text":"Well, A and C is 0.07,"},{"Start":"03:45.740 ","End":"03:47.449","Text":"that\u0027s this area right here."},{"Start":"03:47.449 ","End":"03:49.970","Text":"But we already have 0.05 in the middle,"},{"Start":"03:49.970 ","End":"03:54.960","Text":"so out here, that will be 0.02."},{"Start":"03:55.550 ","End":"03:59.690","Text":"Now, what about the probability of A?"},{"Start":"03:59.690 ","End":"04:03.650","Text":"Well, that\u0027s 0.4, 40 percent."},{"Start":"04:03.650 ","End":"04:10.014","Text":"The area outside of all of these intersect, what would that be?"},{"Start":"04:10.014 ","End":"04:16.835","Text":"That would be 0.1 plus 0.05 plus 0.02."},{"Start":"04:16.835 ","End":"04:19.535","Text":"Well, that\u0027s 0.17."},{"Start":"04:19.535 ","End":"04:22.880","Text":"We have to take away from 0.4,"},{"Start":"04:22.880 ","End":"04:29.155","Text":"0.17, which leaves me with 0.23 out here."},{"Start":"04:29.155 ","End":"04:32.070","Text":"That\u0027s this area right here."},{"Start":"04:32.070 ","End":"04:37.300","Text":"What about the area outside the intersects of B?"},{"Start":"04:37.300 ","End":"04:41.375","Text":"The probability of B is 0.52,"},{"Start":"04:41.375 ","End":"04:43.520","Text":"so we have to take away from that,"},{"Start":"04:43.520 ","End":"04:47.855","Text":"0.1 plus 0.05 plus 0.03."},{"Start":"04:47.855 ","End":"04:53.775","Text":"That leaves me with 0.34 outside of the intersects."},{"Start":"04:53.775 ","End":"04:56.430","Text":"We\u0027ll do the same for C. C,"},{"Start":"04:56.430 ","End":"04:59.025","Text":"probability of C is 0.2,"},{"Start":"04:59.025 ","End":"05:01.055","Text":"so we have to take away from that,"},{"Start":"05:01.055 ","End":"05:06.620","Text":"0.03 plus 0.05 plus 0.02."},{"Start":"05:06.620 ","End":"05:10.890","Text":"That leaves me with 0.1 out here."},{"Start":"05:11.460 ","End":"05:21.380","Text":"Here, what\u0027s the probability of all the events right here?"},{"Start":"05:21.380 ","End":"05:27.455","Text":"Well, if we add all the events up here, we\u0027ll get 0.87."},{"Start":"05:27.455 ","End":"05:34.445","Text":"The area outside of all these 3 events is 0.13."},{"Start":"05:34.445 ","End":"05:36.350","Text":"Now, check me on this,"},{"Start":"05:36.350 ","End":"05:39.800","Text":"but if we add up all the probabilities right now,"},{"Start":"05:39.800 ","End":"05:42.370","Text":"we\u0027ll come up with 1."},{"Start":"05:42.370 ","End":"05:47.060","Text":"Now that we have all the probabilities mapped out in the Venn diagram,"},{"Start":"05:47.060 ","End":"05:50.240","Text":"let\u0027s solve the first section here."},{"Start":"05:50.240 ","End":"05:52.220","Text":"The first section states,"},{"Start":"05:52.220 ","End":"05:54.755","Text":"if a person has a Visa card,"},{"Start":"05:54.755 ","End":"05:58.370","Text":"what are the chances that he does not have a Diners card?"},{"Start":"05:58.370 ","End":"06:00.672","Text":"Well, this is a conditional probability question."},{"Start":"06:00.672 ","End":"06:03.520","Text":"Why is that? Because we\u0027re given something."},{"Start":"06:03.520 ","End":"06:06.950","Text":"We\u0027re given that a person has a Visa card and we\u0027re asked about"},{"Start":"06:06.950 ","End":"06:10.340","Text":"the probability that he does not have a Diners card."},{"Start":"06:10.340 ","End":"06:12.335","Text":"Now let\u0027s write this probability out."},{"Start":"06:12.335 ","End":"06:15.095","Text":"We\u0027re looking for the probability."},{"Start":"06:15.095 ","End":"06:16.940","Text":"Now, what\u0027s given to us?"},{"Start":"06:16.940 ","End":"06:18.710","Text":"That he has a Visa card."},{"Start":"06:18.710 ","End":"06:21.165","Text":"That means event A."},{"Start":"06:21.165 ","End":"06:23.366","Text":"We\u0027re given A."},{"Start":"06:23.366 ","End":"06:27.650","Text":"We\u0027re asked about the probability that he does not have a Diners card."},{"Start":"06:27.650 ","End":"06:30.955","Text":"That means that we\u0027re looking for not B."},{"Start":"06:30.955 ","End":"06:33.935","Text":"This is our conditional probability."},{"Start":"06:33.935 ","End":"06:36.185","Text":"What does that equal to?"},{"Start":"06:36.185 ","End":"06:37.940","Text":"Well, in the denominator,"},{"Start":"06:37.940 ","End":"06:40.960","Text":"we have the probability of what\u0027s given,"},{"Start":"06:40.960 ","End":"06:42.560","Text":"and in the numerator,"},{"Start":"06:42.560 ","End":"06:44.915","Text":"we have the probability of the intersect,"},{"Start":"06:44.915 ","End":"06:49.290","Text":"B bar intersect A."},{"Start":"06:49.660 ","End":"06:52.685","Text":"What\u0027s the probability of A?"},{"Start":"06:52.685 ","End":"06:56.750","Text":"That\u0027s given to us right here, that\u0027s 0.4."},{"Start":"06:56.750 ","End":"07:02.705","Text":"What about the probability of not B and A?"},{"Start":"07:02.705 ","End":"07:06.230","Text":"Now, the probability of not being A,"},{"Start":"07:06.230 ","End":"07:08.735","Text":"that\u0027s the area in yellow right here."},{"Start":"07:08.735 ","End":"07:10.835","Text":"We have all of A,"},{"Start":"07:10.835 ","End":"07:18.037","Text":"but we have to take away from that the part that is common with B. What are we left with?"},{"Start":"07:18.037 ","End":"07:20.330","Text":"We\u0027re left with 0.23,"},{"Start":"07:20.330 ","End":"07:24.035","Text":"that\u0027s right here, and 0.02 here."},{"Start":"07:24.035 ","End":"07:29.110","Text":"That\u0027s 0.23 plus 0.02,"},{"Start":"07:29.110 ","End":"07:31.275","Text":"that\u0027s this area in yellow."},{"Start":"07:31.275 ","End":"07:38.390","Text":"Now, that equals to 0.25 divided by 0.4."},{"Start":"07:38.390 ","End":"07:41.810","Text":"That equals to 5/8,"},{"Start":"07:41.810 ","End":"07:47.470","Text":"or that equals to 0.625."}],"ID":12896},{"Watched":false,"Name":"Exercise 5 - Part b","Duration":"4m 2s","ChapterTopicVideoID":12418,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"In this section, we\u0027re asked,"},{"Start":"00:01.470 ","End":"00:03.390","Text":"if a person has 2 credit cards,"},{"Start":"00:03.390 ","End":"00:06.990","Text":"what are the chances that he does not have a Diners card?"},{"Start":"00:06.990 ","End":"00:09.555","Text":"This is a conditional probability."},{"Start":"00:09.555 ","End":"00:11.550","Text":"Now, what\u0027s given to us?"},{"Start":"00:11.550 ","End":"00:15.060","Text":"We\u0027re given that the person has 2 credit cards."},{"Start":"00:15.060 ","End":"00:17.010","Text":"We\u0027re not told which ones."},{"Start":"00:17.010 ","End":"00:19.020","Text":"It could be a Visa and a Diner,"},{"Start":"00:19.020 ","End":"00:20.760","Text":"a Visa and an American Express,"},{"Start":"00:20.760 ","End":"00:22.905","Text":"or a Diners and an American Express."},{"Start":"00:22.905 ","End":"00:27.915","Text":"We\u0027re asked about the chances that he does not have a Diners card."},{"Start":"00:27.915 ","End":"00:29.850","Text":"Let\u0027s write out this probability."},{"Start":"00:29.850 ","End":"00:36.390","Text":"We\u0027re looking for the probability that the person does not have a Diners card,"},{"Start":"00:36.390 ","End":"00:43.090","Text":"that\u0027s not B, given that he has 2 cards."},{"Start":"00:43.310 ","End":"00:48.335","Text":"Now, what does 2 cards mean?"},{"Start":"00:48.335 ","End":"00:52.490","Text":"Let\u0027s write this out in terms of the events."},{"Start":"00:52.490 ","End":"00:57.185","Text":"We\u0027re looking at the probability of not B given."},{"Start":"00:57.185 ","End":"00:59.765","Text":"Now, 2 cards mean, as I said,"},{"Start":"00:59.765 ","End":"01:04.390","Text":"that he could have a Visa and a Diners and not an American Express."},{"Start":"01:04.390 ","End":"01:11.910","Text":"That\u0027ll be A and B and not C. Or he"},{"Start":"01:11.910 ","End":"01:20.200","Text":"could have a Visa and an American Express and not a Diners."},{"Start":"01:20.200 ","End":"01:25.365","Text":"Or he can have a Diners"},{"Start":"01:25.365 ","End":"01:32.065","Text":"and an American Express and not a Visa card."},{"Start":"01:32.065 ","End":"01:36.330","Text":"This is the meaning of having 2 cards."},{"Start":"01:36.330 ","End":"01:39.150","Text":"A and B and not C,"},{"Start":"01:39.150 ","End":"01:41.745","Text":"or A and C and not B,"},{"Start":"01:41.745 ","End":"01:44.905","Text":"or B and C and not A."},{"Start":"01:44.905 ","End":"01:48.995","Text":"Now, let\u0027s write this condition probability out."},{"Start":"01:48.995 ","End":"01:52.580","Text":"That equals, well, in the denominator,"},{"Start":"01:52.580 ","End":"01:55.130","Text":"we have the probability of what\u0027s given."},{"Start":"01:55.130 ","End":"02:02.535","Text":"We have the probability of all of this and all this means that the person has 2 cards."},{"Start":"02:02.535 ","End":"02:05.175","Text":"In the numerator,"},{"Start":"02:05.175 ","End":"02:06.770","Text":"we have the intersects,"},{"Start":"02:06.770 ","End":"02:16.990","Text":"so that\u0027s the probability of B-bar not having a Diners Club and having 2 cards."},{"Start":"02:17.840 ","End":"02:22.040","Text":"Now that we have this conditional probability,"},{"Start":"02:22.040 ","End":"02:26.057","Text":"let\u0027s go calculate the actual probabilities here."},{"Start":"02:26.057 ","End":"02:29.750","Text":"Let\u0027s take a look at the denominator."},{"Start":"02:29.750 ","End":"02:31.930","Text":"The person has 2 cards."},{"Start":"02:31.930 ","End":"02:33.250","Text":"Now, what does that mean?"},{"Start":"02:33.250 ","End":"02:41.540","Text":"He could have a Visa and a Diner and not an American Express, that\u0027s right here."},{"Start":"02:41.540 ","End":"02:46.010","Text":"That\u0027s the intersect of A and B and not C,"},{"Start":"02:46.010 ","End":"02:49.240","Text":"or A and C and not B."},{"Start":"02:49.240 ","End":"02:51.625","Text":"That\u0027s this area right here."},{"Start":"02:51.625 ","End":"02:53.350","Text":"Or B and C,"},{"Start":"02:53.350 ","End":"02:56.860","Text":"and not A, that\u0027s this area right here."},{"Start":"02:56.860 ","End":"03:04.565","Text":"We\u0027re looking at 0.1 plus 0.02,"},{"Start":"03:04.565 ","End":"03:05.955","Text":"that\u0027s this area,"},{"Start":"03:05.955 ","End":"03:09.945","Text":"plus 0.03, that\u0027s this area."},{"Start":"03:09.945 ","End":"03:13.075","Text":"What about the numerator?"},{"Start":"03:13.075 ","End":"03:20.365","Text":"That\u0027s the area in red intersect not B. What\u0027s not B?"},{"Start":"03:20.365 ","End":"03:27.885","Text":"Well, not B is all this area outside of B."},{"Start":"03:27.885 ","End":"03:35.085","Text":"We have to intersect the green area with the red area."},{"Start":"03:35.085 ","End":"03:39.260","Text":"The only thing that intersects is this thing right here,"},{"Start":"03:39.260 ","End":"03:41.225","Text":"this area right here."},{"Start":"03:41.225 ","End":"03:46.690","Text":"Now, that equals to 0.02."},{"Start":"03:46.690 ","End":"03:56.630","Text":"The probability here is 0.02 divided by 0.15,"},{"Start":"03:56.630 ","End":"04:02.700","Text":"and that equals to 0.133."}],"ID":12897},{"Watched":false,"Name":"Exercise 5 - Part c","Duration":"3m 48s","ChapterTopicVideoID":12419,"CourseChapterTopicPlaylistID":245032,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In this section, we\u0027re asked,"},{"Start":"00:01.260 ","End":"00:03.720","Text":"if a person has at least 1 credit card,"},{"Start":"00:03.720 ","End":"00:07.560","Text":"what are the chances that he doesn\u0027t have a Diners card?"},{"Start":"00:07.560 ","End":"00:12.840","Text":"Again, this is a conditional probability because we\u0027re given that a person has"},{"Start":"00:12.840 ","End":"00:19.475","Text":"at least 1 credit card and we\u0027re asked about the chances of not having a Diners card."},{"Start":"00:19.475 ","End":"00:23.070","Text":"Let\u0027s write the conditional probability."},{"Start":"00:23.070 ","End":"00:27.981","Text":"That\u0027ll be the probability of not having a Diners card,"},{"Start":"00:27.981 ","End":"00:30.540","Text":"that\u0027s not B,"},{"Start":"00:30.540 ","End":"00:34.050","Text":"given that the person has at least 1 credit card."},{"Start":"00:34.050 ","End":"00:43.350","Text":"Now at least 1 credit card would be A or B or C. Now,"},{"Start":"00:43.350 ","End":"00:47.075","Text":"this means that he has either a Visa card,"},{"Start":"00:47.075 ","End":"00:48.920","Text":"a Diners card,"},{"Start":"00:48.920 ","End":"00:50.840","Text":"an American Express,"},{"Start":"00:50.840 ","End":"00:56.210","Text":"or maybe a combination of 2 cards or all 3 of the cards."},{"Start":"00:56.210 ","End":"00:58.940","Text":"That\u0027s what this thing means right here."},{"Start":"00:58.940 ","End":"01:02.450","Text":"Now, let\u0027s write out the conditional probability."},{"Start":"01:02.450 ","End":"01:04.800","Text":"Now that equals 2."},{"Start":"01:04.800 ","End":"01:06.590","Text":"In the denominator, what do we have?"},{"Start":"01:06.590 ","End":"01:09.495","Text":"We have the probability of what\u0027s given,"},{"Start":"01:09.495 ","End":"01:13.590","Text":"A or B or C,"},{"Start":"01:13.590 ","End":"01:14.670","Text":"and in the numerator,"},{"Start":"01:14.670 ","End":"01:16.390","Text":"we have the intersect."},{"Start":"01:16.390 ","End":"01:22.065","Text":"So that\u0027s the probability of B bar intersect A"},{"Start":"01:22.065 ","End":"01:29.140","Text":"or B or C. This is what we have to calculate right now."},{"Start":"01:29.140 ","End":"01:32.160","Text":"Let\u0027s take a look."},{"Start":"01:32.160 ","End":"01:36.830","Text":"In the denominator, A or B or C. Well,"},{"Start":"01:36.830 ","End":"01:39.050","Text":"what is that equal to?"},{"Start":"01:39.050 ","End":"01:44.380","Text":"Here we go. This is the area in yellow right here."},{"Start":"01:44.380 ","End":"01:51.515","Text":"In the denominator, what\u0027s the probability of all the area in yellow?"},{"Start":"01:51.515 ","End":"01:54.740","Text":"That equals to 0.87."},{"Start":"01:54.740 ","End":"01:59.315","Text":"Now, we could have easily calculated this using De Morgan\u0027s rule."},{"Start":"01:59.315 ","End":"02:00.800","Text":"Now, what is that?"},{"Start":"02:00.800 ","End":"02:06.920","Text":"We know that the probability of A union B union C, well,"},{"Start":"02:06.920 ","End":"02:11.585","Text":"that equals to 1 minus the probability of not"},{"Start":"02:11.585 ","End":"02:17.030","Text":"A and not B and not C. Now,"},{"Start":"02:17.030 ","End":"02:19.985","Text":"the probability of not A and not B and not C,"},{"Start":"02:19.985 ","End":"02:23.690","Text":"that\u0027s this probability right here, that\u0027s 0.13."},{"Start":"02:23.690 ","End":"02:26.720","Text":"So that equals to 1 minus 0.13,"},{"Start":"02:26.720 ","End":"02:29.150","Text":"and that equals to 0.87,"},{"Start":"02:29.150 ","End":"02:31.540","Text":"which is exactly what we have here."},{"Start":"02:31.540 ","End":"02:35.150","Text":"Now, what about the numerator?"},{"Start":"02:35.150 ","End":"02:42.590","Text":"That\u0027s the intersect of not B and A or B or C, that\u0027s in yellow."},{"Start":"02:42.590 ","End":"02:46.480","Text":"So what\u0027s not B intersect the area in yellow."},{"Start":"02:46.480 ","End":"02:48.810","Text":"Here we go."},{"Start":"02:48.810 ","End":"02:53.660","Text":"Not B and the area in yellow,"},{"Start":"02:53.660 ","End":"02:56.525","Text":"that\u0027s this area in green right here."},{"Start":"02:56.525 ","End":"02:57.950","Text":"What does that equal to?"},{"Start":"02:57.950 ","End":"03:04.790","Text":"That equals to 0.23 plus 0.02 plus 0.1,"},{"Start":"03:04.790 ","End":"03:06.535","Text":"and it makes sense."},{"Start":"03:06.535 ","End":"03:13.035","Text":"The area A or B or C intersect not B."},{"Start":"03:13.035 ","End":"03:19.265","Text":"That means that I either have a Visa or a Diners or an American Express,"},{"Start":"03:19.265 ","End":"03:21.410","Text":"but I don\u0027t have a Diners."},{"Start":"03:21.410 ","End":"03:25.655","Text":"I can either have a Visa or an American Express or both of them."},{"Start":"03:25.655 ","End":"03:30.480","Text":"This is exactly these probabilities right here,"},{"Start":"03:30.480 ","End":"03:34.635","Text":"0.23, 0.02 or 0.1."},{"Start":"03:34.635 ","End":"03:43.790","Text":"Now, this comes out to 0.35 divided by 0.87,"},{"Start":"03:43.790 ","End":"03:47.280","Text":"and that equals to 0.402."}],"ID":12898}],"Thumbnail":null,"ID":245032},{"Name":"Tree Diagrams the Bayes Formula and the Complete Probability Formula","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"2m 21s","ChapterTopicVideoID":12420,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this chapter, we\u0027ll be talking about tree diagrams,"},{"Start":"00:03.270 ","End":"00:06.705","Text":"the Bayes formula and the complete probability formula."},{"Start":"00:06.705 ","End":"00:11.850","Text":"Now, we use this technique in exercises where the events occur in steps so"},{"Start":"00:11.850 ","End":"00:17.055","Text":"that the results of a specific step depends on the preceding step,"},{"Start":"00:17.055 ","End":"00:20.430","Text":"and all this except the results of the first step."},{"Start":"00:20.430 ","End":"00:23.560","Text":"Now, let\u0027s just take a look at an example here."},{"Start":"00:23.630 ","End":"00:26.670","Text":"Now, in a given company,"},{"Start":"00:26.670 ","End":"00:29.120","Text":"10 percent of the employees are defined as"},{"Start":"00:29.120 ","End":"00:32.450","Text":"senior and the rest are defined as junior employees."},{"Start":"00:32.450 ","End":"00:34.580","Text":"Of all the senior employees,"},{"Start":"00:34.580 ","End":"00:37.295","Text":"70 percent are university graduates,"},{"Start":"00:37.295 ","End":"00:39.705","Text":"and of the junior employees,"},{"Start":"00:39.705 ","End":"00:42.350","Text":"20 percent are university graduates."},{"Start":"00:42.350 ","End":"00:46.475","Text":"Now, we have to draw a tree that describes the data."},{"Start":"00:46.475 ","End":"00:49.870","Text":"Now, the first stage of the tree does not depend on anything"},{"Start":"00:49.870 ","End":"00:53.615","Text":"while the second stage depends on the first stage."},{"Start":"00:53.615 ","End":"00:56.000","Text":"Let\u0027s see how we do that."},{"Start":"00:56.000 ","End":"00:57.950","Text":"Well, here we go."},{"Start":"00:57.950 ","End":"00:59.660","Text":"In the first step,"},{"Start":"00:59.660 ","End":"01:02.960","Text":"we draw the tree like this."},{"Start":"01:02.960 ","End":"01:05.854","Text":"We have seniors and we have juniors."},{"Start":"01:05.854 ","End":"01:10.100","Text":"Now, how do we know that this is the first step?"},{"Start":"01:10.100 ","End":"01:11.735","Text":"Well, we\u0027re given that."},{"Start":"01:11.735 ","End":"01:14.300","Text":"We\u0027re told in the story that 10 percent of"},{"Start":"01:14.300 ","End":"01:17.900","Text":"the employees are defined as senior and the rest are defined as junior."},{"Start":"01:17.900 ","End":"01:19.610","Text":"Here, 10 percent,"},{"Start":"01:19.610 ","End":"01:24.818","Text":"that\u0027s 0.1 and 0.9 is 90 percent,"},{"Start":"01:24.818 ","End":"01:27.935","Text":"so we have 90 percent juniors and 10 percent seniors."},{"Start":"01:27.935 ","End":"01:32.325","Text":"Now, the next step is this. Here we go."},{"Start":"01:32.325 ","End":"01:38.908","Text":"Step number 2 is stating which of the senior people are graduates or not graduates,"},{"Start":"01:38.908 ","End":"01:45.510","Text":"and which are the graduates or not graduates for the junior employees."},{"Start":"01:45.510 ","End":"01:49.620","Text":"We see here that of the senior employees,"},{"Start":"01:49.620 ","End":"01:51.199","Text":"70 percent are graduates,"},{"Start":"01:51.199 ","End":"01:53.803","Text":"that means that 30 percent are not graduate,"},{"Start":"01:53.803 ","End":"01:55.750","Text":"and of the junior employees,"},{"Start":"01:55.750 ","End":"01:59.360","Text":"20 percent are graduates and 80 percent are not graduate."},{"Start":"01:59.360 ","End":"02:02.135","Text":"Now, how do we know that this is the second step?"},{"Start":"02:02.135 ","End":"02:05.810","Text":"Well, we\u0027re told that of all the senior employees,"},{"Start":"02:05.810 ","End":"02:08.495","Text":"given that we have a senior employee,"},{"Start":"02:08.495 ","End":"02:11.330","Text":"well, 70 percent are university graduates."},{"Start":"02:11.330 ","End":"02:14.090","Text":"We know that this is the second step."},{"Start":"02:14.090 ","End":"02:22.050","Text":"This then would be the tree that we\u0027re asked to draw."}],"ID":12899},{"Watched":false,"Name":"Tutorial 2","Duration":"2m 28s","ChapterTopicVideoID":12421,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"Now that we\u0027ve drawn the tree,"},{"Start":"00:02.670 ","End":"00:07.530","Text":"in order to calculate the probability of a given branch on that tree,"},{"Start":"00:07.530 ","End":"00:10.635","Text":"we multiply all the probabilities on that branch."},{"Start":"00:10.635 ","End":"00:15.030","Text":"Now, for example, a person is randomly selected from"},{"Start":"00:15.030 ","End":"00:21.335","Text":"a company and we\u0027re asked what are the chances that he\u0027s a senior university graduate?"},{"Start":"00:21.335 ","End":"00:24.690","Text":"How do we do that? First of all,"},{"Start":"00:24.690 ","End":"00:31.630","Text":"we go to the tree and we\u0027re told that this person is a senior."},{"Start":"00:32.300 ","End":"00:35.390","Text":"This is this part of the tree,"},{"Start":"00:35.390 ","End":"00:36.785","Text":"this is this branch,"},{"Start":"00:36.785 ","End":"00:39.620","Text":"and he\u0027s a university graduate."},{"Start":"00:39.620 ","End":"00:42.850","Text":"We have this thing right here."},{"Start":"00:42.850 ","End":"00:47.735","Text":"Excellent. Now, what\u0027s the probability of being a senior?"},{"Start":"00:47.735 ","End":"00:56.805","Text":"The probability of being a senior and a university graduate."},{"Start":"00:56.805 ","End":"01:01.815","Text":"The probability of being a senior that\u0027s 0.1."},{"Start":"01:01.815 ","End":"01:06.560","Text":"Now that we want that senior to be a university graduate."},{"Start":"01:06.560 ","End":"01:10.700","Text":"Of the seniors, 0.7 or 70 percent of"},{"Start":"01:10.700 ","End":"01:16.390","Text":"them are university graduates so we have to multiply that by 0.7."},{"Start":"01:16.390 ","End":"01:20.215","Text":"That equals to 0.07."},{"Start":"01:20.215 ","End":"01:30.205","Text":"This is the probability of randomly chosen person being a senior employee and a graduate."},{"Start":"01:30.205 ","End":"01:39.120","Text":"Now, what about the chances that that person is a junior non-university graduate?"},{"Start":"01:39.120 ","End":"01:42.920","Text":"Again, we have a junior,"},{"Start":"01:42.920 ","End":"01:45.755","Text":"so we\u0027re taking this branch of the tree,"},{"Start":"01:45.755 ","End":"01:48.710","Text":"and now he\u0027s a non-university graduate,"},{"Start":"01:48.710 ","End":"01:52.605","Text":"he\u0027s not a graduate so this part of the tree."},{"Start":"01:52.605 ","End":"01:54.500","Text":"If that\u0027s the case,"},{"Start":"01:54.500 ","End":"02:02.340","Text":"we\u0027re looking for the probability of a junior and not a graduate."},{"Start":"02:03.710 ","End":"02:10.145","Text":"The probability of that person being a junior that\u0027s 0.9, that\u0027s 90 percent."},{"Start":"02:10.145 ","End":"02:12.290","Text":"Now, out of those 90 percent,"},{"Start":"02:12.290 ","End":"02:16.910","Text":"80 percent are not a graduate employee, so that would be 0.8."},{"Start":"02:16.910 ","End":"02:20.780","Text":"The probability here would be 0.72."},{"Start":"02:20.780 ","End":"02:23.495","Text":"This then would be the probability"},{"Start":"02:23.495 ","End":"02:28.740","Text":"of the selected person being a junior and not a graduate."}],"ID":12900},{"Watched":false,"Name":"Tutorial 3","Duration":"3m 49s","ChapterTopicVideoID":12422,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"Now, in order to calculate the probabilities of several branches,"},{"Start":"00:04.410 ","End":"00:06.555","Text":"we add the chances of each branch,"},{"Start":"00:06.555 ","End":"00:10.245","Text":"after multiplying the probabilities within the branch, for example,"},{"Start":"00:10.245 ","End":"00:15.225","Text":"we\u0027re asked, what are the chances that that person is a university graduate?"},{"Start":"00:15.225 ","End":"00:17.130","Text":"When looking at the tree,"},{"Start":"00:17.130 ","End":"00:21.690","Text":"you can be university graduate here or here."},{"Start":"00:21.690 ","End":"00:23.610","Text":"Now how do we get to here?"},{"Start":"00:23.610 ","End":"00:26.040","Text":"Well, we can go like this."},{"Start":"00:26.040 ","End":"00:30.389","Text":"It can be a senior and a university graduate,"},{"Start":"00:30.389 ","End":"00:34.410","Text":"or we can be a junior and a university graduate."},{"Start":"00:34.410 ","End":"00:37.080","Text":"Now, let\u0027s just calculate these probabilities."},{"Start":"00:37.080 ","End":"00:41.745","Text":"The probability of being a graduate, that equals,"},{"Start":"00:41.745 ","End":"00:43.530","Text":"now let\u0027s take this path right here,"},{"Start":"00:43.530 ","End":"00:44.720","Text":"senior and a graduate,"},{"Start":"00:44.720 ","End":"00:48.025","Text":"well that\u0027s 0.1 times 0.7."},{"Start":"00:48.025 ","End":"00:50.260","Text":"We have to add to that,"},{"Start":"00:50.260 ","End":"00:53.090","Text":"the probability of being a junior and a graduate,"},{"Start":"00:53.090 ","End":"00:56.630","Text":"so that\u0027ll be 0.9 times 0.2."},{"Start":"00:56.630 ","End":"01:00.590","Text":"Now, why are we adding these probabilities?"},{"Start":"01:00.590 ","End":"01:03.275","Text":"What\u0027s the justification for that?"},{"Start":"01:03.275 ","End":"01:05.435","Text":"Each path here,"},{"Start":"01:05.435 ","End":"01:07.940","Text":"is an exclusive event."},{"Start":"01:07.940 ","End":"01:15.800","Text":"For example, you can\u0027t be a senior graduate and a junior graduate at the same time."},{"Start":"01:15.800 ","End":"01:19.130","Text":"That means that these are exclusive events."},{"Start":"01:19.130 ","End":"01:22.350","Text":"That\u0027s true for all the paths here."},{"Start":"01:22.390 ","End":"01:28.380","Text":"This comes out to 0.25,"},{"Start":"01:28.380 ","End":"01:33.110","Text":"so the probability being a graduate is this probability,"},{"Start":"01:33.110 ","End":"01:35.090","Text":"0.1 times 0.7,"},{"Start":"01:35.090 ","End":"01:37.400","Text":"plus this probability,"},{"Start":"01:37.400 ","End":"01:41.095","Text":"0.9 times 0.2 and again,"},{"Start":"01:41.095 ","End":"01:44.220","Text":"that comes out to 0.25."},{"Start":"01:44.220 ","End":"01:46.805","Text":"Now let\u0027s look at this section."},{"Start":"01:46.805 ","End":"01:50.270","Text":"A university graduate is selected and we\u0027re asked,"},{"Start":"01:50.270 ","End":"01:51.665","Text":"what\u0027s the probability,"},{"Start":"01:51.665 ","End":"01:54.095","Text":"that he\u0027s a junior employee, hold on,"},{"Start":"01:54.095 ","End":"01:57.980","Text":"this question involves conditional probability,"},{"Start":"01:57.980 ","End":"02:01.820","Text":"so we have to use the conditional probability principles."},{"Start":"02:01.820 ","End":"02:11.950","Text":"Let\u0027s get to it. We\u0027re given that a person is a university graduate and we\u0027re asked,"},{"Start":"02:11.950 ","End":"02:14.825","Text":"what\u0027s the probability that he\u0027s a junior?"},{"Start":"02:14.825 ","End":"02:16.670","Text":"We know how to solve this."},{"Start":"02:16.670 ","End":"02:24.035","Text":"This is the probability that a person is a junior and that he\u0027s a graduate,"},{"Start":"02:24.035 ","End":"02:29.960","Text":"divided by, the probability that that person is a graduate."},{"Start":"02:29.960 ","End":"02:33.060","Text":"Let\u0027s just calculate that."},{"Start":"02:33.220 ","End":"02:36.300","Text":"Being a junior and a graduate, again,"},{"Start":"02:36.300 ","End":"02:40.680","Text":"this path right here."},{"Start":"02:40.680 ","End":"02:42.255","Text":"What does that equal to?"},{"Start":"02:42.255 ","End":"02:46.480","Text":"That\u0027s 0.9 times 0.2."},{"Start":"02:47.190 ","End":"02:52.750","Text":"This is the path where a person is a junior and a graduate."},{"Start":"02:52.750 ","End":"02:54.760","Text":"Now we have to divide that,"},{"Start":"02:54.760 ","End":"02:57.460","Text":"by the probability that a person is a graduate."},{"Start":"02:57.460 ","End":"03:01.540","Text":"Now, a person\u0027s a graduate here and here."},{"Start":"03:01.540 ","End":"03:06.490","Text":"That means, that we\u0027re looking to calculate this path right"},{"Start":"03:06.490 ","End":"03:12.180","Text":"here and this path right here. Let\u0027s do that."},{"Start":"03:12.180 ","End":"03:13.515","Text":"Well, this path right here,"},{"Start":"03:13.515 ","End":"03:18.675","Text":"that\u0027s 0.1 times 0.7 and we have to add to that,"},{"Start":"03:18.675 ","End":"03:23.070","Text":"this path, 0.9 times 0.2."},{"Start":"03:23.070 ","End":"03:33.075","Text":"Now this comes out to 0.18 divided by 0.25."},{"Start":"03:33.075 ","End":"03:36.230","Text":"We\u0027ve calculated this in the last section,"},{"Start":"03:36.230 ","End":"03:40.500","Text":"so that comes out to 0.72."},{"Start":"03:40.640 ","End":"03:44.945","Text":"This then is the probability of a person being"},{"Start":"03:44.945 ","End":"03:49.650","Text":"a junior employee given that he is a graduate."}],"ID":12901},{"Watched":false,"Name":"Tutorial 4","Duration":"3m 52s","ChapterTopicVideoID":12423,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"Most of this chapter deals with 2 fundamental principles or formulas."},{"Start":"00:05.085 ","End":"00:09.990","Text":"One is called the complete probability formula or the law of total probability,"},{"Start":"00:09.990 ","End":"00:13.080","Text":"and the other one is called the Bayes formula."},{"Start":"00:13.080 ","End":"00:17.340","Text":"Now, both of these things were used in the previous sections."},{"Start":"00:17.340 ","End":"00:21.660","Text":"The complete probability formula was used in Section C"},{"Start":"00:21.660 ","End":"00:26.250","Text":"and the Bayes formula was used in Section D. You didn\u0027t know it,"},{"Start":"00:26.250 ","End":"00:27.585","Text":"but you were using it."},{"Start":"00:27.585 ","End":"00:33.015","Text":"But it\u0027s not really necessary to know the actual names of these formulas,"},{"Start":"00:33.015 ","End":"00:37.100","Text":"but I want you to understand the principle behind them."},{"Start":"00:37.100 ","End":"00:39.670","Text":"Let\u0027s just try to understand what\u0027s going on here."},{"Start":"00:39.670 ","End":"00:43.835","Text":"The first thing is the complete probability formula."},{"Start":"00:43.835 ","End":"00:46.300","Text":"Let\u0027s take a look at that."},{"Start":"00:46.300 ","End":"00:51.770","Text":"Assume that you have this type of tree where you have A_1,"},{"Start":"00:51.770 ","End":"00:53.935","Text":"A_2, and A_3."},{"Start":"00:53.935 ","End":"00:58.380","Text":"These are events covering all of the sample space."},{"Start":"00:58.380 ","End":"01:00.615","Text":"Then you have B,"},{"Start":"01:00.615 ","End":"01:05.340","Text":"an event B within each one of the A_1, A_2, and 3."},{"Start":"01:05.340 ","End":"01:07.965","Text":"If you have B, you also have naught B."},{"Start":"01:07.965 ","End":"01:11.025","Text":"In A_1 you have B, and naught B,"},{"Start":"01:11.025 ","End":"01:14.610","Text":"A_2 B and naught B and A_3 B and naught B."},{"Start":"01:14.610 ","End":"01:16.400","Text":"Now, you\u0027re asked,"},{"Start":"01:16.400 ","End":"01:20.410","Text":"what\u0027s the probability of B?"},{"Start":"01:20.410 ","End":"01:22.934","Text":"Well, if you recall,"},{"Start":"01:22.934 ","End":"01:24.587","Text":"in Section C, what happened?"},{"Start":"01:24.587 ","End":"01:27.740","Text":"We identified all the branches where we had B,"},{"Start":"01:27.740 ","End":"01:36.990","Text":"and then we multiplied the probabilities of each branch to get to B,"},{"Start":"01:36.990 ","End":"01:42.295","Text":"and then we added these probabilities for all the various paths."},{"Start":"01:42.295 ","End":"01:45.665","Text":"Let\u0027s just put this in mathematical terms."},{"Start":"01:45.665 ","End":"01:49.160","Text":"That means that we\u0027re looking at the probability of A_1,"},{"Start":"01:49.160 ","End":"01:51.395","Text":"let\u0027s take a look at this,"},{"Start":"01:51.395 ","End":"01:54.080","Text":"times the probability of what?"},{"Start":"01:54.080 ","End":"01:56.612","Text":"Now here, this isn\u0027t the probability of B,"},{"Start":"01:56.612 ","End":"02:01.260","Text":"this is the probability of B given A_1."},{"Start":"02:01.260 ","End":"02:03.020","Text":"Given that you\u0027re in A_1,"},{"Start":"02:03.020 ","End":"02:04.460","Text":"what\u0027s the probability of B?"},{"Start":"02:04.460 ","End":"02:06.245","Text":"Well, that\u0027s this thing right here."},{"Start":"02:06.245 ","End":"02:09.170","Text":"We have to add to that this path,"},{"Start":"02:09.170 ","End":"02:11.240","Text":"the probability of this path right here,"},{"Start":"02:11.240 ","End":"02:20.175","Text":"which is the probability of A_2 times the probability of B given A_2,"},{"Start":"02:20.175 ","End":"02:22.445","Text":"plus now this path,"},{"Start":"02:22.445 ","End":"02:30.955","Text":"the probability of A_3 times the probability of B given A_3."},{"Start":"02:30.955 ","End":"02:33.450","Text":"Now, we can simplify that."},{"Start":"02:33.450 ","End":"02:38.190","Text":"This is the sum where i goes from 1 to n,"},{"Start":"02:38.190 ","End":"02:42.825","Text":"because we can have n events,"},{"Start":"02:42.825 ","End":"02:45.840","Text":"A_1, A_2, A_3 up to A_n,"},{"Start":"02:45.840 ","End":"02:48.540","Text":"and we\u0027re summing that up."},{"Start":"02:48.540 ","End":"02:57.765","Text":"What are we summing? The probability of A_i times the probability of B given A_i."},{"Start":"02:57.765 ","End":"03:00.615","Text":"This is what we\u0027ve written up here."},{"Start":"03:00.615 ","End":"03:02.300","Text":"This is a short form."},{"Start":"03:02.300 ","End":"03:03.710","Text":"This is how it looks."},{"Start":"03:03.710 ","End":"03:05.090","Text":"Now, let\u0027s take a look."},{"Start":"03:05.090 ","End":"03:09.850","Text":"Lets just remind ourselves what we did in Section C above."},{"Start":"03:09.850 ","End":"03:12.210","Text":"This is Section C,"},{"Start":"03:12.210 ","End":"03:17.233","Text":"and here we were asked about the probability of being a graduate,"},{"Start":"03:17.233 ","End":"03:19.609","Text":"that the employee would be a graduate."},{"Start":"03:19.609 ","End":"03:21.860","Text":"Now, it would be this path,"},{"Start":"03:21.860 ","End":"03:25.115","Text":"being a senior undergraduate and being a junior undergraduate."},{"Start":"03:25.115 ","End":"03:29.390","Text":"We identified all the places where we had graduates."},{"Start":"03:29.390 ","End":"03:33.560","Text":"We\u0027ve identified the paths leading to the graduates,"},{"Start":"03:33.560 ","End":"03:37.310","Text":"and then we\u0027ve just calculated the probabilities of each path."},{"Start":"03:37.310 ","End":"03:42.290","Text":"For example, at senior graduate would be 0.1 times 0.7."},{"Start":"03:42.290 ","End":"03:43.610","Text":"That\u0027s this path right here."},{"Start":"03:43.610 ","End":"03:47.570","Text":"A junior graduate would be 0.9 times 0.2."},{"Start":"03:47.570 ","End":"03:52.740","Text":"Well, this exactly the complete probability formula."}],"ID":12902},{"Watched":false,"Name":"Tutorial 5","Duration":"2m 59s","ChapterTopicVideoID":12424,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"Let\u0027s take a look at the base formula right now."},{"Start":"00:02.940 ","End":"00:08.445","Text":"The base formula asks what\u0027s the probability of A_j given B?"},{"Start":"00:08.445 ","End":"00:11.040","Text":"Well, this is a conditional probability."},{"Start":"00:11.040 ","End":"00:15.315","Text":"In the denominator we have the probability of what\u0027s given,"},{"Start":"00:15.315 ","End":"00:19.350","Text":"and in the numerator we have the probability of the intersect."},{"Start":"00:19.350 ","End":"00:23.820","Text":"That will be the probability of A_j intersect B."},{"Start":"00:23.820 ","End":"00:25.935","Text":"Well, in the denominator,"},{"Start":"00:25.935 ","End":"00:27.320","Text":"the probability of B."},{"Start":"00:27.320 ","End":"00:30.570","Text":"Well this is exactly the complete probability formula."},{"Start":"00:30.570 ","End":"00:32.399","Text":"That\u0027s this guy right here."},{"Start":"00:32.399 ","End":"00:34.595","Text":"Let\u0027s write this down right here."},{"Start":"00:34.595 ","End":"00:43.835","Text":"That will be the sum of the probability of A times the probability of B given A."},{"Start":"00:43.835 ","End":"00:49.605","Text":"This is i going from 1 to n so that will be A_i right here."},{"Start":"00:49.605 ","End":"00:51.169","Text":"What about the numerator?"},{"Start":"00:51.169 ","End":"00:57.455","Text":"Well, the numerator is the probability of A_j and B. What does that mean?"},{"Start":"00:57.455 ","End":"01:03.575","Text":"That means that we\u0027re looking at a specific path leading to B."},{"Start":"01:03.575 ","End":"01:07.310","Text":"We have here this thing right here."},{"Start":"01:07.310 ","End":"01:09.230","Text":"This would be a specific path."},{"Start":"01:09.230 ","End":"01:10.550","Text":"Assume that j,"},{"Start":"01:10.550 ","End":"01:12.755","Text":"for example is 2."},{"Start":"01:12.755 ","End":"01:21.575","Text":"We\u0027re looking then that the probability of A_j times,"},{"Start":"01:21.575 ","End":"01:24.500","Text":"now, what\u0027s this probability right here?"},{"Start":"01:24.500 ","End":"01:30.100","Text":"Well, that\u0027s the probability of B given A_j."},{"Start":"01:30.100 ","End":"01:32.735","Text":"This then is the base formula."},{"Start":"01:32.735 ","End":"01:37.190","Text":"Now as I said, we\u0027ve already done this in Section D above,"},{"Start":"01:37.190 ","End":"01:41.030","Text":"so let\u0027s just recall what we did there."},{"Start":"01:41.030 ","End":"01:45.190","Text":"This Section D, and if we recall,"},{"Start":"01:45.190 ","End":"01:49.505","Text":"we are asked about the probability of being a junior employee"},{"Start":"01:49.505 ","End":"01:55.235","Text":"given that person selected is a university graduate."},{"Start":"01:55.235 ","End":"01:57.740","Text":"Now, in the numerator we have"},{"Start":"01:57.740 ","End":"02:01.880","Text":"the intersect a probability being a junior and being a graduate,"},{"Start":"02:01.880 ","End":"02:06.125","Text":"and in the denominator we have the probability of being a graduate."},{"Start":"02:06.125 ","End":"02:08.720","Text":"Now the probability of being a junior and"},{"Start":"02:08.720 ","End":"02:11.660","Text":"a undergraduate would be that\u0027s this yellow path right here,"},{"Start":"02:11.660 ","End":"02:14.900","Text":"that\u0027s 0.9 times 0.2."},{"Start":"02:14.900 ","End":"02:17.285","Text":"That\u0027s the probability of being a graduate,"},{"Start":"02:17.285 ","End":"02:25.700","Text":"given that you\u0027re a junior or the probability of B given A_j divided by,"},{"Start":"02:25.700 ","End":"02:28.525","Text":"that\u0027s the total probability formula."},{"Start":"02:28.525 ","End":"02:30.470","Text":"The probability of being a graduate."},{"Start":"02:30.470 ","End":"02:34.100","Text":"Now we have to identify all the paths leading to being a graduate,"},{"Start":"02:34.100 ","End":"02:36.740","Text":"and that\u0027s here and here."},{"Start":"02:36.740 ","End":"02:38.765","Text":"We have to add this,"},{"Start":"02:38.765 ","End":"02:42.830","Text":"the probability of this path and the probability of this path,"},{"Start":"02:42.830 ","End":"02:46.300","Text":"so that\u0027ll be 0.9 times 0.2,"},{"Start":"02:46.300 ","End":"02:50.625","Text":"and then be 0.1 times 0.7."},{"Start":"02:50.625 ","End":"02:53.525","Text":"We have to add these probabilities up."},{"Start":"02:53.525 ","End":"02:58.380","Text":"That will be the Bayesian formula."}],"ID":12903},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"3m 40s","ChapterTopicVideoID":12425,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this question, there are 4 strawberry candies and 3 lemon candies in a bag of candies."},{"Start":"00:04.620 ","End":"00:07.170","Text":"A candy is removed randomly from the bag."},{"Start":"00:07.170 ","End":"00:09.000","Text":"If it\u0027s a strawberry candy,"},{"Start":"00:09.000 ","End":"00:11.490","Text":"it\u0027s eaten, and another candy is removed."},{"Start":"00:11.490 ","End":"00:13.500","Text":"If it\u0027s a lemon candy, however,"},{"Start":"00:13.500 ","End":"00:16.080","Text":"is returned to the bag and another candy is removed."},{"Start":"00:16.080 ","End":"00:19.920","Text":"We\u0027re asked, what\u0027s the probability that the first candy"},{"Start":"00:19.920 ","End":"00:24.885","Text":"removed is a strawberry candy and the second 1 removed is a lemon candy?"},{"Start":"00:24.885 ","End":"00:30.600","Text":"The first thing that we need to do is recognize that we\u0027re dealing with a tree diagram,"},{"Start":"00:30.600 ","End":"00:32.400","Text":"and why is that?"},{"Start":"00:32.400 ","End":"00:36.660","Text":"That\u0027s because we have 2 steps here."},{"Start":"00:36.660 ","End":"00:40.649","Text":"The first draw and the second draw of the candies."},{"Start":"00:40.649 ","End":"00:44.240","Text":"Let\u0027s just draw our probability tree here."},{"Start":"00:44.240 ","End":"00:46.610","Text":"Here\u0027s our first step here,"},{"Start":"00:46.610 ","End":"00:50.245","Text":"and we draw either a strawberry candy or a lemon candy."},{"Start":"00:50.245 ","End":"00:53.475","Text":"What\u0027s the probability of drawing a strawberry candy?"},{"Start":"00:53.475 ","End":"00:58.290","Text":"We have 4 strawberry candies and we have 4 plus 3,"},{"Start":"00:58.290 ","End":"01:03.225","Text":"a total of 7 candies in the bag."},{"Start":"01:03.225 ","End":"01:08.700","Text":"The probability of drawing a strawberry candy is 4 divided by 7,"},{"Start":"01:08.700 ","End":"01:14.660","Text":"and the probability drawing a lemon candy is 3 divided by 7."},{"Start":"01:14.660 ","End":"01:17.095","Text":"What about the next step?"},{"Start":"01:17.095 ","End":"01:19.545","Text":"Here\u0027s the next step."},{"Start":"01:19.545 ","End":"01:23.810","Text":"If you drew a strawberry candy right in the first step,"},{"Start":"01:23.810 ","End":"01:27.350","Text":"then how many candies do you have left in the bag?"},{"Start":"01:27.350 ","End":"01:29.930","Text":"You have total of 6 candies left in the bag."},{"Start":"01:29.930 ","End":"01:35.905","Text":"Why is that? Because you\u0027ve eaten the first strawberry candy."},{"Start":"01:35.905 ","End":"01:43.190","Text":"So you have a total 6 candies in the bag and how many strawberry candies remain?"},{"Start":"01:43.190 ","End":"01:46.325","Text":"You had originally 4, you\u0027ve eaten 1,"},{"Start":"01:46.325 ","End":"01:50.900","Text":"so you have 3 strawberry candies remaining in the bag."},{"Start":"01:50.900 ","End":"01:57.500","Text":"The probability of drawing a strawberry candy on the second draw is 3 out of 6."},{"Start":"01:57.500 ","End":"01:59.690","Text":"What\u0027s the probability of drawing"},{"Start":"01:59.690 ","End":"02:05.515","Text":"a lemon candy after you\u0027ve drawn a strawberry candy in the first step?"},{"Start":"02:05.515 ","End":"02:08.465","Text":"You still have 6 candies in the bag,"},{"Start":"02:08.465 ","End":"02:11.045","Text":"but you didn\u0027t take a lemon candy."},{"Start":"02:11.045 ","End":"02:13.460","Text":"There\u0027s still 3 lemon candies in the bag."},{"Start":"02:13.460 ","End":"02:18.700","Text":"The probability here would be 3 divided by 6."},{"Start":"02:18.700 ","End":"02:21.320","Text":"If you drew a lemon candy first,"},{"Start":"02:21.320 ","End":"02:23.735","Text":"then you put it back,"},{"Start":"02:23.735 ","End":"02:26.990","Text":"and then you draw for the second time."},{"Start":"02:26.990 ","End":"02:31.240","Text":"On the second try, you still have 7 candies in the bag"},{"Start":"02:31.240 ","End":"02:36.575","Text":"and the probability of drawing a strawberry candy is 4 out of 7,"},{"Start":"02:36.575 ","End":"02:41.350","Text":"and the probability of drawing a lemon candy is 3 out of 7."},{"Start":"02:41.350 ","End":"02:45.856","Text":"This then would be the probability tree and we\u0027re asked,"},{"Start":"02:45.856 ","End":"02:49.100","Text":"what\u0027s the probability that the first candy removed is"},{"Start":"02:49.100 ","End":"02:53.320","Text":"a strawberry candy and the second 1 removed is a lemon candy?"},{"Start":"02:53.320 ","End":"02:55.610","Text":"This is what we\u0027re looking for,"},{"Start":"02:55.610 ","End":"02:58.085","Text":"this probability right here."},{"Start":"02:58.085 ","End":"03:05.300","Text":"The first step would be to draw a strawberry candy,"},{"Start":"03:05.300 ","End":"03:08.480","Text":"so we\u0027re here on this branch."},{"Start":"03:08.480 ","End":"03:11.455","Text":"The second step is to draw a lemon candy,"},{"Start":"03:11.455 ","End":"03:13.400","Text":"we\u0027re right here on this branch."},{"Start":"03:13.400 ","End":"03:17.195","Text":"This is probability that we\u0027re asked to calculate."},{"Start":"03:17.195 ","End":"03:21.830","Text":"That equals to 4 divided by 7."},{"Start":"03:21.830 ","End":"03:23.150","Text":"That\u0027s this guy right here,"},{"Start":"03:23.150 ","End":"03:25.970","Text":"the probability of drawing a strawberry candy first."},{"Start":"03:25.970 ","End":"03:31.370","Text":"Then we have to multiply that by 3 divided by 6,"},{"Start":"03:31.370 ","End":"03:35.014","Text":"that\u0027s the probability of drawing a lemon candy, second."},{"Start":"03:35.014 ","End":"03:41.170","Text":"That equals to 2 divided by 7."}],"ID":12904},{"Watched":false,"Name":"Exercise 1 - Part b","Duration":"1m 8s","ChapterTopicVideoID":12426,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"In the next section, we\u0027re asked,"},{"Start":"00:01.470 ","End":"00:05.670","Text":"what\u0027s the probability that the second candy is a lemon candy?"},{"Start":"00:05.670 ","End":"00:11.880","Text":"Okay. That means that we\u0027re looking at these branches right here."},{"Start":"00:11.880 ","End":"00:15.360","Text":"If the second candy is a lemon candy,"},{"Start":"00:15.360 ","End":"00:19.245","Text":"then we\u0027re either here or we\u0027re here."},{"Start":"00:19.245 ","End":"00:24.630","Text":"So we have to find the probabilities of these branches,"},{"Start":"00:24.630 ","End":"00:30.150","Text":"these paths right here and this path right here."},{"Start":"00:30.150 ","End":"00:32.025","Text":"Let\u0027s get to it."},{"Start":"00:32.025 ","End":"00:38.725","Text":"We\u0027re looking for the probability that the second draw would be a lemon."},{"Start":"00:38.725 ","End":"00:42.770","Text":"Now, that equals to the probability of this path,"},{"Start":"00:42.770 ","End":"00:47.380","Text":"which is 4 divided by 7 times 3 divided by 6,"},{"Start":"00:47.380 ","End":"00:49.175","Text":"that\u0027s this guy right here,"},{"Start":"00:49.175 ","End":"00:57.225","Text":"plus 3 divided by 7 times 3 divided by 7."},{"Start":"00:57.225 ","End":"01:00.665","Text":"Now, that\u0027s this path right here."},{"Start":"01:00.665 ","End":"01:08.490","Text":"The total comes out to 23 divided by 49."}],"ID":12905},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"3m 24s","ChapterTopicVideoID":12427,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this question, we have a population that\u0027s composed of 30 percent children,"},{"Start":"00:03.810 ","End":"00:07.079","Text":"50 percent adults, and the rest are senior citizens."},{"Start":"00:07.079 ","End":"00:09.884","Text":"Now, according to the Ministry of Health figures,"},{"Start":"00:09.884 ","End":"00:13.920","Text":"the chances that a child would get the flu in winter are 80 percent,"},{"Start":"00:13.920 ","End":"00:17.085","Text":"the chances that an adult would get the flu are 40 percent,"},{"Start":"00:17.085 ","End":"00:21.348","Text":"and the chances that a senior citizen will get the flu are 70 percent,"},{"Start":"00:21.348 ","End":"00:25.230","Text":"and we\u0027re asked what\u0027s the proportion of the population that"},{"Start":"00:25.230 ","End":"00:29.835","Text":"consists of senior citizens who do not get the flu during winter?"},{"Start":"00:29.835 ","End":"00:34.875","Text":"Okay, so the first thing that we need to know is that we\u0027re dealing"},{"Start":"00:34.875 ","End":"00:39.985","Text":"with a probability tree and we have to draw a tree diagram. Now why is that?"},{"Start":"00:39.985 ","End":"00:42.710","Text":"Well, here we have 2 levels."},{"Start":"00:42.710 ","End":"00:46.910","Text":"The first level is the type of citizen that you are."},{"Start":"00:46.910 ","End":"00:50.705","Text":"You\u0027re either a child, an adult or a senior citizen."},{"Start":"00:50.705 ","End":"00:54.575","Text":"This level isn\u0027t dependent on anything."},{"Start":"00:54.575 ","End":"00:58.460","Text":"The second level is whether you have the flu or not,"},{"Start":"00:58.460 ","End":"01:02.405","Text":"now that\u0027s dependent on the age group that you\u0027re in."},{"Start":"01:02.405 ","End":"01:03.590","Text":"You\u0027re either a child,"},{"Start":"01:03.590 ","End":"01:05.915","Text":"an adult, or a senior citizen."},{"Start":"01:05.915 ","End":"01:08.705","Text":"The probabilities whether you get the flu or not,"},{"Start":"01:08.705 ","End":"01:12.620","Text":"are dependent on the age group that you\u0027re in."},{"Start":"01:12.620 ","End":"01:18.020","Text":"The first thing that we need to do then is to draw our probability tree."},{"Start":"01:18.020 ","End":"01:21.230","Text":"Here\u0027s our probability tree and that\u0027s"},{"Start":"01:21.230 ","End":"01:25.190","Text":"the first level and in the first level we have the 3 age groups."},{"Start":"01:25.190 ","End":"01:26.480","Text":"We have the child,"},{"Start":"01:26.480 ","End":"01:28.865","Text":"the adult, and the senior citizen."},{"Start":"01:28.865 ","End":"01:31.790","Text":"Now, the proportion of children,"},{"Start":"01:31.790 ","End":"01:33.770","Text":"the population is 30 percent,"},{"Start":"01:33.770 ","End":"01:37.205","Text":"so the probability of being a child is 0.3."},{"Start":"01:37.205 ","End":"01:39.557","Text":"The proportion of adult is 50 percent,"},{"Start":"01:39.557 ","End":"01:45.800","Text":"so the probability of being adult is 0.5 and the proportion of senior citizens,"},{"Start":"01:45.800 ","End":"01:47.195","Text":"so that\u0027s 20 percent,"},{"Start":"01:47.195 ","End":"01:50.840","Text":"so the probability here would be 0.2."},{"Start":"01:50.840 ","End":"01:56.165","Text":"The second level deals with the probabilities of getting the flu or not."},{"Start":"01:56.165 ","End":"01:58.430","Text":"Let\u0027s draw the second level."},{"Start":"01:58.430 ","End":"02:00.560","Text":"This is at right here."},{"Start":"02:00.560 ","End":"02:03.590","Text":"If you\u0027re a child,"},{"Start":"02:03.590 ","End":"02:08.425","Text":"then the probability of getting a flu is 0.8, that\u0027s 80 percent."},{"Start":"02:08.425 ","End":"02:09.740","Text":"If you\u0027re an adult,"},{"Start":"02:09.740 ","End":"02:12.640","Text":"the probability of getting the flu are 40 percent,"},{"Start":"02:12.640 ","End":"02:15.739","Text":"that\u0027s right here and if you\u0027re a senior citizen,"},{"Start":"02:15.739 ","End":"02:18.805","Text":"then the probability of getting the flu 0.7."},{"Start":"02:18.805 ","End":"02:26.495","Text":"Now we\u0027ve drawn a probability tree and we have other probabilities on there."},{"Start":"02:26.495 ","End":"02:30.740","Text":"Now we\u0027re asked what proportion of the population consists of"},{"Start":"02:30.740 ","End":"02:35.090","Text":"senior citizens who do not get the flu during winter?"},{"Start":"02:35.090 ","End":"02:38.620","Text":"That means that we\u0027re dealing with senior citizens,"},{"Start":"02:38.620 ","End":"02:42.785","Text":"and they do not get the flu."},{"Start":"02:42.785 ","End":"02:51.770","Text":"We\u0027re looking for the probability of senior citizens and no flu."},{"Start":"02:51.770 ","End":"02:54.170","Text":"Well, if you\u0027re a senior citizen,"},{"Start":"02:54.170 ","End":"02:57.845","Text":"the probability of being a senior citizen, that\u0027s 0.2,"},{"Start":"02:57.845 ","End":"03:00.035","Text":"and if you don\u0027t get the flu,"},{"Start":"03:00.035 ","End":"03:03.650","Text":"if you\u0027re a senior citizen,"},{"Start":"03:03.650 ","End":"03:06.740","Text":"well, that\u0027s what the probability of 0.3."},{"Start":"03:06.740 ","End":"03:10.310","Text":"We have here 0.2 times 0.3,"},{"Start":"03:10.310 ","End":"03:15.420","Text":"that\u0027s 0.06 or we\u0027re dealing with 6 percent."},{"Start":"03:15.420 ","End":"03:24.600","Text":"So 6 percent of the population would be senior citizens who do not get the flu."}],"ID":12906},{"Watched":false,"Name":"Exercise 2 - Part b","Duration":"1m 18s","ChapterTopicVideoID":12428,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"In this section we\u0027re asked,"},{"Start":"00:01.470 ","End":"00:05.040","Text":"what proportion of the people get flu during the winter?"},{"Start":"00:05.040 ","End":"00:07.065","Text":"Well, first of all,"},{"Start":"00:07.065 ","End":"00:10.695","Text":"let\u0027s identify the path to getting the flu."},{"Start":"00:10.695 ","End":"00:12.120","Text":"Well, that\u0027s right here, getting the flu, getting"},{"Start":"00:12.120 ","End":"00:15.960","Text":"the flu here, I\u0027m getting the flu here."},{"Start":"00:15.960 ","End":"00:20.010","Text":"Okay, so what we need to do is we need to calculate"},{"Start":"00:20.010 ","End":"00:25.515","Text":"the probabilities that bring us to getting the flu."},{"Start":"00:25.515 ","End":"00:27.030","Text":"Now, how do we do that?"},{"Start":"00:27.030 ","End":"00:30.405","Text":"Well, we can be a child and get the flu."},{"Start":"00:30.405 ","End":"00:33.120","Text":"We can be an adult and get the flu,"},{"Start":"00:33.120 ","End":"00:36.820","Text":"or we can be a senior citizen that gets the flu."},{"Start":"00:36.820 ","End":"00:41.270","Text":"What\u0027s the probability then of getting the flu?"},{"Start":"00:41.270 ","End":"00:46.490","Text":"Well, that equals to the sum of the probabilities of each one of the pairs."},{"Start":"00:46.490 ","End":"00:48.515","Text":"So let\u0027s calculate that."},{"Start":"00:48.515 ","End":"00:50.645","Text":"Being a child and getting the flu,"},{"Start":"00:50.645 ","End":"00:54.380","Text":"that\u0027s 0.3 times 0.8,"},{"Start":"00:54.380 ","End":"00:59.270","Text":"and we have to add to that the probability of being an adult and getting the flu,"},{"Start":"00:59.270 ","End":"01:03.260","Text":"that\u0027s 0.5 times 0.4; plus,"},{"Start":"01:03.260 ","End":"01:06.590","Text":"the probability of being a senior citizen getting the flu,"},{"Start":"01:06.590 ","End":"01:10.115","Text":"that\u0027s 0.2 times 0.7."},{"Start":"01:10.115 ","End":"01:18.390","Text":"Now, that comes out to 0.58 or 58 percent."}],"ID":12907},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"1m 56s","ChapterTopicVideoID":12429,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":false,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[],"ID":12908},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"2m 7s","ChapterTopicVideoID":12430,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this section, we\u0027re told that a child is selected and we\u0027re asked,"},{"Start":"00:03.870 ","End":"00:09.285","Text":"what\u0027s the probability that he will not get the flu during the winter?"},{"Start":"00:09.285 ","End":"00:12.570","Text":"We\u0027re looking for the probability."},{"Start":"00:12.570 ","End":"00:17.460","Text":"Now, since it\u0027s a conditional probability, then what are we told?"},{"Start":"00:17.460 ","End":"00:19.980","Text":"We\u0027re told that a child is selected."},{"Start":"00:19.980 ","End":"00:21.720","Text":"We have this."},{"Start":"00:21.720 ","End":"00:23.850","Text":"We\u0027re given that this person is a child,"},{"Start":"00:23.850 ","End":"00:30.160","Text":"and we\u0027re asked what\u0027s the probability that he will not get the flu?"},{"Start":"00:30.350 ","End":"00:37.670","Text":"This probability can be derived instantaneously from the tree,"},{"Start":"00:37.670 ","End":"00:41.320","Text":"but let\u0027s do it the long way around first."},{"Start":"00:41.320 ","End":"00:46.525","Text":"In the denominator we have the probability of what\u0027s given,"},{"Start":"00:46.525 ","End":"00:50.630","Text":"and in the numerator we have the probability of the intersect;"},{"Start":"00:50.630 ","End":"00:56.220","Text":"not flu and being a child."},{"Start":"00:57.410 ","End":"00:59.960","Text":"Well, that equals to the following."},{"Start":"00:59.960 ","End":"01:01.910","Text":"The probability of being a child,"},{"Start":"01:01.910 ","End":"01:02.960","Text":"we were given that,"},{"Start":"01:02.960 ","End":"01:12.030","Text":"that\u0027s 0.3 and the probability of not having the flu and being a child,"},{"Start":"01:12.030 ","End":"01:15.910","Text":"well, that\u0027s this path right here."},{"Start":"01:15.910 ","End":"01:20.705","Text":"That equals to 0.3 times 0.2."},{"Start":"01:20.705 ","End":"01:28.790","Text":"0.3 cancels each other out and we\u0027re left with 0.2 and this is exactly what we have here."},{"Start":"01:28.790 ","End":"01:31.025","Text":"What are these probabilities?"},{"Start":"01:31.025 ","End":"01:34.220","Text":"These are conditional probabilities."},{"Start":"01:34.220 ","End":"01:37.415","Text":"For example, this 0.2,"},{"Start":"01:37.415 ","End":"01:44.245","Text":"that means that this is a probability of not getting the flu given that you\u0027re a child."},{"Start":"01:44.245 ","End":"01:47.970","Text":"Or what\u0027s this probability right here?"},{"Start":"01:47.970 ","End":"01:54.130","Text":"Well, this is the probability of getting the flu given that you\u0027re an adult."},{"Start":"01:54.130 ","End":"01:57.605","Text":"Whenever you see a question like this,"},{"Start":"01:57.605 ","End":"02:02.465","Text":"where what\u0027s given to you is the first level."},{"Start":"02:02.465 ","End":"02:07.200","Text":"Well, then you know that the answer is almost immediate."}],"ID":12909},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"6m 1s","ChapterTopicVideoID":12431,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.265","Text":"In this question, there are 5 blue balls and 5 red balls in Container A,"},{"Start":"00:05.265 ","End":"00:08.775","Text":"and 6 blue balls and 4 red balls in Container B."},{"Start":"00:08.775 ","End":"00:11.955","Text":"Now, a container is randomly selected,"},{"Start":"00:11.955 ","End":"00:13.860","Text":"a ball is taken out,"},{"Start":"00:13.860 ","End":"00:17.595","Text":"and another ball is removed without returning the first 1."},{"Start":"00:17.595 ","End":"00:24.075","Text":"We\u0027re asked what\u0027s the probability that the 2 balls removed will be of different colors."},{"Start":"00:24.075 ","End":"00:29.605","Text":"The first thing that we need to know is that we\u0027re dealing with a probability tree."},{"Start":"00:29.605 ","End":"00:36.925","Text":"We have to draw a tree diagram to describe the steps here."},{"Start":"00:36.925 ","End":"00:38.720","Text":"Now, why is that?"},{"Start":"00:38.720 ","End":"00:43.610","Text":"Well, first of all, I see that I have 3 levels to deal with."},{"Start":"00:43.610 ","End":"00:49.610","Text":"First of all, the first level is choosing a container."},{"Start":"00:49.610 ","End":"00:51.890","Text":"Now, we choose it randomly."},{"Start":"00:51.890 ","End":"00:57.275","Text":"The second level is choosing the first ball,"},{"Start":"00:57.275 ","End":"01:04.025","Text":"and then the third level is choosing another ball."},{"Start":"01:04.025 ","End":"01:08.210","Text":"Now, with this type of logic,"},{"Start":"01:08.210 ","End":"01:10.085","Text":"we need a tree diagram."},{"Start":"01:10.085 ","End":"01:13.115","Text":"Let\u0027s start building this diagram."},{"Start":"01:13.115 ","End":"01:18.920","Text":"We see here that we have Container A and Container B."},{"Start":"01:18.920 ","End":"01:23.120","Text":"Now, we\u0027re told that the container is randomly selected."},{"Start":"01:23.120 ","End":"01:26.900","Text":"That means that the probability of choosing Container A is"},{"Start":"01:26.900 ","End":"01:32.465","Text":"50 percent and the same probability for Container B."},{"Start":"01:32.465 ","End":"01:36.480","Text":"Now, what about the second level?"},{"Start":"01:36.480 ","End":"01:38.250","Text":"Here\u0027s the second level."},{"Start":"01:38.250 ","End":"01:40.980","Text":"Now in Container A, what are we told?"},{"Start":"01:40.980 ","End":"01:45.830","Text":"There are 5 blue balls and 5 red balls in Container A."},{"Start":"01:45.830 ","End":"01:49.635","Text":"Out of a total of 10 balls in Container A,"},{"Start":"01:49.635 ","End":"01:52.655","Text":"we have 5 blue balls and 5 red balls."},{"Start":"01:52.655 ","End":"01:54.755","Text":"Now what about Container B?"},{"Start":"01:54.755 ","End":"01:59.210","Text":"There are 6 blue balls and 4 red balls in Container B,"},{"Start":"01:59.210 ","End":"02:01.070","Text":"again, 10 balls in the container,"},{"Start":"02:01.070 ","End":"02:04.280","Text":"and we have 6 blue and 4 red."},{"Start":"02:04.280 ","End":"02:11.195","Text":"These are the probabilities of getting a red ball or a blue ball."},{"Start":"02:11.195 ","End":"02:13.190","Text":"In each 1 of the containers."},{"Start":"02:13.190 ","End":"02:16.615","Text":"Now, what about the third level?"},{"Start":"02:16.615 ","End":"02:19.790","Text":"Here we go with the third level."},{"Start":"02:19.790 ","End":"02:24.275","Text":"Let\u0027s say I\u0027m in Container A and I chose a blue ball."},{"Start":"02:24.275 ","End":"02:26.600","Text":"Well, how many balls now do I have in the container?"},{"Start":"02:26.600 ","End":"02:27.895","Text":"I have 9 balls."},{"Start":"02:27.895 ","End":"02:31.435","Text":"how many blue balls do I have in the container?"},{"Start":"02:31.435 ","End":"02:37.270","Text":"4 blue balls because I\u0027ve already chosen a blue ball in the first draw."},{"Start":"02:37.270 ","End":"02:39.485","Text":"How many red balls do I have?"},{"Start":"02:39.485 ","End":"02:42.445","Text":"I have 5 red balls, I didn\u0027t touch them."},{"Start":"02:42.445 ","End":"02:46.686","Text":"Now, if I chose a red ball in the Container A, well,"},{"Start":"02:46.686 ","End":"02:50.055","Text":"then I have 5 blue balls out of 9,"},{"Start":"02:50.055 ","End":"02:53.320","Text":"and I have 4 red balls out of 9."},{"Start":"02:53.320 ","End":"02:55.715","Text":"Now, what about Container B?"},{"Start":"02:55.715 ","End":"02:58.490","Text":"Well, if I chose a blue ball first,"},{"Start":"02:58.490 ","End":"02:59.915","Text":"while the second choice,"},{"Start":"02:59.915 ","End":"03:03.070","Text":"I would have 5 ball to choose from,"},{"Start":"03:03.070 ","End":"03:05.945","Text":"5 out of 9 balls to choose from."},{"Start":"03:05.945 ","End":"03:08.660","Text":"If I chose a blue ball,"},{"Start":"03:08.660 ","End":"03:10.400","Text":"and if I shows a red ball,"},{"Start":"03:10.400 ","End":"03:12.965","Text":"that would be 4 out of 9."},{"Start":"03:12.965 ","End":"03:15.110","Text":"In the second container,"},{"Start":"03:15.110 ","End":"03:17.300","Text":"if I chose a red ball first,"},{"Start":"03:17.300 ","End":"03:18.950","Text":"well, the second choice,"},{"Start":"03:18.950 ","End":"03:21.125","Text":"I would have 6 out of 9,"},{"Start":"03:21.125 ","End":"03:24.110","Text":"a chance of 6 out of 9 to choose a blue ball,"},{"Start":"03:24.110 ","End":"03:27.170","Text":"and 3 out of 9 to choose a red ball."},{"Start":"03:27.170 ","End":"03:30.610","Text":"Why is that? Because I\u0027ve already chosen a red ball here."},{"Start":"03:30.610 ","End":"03:35.315","Text":"This would be the probability tree here and we\u0027re asked,"},{"Start":"03:35.315 ","End":"03:42.260","Text":"what\u0027s the probability that the 2 balls removed will be of different colors?"},{"Start":"03:42.260 ","End":"03:44.570","Text":"Let\u0027s see what happens here."},{"Start":"03:44.570 ","End":"03:47.180","Text":"What does it mean that there are of different colors?"},{"Start":"03:47.180 ","End":"03:54.395","Text":"Well, if I chose a blue ball in the first draw,"},{"Start":"03:54.395 ","End":"04:00.000","Text":"then that means I have to choose a red ball in the second draw, and vice versa."},{"Start":"04:00.000 ","End":"04:03.410","Text":"Let\u0027s start drawing the paths here."},{"Start":"04:03.410 ","End":"04:05.495","Text":"Well, in Container A,"},{"Start":"04:05.495 ","End":"04:09.050","Text":"if I chose a blue ball,"},{"Start":"04:09.050 ","End":"04:11.330","Text":"then I have to choose a red ball."},{"Start":"04:11.330 ","End":"04:16.265","Text":"That means that the 2 balls that were removed are of different colors."},{"Start":"04:16.265 ","End":"04:23.190","Text":"But I could be in Container A and choose a red ball first and then a blue ball second."},{"Start":"04:23.190 ","End":"04:29.520","Text":"Now, if I go to Container B and I choose a blue ball first,"},{"Start":"04:29.520 ","End":"04:32.235","Text":"then I have to choose a red ball second."},{"Start":"04:32.235 ","End":"04:35.690","Text":"In Container B, if I choose a red ball first,"},{"Start":"04:35.690 ","End":"04:39.500","Text":"then that means I have to choose a blue ball second."},{"Start":"04:39.500 ","End":"04:43.100","Text":"These are all the various paths that I have"},{"Start":"04:43.100 ","End":"04:47.880","Text":"to deal with and I have to calculate their probabilities."},{"Start":"04:47.880 ","End":"04:54.079","Text":"The probability of getting 2 balls of different colors,"},{"Start":"04:54.079 ","End":"04:56.090","Text":"well, that would be the following."},{"Start":"04:56.090 ","End":"04:58.430","Text":"Well, let\u0027s start with Container A."},{"Start":"04:58.430 ","End":"05:04.710","Text":"That\u0027ll be 0.5 times 5/10 times 5 divided by 9."},{"Start":"05:04.710 ","End":"05:13.275","Text":"That\u0027ll be 0.5 times 5 divided by 10 times 5 divided by 9."},{"Start":"05:13.275 ","End":"05:14.945","Text":"Plus, now,"},{"Start":"05:14.945 ","End":"05:18.215","Text":"I have to calculate this path right here."},{"Start":"05:18.215 ","End":"05:24.612","Text":"That\u0027ll be 0.5 times 5 divided by 10 times,"},{"Start":"05:24.612 ","End":"05:27.955","Text":"again, 5 divided by 9."},{"Start":"05:27.955 ","End":"05:31.620","Text":"Plus, now I go to Container B."},{"Start":"05:31.620 ","End":"05:34.545","Text":"That\u0027ll be 0.5 times,"},{"Start":"05:34.545 ","End":"05:37.980","Text":"choosing a blue ball that\u0027s 6 divided by 10."},{"Start":"05:37.980 ","End":"05:43.800","Text":"Then I have to choose a red ball that\u0027s 4 divided by 9 plus in Container B again,"},{"Start":"05:43.800 ","End":"05:46.230","Text":"that\u0027s 0.5 times,"},{"Start":"05:46.230 ","End":"05:47.489","Text":"now choosing a red ball,"},{"Start":"05:47.489 ","End":"05:49.730","Text":"that\u0027s 4 divided by 10 times,"},{"Start":"05:49.730 ","End":"05:51.350","Text":"and now choosing a blue ball next,"},{"Start":"05:51.350 ","End":"05:53.600","Text":"6 divided by 9."},{"Start":"05:53.600 ","End":"06:00.750","Text":"Now, all of this comes out to 0.544."}],"ID":12910},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"2m 57s","ChapterTopicVideoID":12432,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.425","Text":"In this section, we\u0027re asked if the balls removed are of different colors,"},{"Start":"00:04.425 ","End":"00:08.610","Text":"what\u0027s the probability that the second ball is red?"},{"Start":"00:08.610 ","End":"00:12.060","Text":"This is a conditional probability,"},{"Start":"00:12.060 ","End":"00:13.995","Text":"and it looks like this."},{"Start":"00:13.995 ","End":"00:16.545","Text":"We have the probability."},{"Start":"00:16.545 ","End":"00:19.560","Text":"Now, what\u0027s given to us that the balls are of different colors,"},{"Start":"00:19.560 ","End":"00:22.935","Text":"it says so, if the balls removed are of different colors."},{"Start":"00:22.935 ","End":"00:25.425","Text":"That\u0027s what\u0027s given, and what are we looking for?"},{"Start":"00:25.425 ","End":"00:29.670","Text":"The probability that the second ball is red."},{"Start":"00:29.670 ","End":"00:32.190","Text":"Now, we know how to solve that."},{"Start":"00:32.190 ","End":"00:36.800","Text":"In the denominator, we have the probability of what\u0027s given,"},{"Start":"00:36.800 ","End":"00:39.470","Text":"and in the numerator,"},{"Start":"00:39.470 ","End":"00:43.010","Text":"we have the intersect of the 2 events here."},{"Start":"00:43.010 ","End":"00:45.290","Text":"That means the probability of"},{"Start":"00:45.290 ","End":"00:52.560","Text":"the second ball being red and the balls being different colors."},{"Start":"00:52.640 ","End":"00:56.450","Text":"Let\u0027s take a look at what we have here."},{"Start":"00:56.450 ","End":"00:57.995","Text":"In the denominator,"},{"Start":"00:57.995 ","End":"01:03.155","Text":"we have the probability that the balls are of different colors."},{"Start":"01:03.155 ","End":"01:07.010","Text":"Well, we\u0027ve calculated this probability in the last section,"},{"Start":"01:07.010 ","End":"01:12.965","Text":"that equal to 0.544."},{"Start":"01:12.965 ","End":"01:15.154","Text":"Now, what about the numerator?"},{"Start":"01:15.154 ","End":"01:20.840","Text":"That the second ball is red and that the balls are different colors."},{"Start":"01:20.840 ","End":"01:22.295","Text":"What does that mean?"},{"Start":"01:22.295 ","End":"01:25.130","Text":"Well, let\u0027s take a look at the tree here."},{"Start":"01:25.130 ","End":"01:27.725","Text":"If the second ball is red,"},{"Start":"01:27.725 ","End":"01:32.750","Text":"that means that the first all has to be blue."},{"Start":"01:32.750 ","End":"01:42.280","Text":"So we\u0027re looking at the probability that the first ball is blue and the second is red."},{"Start":"01:42.280 ","End":"01:46.175","Text":"Let\u0027s calculate this probability right here."},{"Start":"01:46.175 ","End":"01:48.620","Text":"The first ball has to be blue."},{"Start":"01:48.620 ","End":"01:51.395","Text":"We could go to container A,"},{"Start":"01:51.395 ","End":"01:55.670","Text":"the first ball is blue and the second ball is red,"},{"Start":"01:55.670 ","End":"01:59.485","Text":"or we can go to the container B,"},{"Start":"01:59.485 ","End":"02:05.240","Text":"and the first ball would be blue and the second ball would be red as well."},{"Start":"02:05.240 ","End":"02:08.930","Text":"What would be this probability right here?"},{"Start":"02:08.930 ","End":"02:12.740","Text":"Well, that\u0027s the sum of the probabilities of the 2 paths here."},{"Start":"02:12.740 ","End":"02:14.150","Text":"Let\u0027s calculate that."},{"Start":"02:14.150 ","End":"02:15.950","Text":"In this path right here,"},{"Start":"02:15.950 ","End":"02:23.420","Text":"that\u0027ll be 0.5 times 5 divided by 10 times 5 divided by 9,"},{"Start":"02:23.420 ","End":"02:25.685","Text":"that\u0027s this path right here,"},{"Start":"02:25.685 ","End":"02:30.545","Text":"plus this path that\u0027s 0.5"},{"Start":"02:30.545 ","End":"02:36.815","Text":"times 6 divided by 10 times 4 divided by 9."},{"Start":"02:36.815 ","End":"02:42.880","Text":"All of this has to be over 0.544."},{"Start":"02:42.880 ","End":"02:48.480","Text":"Now, that comes out to 0.5."},{"Start":"02:48.480 ","End":"02:53.180","Text":"That\u0027s the probability that the second ball is red,"},{"Start":"02:53.180 ","End":"02:57.960","Text":"given that the balls are of different colors."}],"ID":12911},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"2m 44s","ChapterTopicVideoID":12433,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.430","Text":"In this question, a cellular company classifies its customers according to 3 age groups,"},{"Start":"00:05.430 ","End":"00:08.625","Text":"young people, adults, and retirees."},{"Start":"00:08.625 ","End":"00:12.135","Text":"Now it\u0027s given that 10 percent of the customer, say young,"},{"Start":"00:12.135 ","End":"00:16.395","Text":"70 percent adults and the rest are retired people,"},{"Start":"00:16.395 ","End":"00:18.525","Text":"90 percent of the young people,"},{"Start":"00:18.525 ","End":"00:24.150","Text":"70 percent of the adults and 30 percent of retirees have smartphones and were"},{"Start":"00:24.150 ","End":"00:30.450","Text":"asked what proportion of the company\u0027s customers are young people with smartphones?"},{"Start":"00:30.450 ","End":"00:33.600","Text":"The first thing we need to know is that we\u0027re"},{"Start":"00:33.600 ","End":"00:36.900","Text":"dealing with a tree diagram and why is that?"},{"Start":"00:36.900 ","End":"00:38.595","Text":"Well, we have 2 levels."},{"Start":"00:38.595 ","End":"00:42.650","Text":"The first level is the level of the age group."},{"Start":"00:42.650 ","End":"00:47.435","Text":"That\u0027s right here, young, adult or retired."},{"Start":"00:47.435 ","End":"00:51.470","Text":"This level isn\u0027t dependent on anything."},{"Start":"00:51.470 ","End":"00:58.070","Text":"Now, the next level is this right here."},{"Start":"00:58.070 ","End":"01:02.375","Text":"That\u0027s the level where the people have smartphones or not."},{"Start":"01:02.375 ","End":"01:09.530","Text":"This level is definitely dependent on the first level, the age group."},{"Start":"01:09.530 ","End":"01:14.065","Text":"Let\u0027s just draw our tree diagram here."},{"Start":"01:14.065 ","End":"01:16.370","Text":"Here we have a first-level,"},{"Start":"01:16.370 ","End":"01:19.460","Text":"we have young adults and retirees here."},{"Start":"01:19.460 ","End":"01:22.885","Text":"The percentage of young people, that\u0027s 0.1,"},{"Start":"01:22.885 ","End":"01:24.285","Text":"that\u0027s 10 percent,"},{"Start":"01:24.285 ","End":"01:25.875","Text":"70 percent for adults,"},{"Start":"01:25.875 ","End":"01:28.760","Text":"and 20 percent for the retirees."},{"Start":"01:28.760 ","End":"01:31.070","Text":"Now, what about the second level?"},{"Start":"01:31.070 ","End":"01:33.470","Text":"Well, here\u0027s the second level."},{"Start":"01:33.470 ","End":"01:37.430","Text":"For each of the age group here we have"},{"Start":"01:37.430 ","End":"01:42.195","Text":"a population that has a smartphone and does not have a smartphone."},{"Start":"01:42.195 ","End":"01:44.840","Text":"For example, for young people,"},{"Start":"01:44.840 ","End":"01:49.735","Text":"when 90 percent of the young people have a smartphone and 10 percent do not."},{"Start":"01:49.735 ","End":"01:53.655","Text":"70 percent of the adults have a smartphone and 30 do not."},{"Start":"01:53.655 ","End":"01:58.735","Text":"30 percent of the retirees have smartphones and 70 percent do not."},{"Start":"01:58.735 ","End":"02:02.705","Text":"We\u0027re asked what proportion of the company\u0027s customers,"},{"Start":"02:02.705 ","End":"02:05.120","Text":"are young people with smartphones."},{"Start":"02:05.120 ","End":"02:07.115","Text":"We\u0027re looking at this branch."},{"Start":"02:07.115 ","End":"02:12.320","Text":"We\u0027re looking at the young people and young people that have a smartphone right here."},{"Start":"02:12.320 ","End":"02:19.580","Text":"We\u0027re looking for the probability of young and a smartphone."},{"Start":"02:19.580 ","End":"02:21.260","Text":"They have a smartphone."},{"Start":"02:21.260 ","End":"02:29.855","Text":"Well, that means that being young population here that\u0027s 0.1 times,"},{"Start":"02:29.855 ","End":"02:37.250","Text":"having a smartphone, 0.9 or that equals to 0.09 or 9 percent."},{"Start":"02:37.250 ","End":"02:45.120","Text":"9 percent of the population are young people with smartphones."}],"ID":12912},{"Watched":false,"Name":"Exercise 4 - Part b","Duration":"2m 17s","ChapterTopicVideoID":12434,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.725","Text":"In this section we\u0027re asked,"},{"Start":"00:01.725 ","End":"00:06.975","Text":"a customer is randomly selected and it\u0027s given that he has a smart phone."},{"Start":"00:06.975 ","End":"00:08.250","Text":"Now, we\u0027re asked,"},{"Start":"00:08.250 ","End":"00:11.565","Text":"what\u0027s the probability that he\u0027s a retired person?"},{"Start":"00:11.565 ","End":"00:16.155","Text":"Again, we have here a conditional probability."},{"Start":"00:16.155 ","End":"00:18.870","Text":"Now, what\u0027s given to us?"},{"Start":"00:18.870 ","End":"00:22.230","Text":"It\u0027s given that he has a smart phone."},{"Start":"00:22.230 ","End":"00:24.450","Text":"That\u0027s a yes here."},{"Start":"00:24.450 ","End":"00:31.000","Text":"We\u0027re asked about the probability that that person is retired."},{"Start":"00:31.100 ","End":"00:34.145","Text":"Now, we know how to solve this."},{"Start":"00:34.145 ","End":"00:41.150","Text":"In the denominator, we have the probability of having a smart phone,"},{"Start":"00:41.150 ","End":"00:46.460","Text":"and in the numerator we have the probability of the intersect."},{"Start":"00:46.460 ","End":"00:52.440","Text":"Retired and having a smart phone."},{"Start":"00:52.850 ","End":"00:57.770","Text":"What\u0027s the probability of having a smart phone?"},{"Start":"00:57.770 ","End":"01:01.745","Text":"Well, you can be young and have a smart phone,"},{"Start":"01:01.745 ","End":"01:04.760","Text":"you can be an adult and have a smart phone,"},{"Start":"01:04.760 ","End":"01:08.850","Text":"or you can be retired and have a smart phone."},{"Start":"01:08.850 ","End":"01:12.555","Text":"Let\u0027s write this probability down."},{"Start":"01:12.555 ","End":"01:17.510","Text":"That\u0027ll be the sum of the probabilities of each of the paths."},{"Start":"01:17.510 ","End":"01:19.115","Text":"Let\u0027s take this path."},{"Start":"01:19.115 ","End":"01:23.225","Text":"That\u0027ll be 0.1 times 0.9."},{"Start":"01:23.225 ","End":"01:26.320","Text":"That\u0027s being young and having a smart phone."},{"Start":"01:26.320 ","End":"01:30.525","Text":"The probability of being an adult and having a smart phone,"},{"Start":"01:30.525 ","End":"01:38.780","Text":"that\u0027s 0.7 times 0.7 plus being retired and having a smart phone,"},{"Start":"01:38.780 ","End":"01:42.125","Text":"that\u0027s 0.2 times 0.3."},{"Start":"01:42.125 ","End":"01:45.500","Text":"Now, what about the probability in the numerator?"},{"Start":"01:45.500 ","End":"01:48.515","Text":"That\u0027s being retired and having a smart phone."},{"Start":"01:48.515 ","End":"01:56.645","Text":"That means that we have this path to calculate."},{"Start":"01:56.645 ","End":"01:58.370","Text":"Now, what\u0027s this path?"},{"Start":"01:58.370 ","End":"02:00.860","Text":"Again, retired having a smart phone."},{"Start":"02:00.860 ","End":"02:06.250","Text":"That\u0027s 0.2 times 0.3."},{"Start":"02:06.250 ","End":"02:18.330","Text":"This comes out to 0.09375."}],"ID":12913},{"Watched":false,"Name":"Exercise 4 - Part c","Duration":"2m 26s","ChapterTopicVideoID":12435,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.035","Text":"In this section, we\u0027re asked if a customer does not have a smart phone,"},{"Start":"00:04.035 ","End":"00:07.320","Text":"what\u0027s the probability that he\u0027s not young?"},{"Start":"00:07.320 ","End":"00:10.365","Text":"Again, this is a conditional probability."},{"Start":"00:10.365 ","End":"00:12.420","Text":"We\u0027re asked for the probability."},{"Start":"00:12.420 ","End":"00:14.235","Text":"Now, what\u0027s given to us?"},{"Start":"00:14.235 ","End":"00:17.460","Text":"The customer does not have a smart phone."},{"Start":"00:17.460 ","End":"00:19.710","Text":"What probability are we asked for?"},{"Start":"00:19.710 ","End":"00:26.560","Text":"The probability that that person is not young."},{"Start":"00:26.570 ","End":"00:30.200","Text":"Let\u0027s just write this probability out."},{"Start":"00:30.200 ","End":"00:34.410","Text":"In the denominator we have the probability of not having"},{"Start":"00:34.410 ","End":"00:39.585","Text":"a smart phone and in the numerator we have the probability of the intersect,"},{"Start":"00:39.585 ","End":"00:48.100","Text":"not young and that he does not have a smart phone."},{"Start":"00:48.290 ","End":"00:52.830","Text":"Well, let\u0027s calculate this thing."},{"Start":"00:52.830 ","End":"00:58.370","Text":"In the denominator, the probability that a person does not have a smart phone,"},{"Start":"00:58.370 ","End":"01:00.340","Text":"well, this branch right here,"},{"Start":"01:00.340 ","End":"01:01.560","Text":"this branch right here,"},{"Start":"01:01.560 ","End":"01:03.390","Text":"and this branch right here."},{"Start":"01:03.390 ","End":"01:07.880","Text":"We\u0027re looking at being young and not having a smart phone,"},{"Start":"01:07.880 ","End":"01:10.820","Text":"being an adult and not having a smart phone"},{"Start":"01:10.820 ","End":"01:14.540","Text":"and being retired and not having a smart phone."},{"Start":"01:14.540 ","End":"01:17.060","Text":"The probabilities here, well,"},{"Start":"01:17.060 ","End":"01:22.470","Text":"that\u0027s 0.1 times 0.1,"},{"Start":"01:22.470 ","End":"01:29.820","Text":"young and not having a smart phone plus 0.7 times 0.3,"},{"Start":"01:29.820 ","End":"01:33.110","Text":"that\u0027s adult not having a smart phone plus"},{"Start":"01:33.110 ","End":"01:39.220","Text":"retired and not having a smart phone that\u0027s 0.2 times 0.7."},{"Start":"01:39.220 ","End":"01:41.330","Text":"What about the numerator?"},{"Start":"01:41.330 ","End":"01:46.385","Text":"Not young and not having a smart phone."},{"Start":"01:46.385 ","End":"01:50.260","Text":"Well, not young means that we\u0027re here."},{"Start":"01:50.260 ","End":"01:54.890","Text":"We\u0027re dealing with not young and not having a smart phone,"},{"Start":"01:54.890 ","End":"01:58.400","Text":"so you are an adult and don\u0027t have a smart phone"},{"Start":"01:58.400 ","End":"02:03.105","Text":"and you\u0027re a retired person and not having a smart phone."},{"Start":"02:03.105 ","End":"02:05.120","Text":"Let\u0027s calculate these guys."},{"Start":"02:05.120 ","End":"02:10.925","Text":"Well, adult and not having a smart phone that\u0027s 0.7 times 0.3,"},{"Start":"02:10.925 ","End":"02:17.810","Text":"plus retired not having a smart phone that\u0027s 0.2 times 0.7."},{"Start":"02:17.810 ","End":"02:26.190","Text":"Now, all this comes out to 0.9722."}],"ID":12914},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"2m 48s","ChapterTopicVideoID":12436,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.695","Text":"In this question, we\u0027re told the 3 exams must be passed in order to get a job."},{"Start":"00:04.695 ","End":"00:07.260","Text":"The exams are in stages,"},{"Start":"00:07.260 ","End":"00:09.330","Text":"meaning that if you fail an exam,"},{"Start":"00:09.330 ","End":"00:11.730","Text":"you do not take the next exam."},{"Start":"00:11.730 ","End":"00:15.525","Text":"70 percent of the candidates pass the first exam."},{"Start":"00:15.525 ","End":"00:20.580","Text":"50 percent of those who passed the first exam also pass the second exam,"},{"Start":"00:20.580 ","End":"00:26.425","Text":"and 40 percent of those who pass the second exam also pass the third exam."},{"Start":"00:26.425 ","End":"00:29.810","Text":"We\u0027re asked what\u0027s the probability of getting the job?"},{"Start":"00:29.810 ","End":"00:33.500","Text":"By now I hope you\u0027ve realized that we"},{"Start":"00:33.500 ","End":"00:38.330","Text":"need to use a tree diagram in order to describe the data,"},{"Start":"00:38.330 ","End":"00:40.660","Text":"to describe the probabilities."},{"Start":"00:40.660 ","End":"00:43.490","Text":"Why is that? Again because we have stages,"},{"Start":"00:43.490 ","End":"00:46.325","Text":"and the stages are the first exam,"},{"Start":"00:46.325 ","End":"00:47.590","Text":"then the second exam,"},{"Start":"00:47.590 ","End":"00:48.740","Text":"then the third exam,"},{"Start":"00:48.740 ","End":"00:52.700","Text":"you can\u0027t take the second exam if you don\u0027t pass the first exam,"},{"Start":"00:52.700 ","End":"00:54.350","Text":"and so on and so forth."},{"Start":"00:54.350 ","End":"00:58.810","Text":"Let\u0027s draw our tree diagram here."},{"Start":"00:58.810 ","End":"01:01.140","Text":"This is a first exam."},{"Start":"01:01.140 ","End":"01:08.810","Text":"You pass with a probability of 70 percent and you fail with a probability of 30 percent."},{"Start":"01:08.810 ","End":"01:12.315","Text":"If you fail, then that\u0027s it,"},{"Start":"01:12.315 ","End":"01:14.920","Text":"you don\u0027t pass on to the second exam."},{"Start":"01:14.920 ","End":"01:18.880","Text":"Let\u0027s go to the second stage."},{"Start":"01:18.880 ","End":"01:22.000","Text":"Here\u0027s the second stage and you pass on to"},{"Start":"01:22.000 ","End":"01:25.795","Text":"the second stage only if you pass the first exam."},{"Start":"01:25.795 ","End":"01:27.760","Text":"In the second exam,"},{"Start":"01:27.760 ","End":"01:31.540","Text":"you either pass with a probability of 50 percent"},{"Start":"01:31.540 ","End":"01:35.740","Text":"or you fail with a probability of 50 percent."},{"Start":"01:35.740 ","End":"01:37.420","Text":"Again, if you fail,"},{"Start":"01:37.420 ","End":"01:39.145","Text":"then you don\u0027t continue on."},{"Start":"01:39.145 ","End":"01:43.210","Text":"Let\u0027s go on to the third and final exam."},{"Start":"01:43.210 ","End":"01:45.145","Text":"This is it right here."},{"Start":"01:45.145 ","End":"01:47.740","Text":"If you pass the second exam,"},{"Start":"01:47.740 ","End":"01:55.040","Text":"then the probability of passing the third exam is 40 percent and then you stop or you"},{"Start":"01:55.040 ","End":"02:02.615","Text":"fail with a probability of 0.6 or 60 percent and then again you stop."},{"Start":"02:02.615 ","End":"02:07.763","Text":"Only here when you pass all 3 exams do you get the job,"},{"Start":"02:07.763 ","End":"02:10.115","Text":"and each time that you fail,"},{"Start":"02:10.115 ","End":"02:15.215","Text":"then you don\u0027t go on to the next stage and obviously you do not get the job."},{"Start":"02:15.215 ","End":"02:21.185","Text":"We\u0027re asked what\u0027s the probability of getting the job?"},{"Start":"02:21.185 ","End":"02:28.580","Text":"The probability of getting a job is to pass the first exam and then the second exam,"},{"Start":"02:28.580 ","End":"02:32.630","Text":"and then the third exam only then will you get the job."},{"Start":"02:32.630 ","End":"02:38.280","Text":"The probability of passing that equals to"},{"Start":"02:38.280 ","End":"02:44.648","Text":"0.7 times 0.5 times 0.4,"},{"Start":"02:44.648 ","End":"02:47.800","Text":"and that comes out to 0.14."}],"ID":12915},{"Watched":false,"Name":"Exercise 5 - Part b","Duration":"2m 14s","ChapterTopicVideoID":12437,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this section, we\u0027re given that the candidate did not get the job."},{"Start":"00:04.500 ","End":"00:08.355","Text":"We\u0027re asked what\u0027s the probability that he failed on the first exam?"},{"Start":"00:08.355 ","End":"00:11.490","Text":"We have here conditional probability."},{"Start":"00:11.490 ","End":"00:12.990","Text":"Let\u0027s write it out."},{"Start":"00:12.990 ","End":"00:21.310","Text":"We\u0027re looking for the probability that the candidate failed on the first exam,"},{"Start":"00:21.380 ","End":"00:24.975","Text":"given that he failed."},{"Start":"00:24.975 ","End":"00:27.930","Text":"Now, what does that equal to?"},{"Start":"00:27.930 ","End":"00:33.565","Text":"Well, in the denominator we have the probability of failing."},{"Start":"00:33.565 ","End":"00:39.270","Text":"In the numerator we have the probability of fail on"},{"Start":"00:39.270 ","End":"00:46.369","Text":"the first exam intersect that the candidate failed."},{"Start":"00:46.369 ","End":"00:52.805","Text":"Now, obviously, if he failed on the first exam, then he failed,"},{"Start":"00:52.805 ","End":"00:55.850","Text":"that means that in the numerator,"},{"Start":"00:55.850 ","End":"01:01.190","Text":"the probability of the intersect is actually the probability"},{"Start":"01:01.190 ","End":"01:06.500","Text":"of the candidate failing on the first exam."},{"Start":"01:06.500 ","End":"01:14.900","Text":"That\u0027s the same probability divided by the probability that the candidate failed."},{"Start":"01:14.900 ","End":"01:19.460","Text":"Now, let\u0027s take a look at the denominator."},{"Start":"01:19.460 ","End":"01:23.600","Text":"What\u0027s the probability of the candidate failing?"},{"Start":"01:23.600 ","End":"01:29.720","Text":"Well, that\u0027s 1 minus the probability of the candidate passing."},{"Start":"01:29.720 ","End":"01:33.635","Text":"Now, we\u0027ve calculated that in the last section."},{"Start":"01:33.635 ","End":"01:40.550","Text":"Now, what\u0027s the probability of the candidate failing on the first exam?"},{"Start":"01:40.550 ","End":"01:44.630","Text":"Well, we have that, that\u0027s right here, that\u0027s 0.3."},{"Start":"01:44.630 ","End":"01:51.440","Text":"We have this, that\u0027ll be 0.3 divided by 1 minus."},{"Start":"01:51.440 ","End":"01:53.360","Text":"Now, in the last section,"},{"Start":"01:53.360 ","End":"01:58.100","Text":"the probability of the candidate passing was 0.14."},{"Start":"01:58.100 ","End":"02:00.710","Text":"So this is what we have to calculate now,"},{"Start":"02:00.710 ","End":"02:06.500","Text":"that\u0027ll be 0.3 divided by 0.86."},{"Start":"02:06.500 ","End":"02:12.990","Text":"That equals to 0.3488."}],"ID":12916},{"Watched":false,"Name":"Exercise 5 - Part c","Duration":"2m 13s","ChapterTopicVideoID":12438,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this section, we\u0027re told that the candidate did not get a job,"},{"Start":"00:04.335 ","End":"00:08.980","Text":"and we\u0027re asked what\u0027s the probability that he passed the second exam?"},{"Start":"00:08.980 ","End":"00:11.670","Text":"Again, there\u0027s a conditional probability,"},{"Start":"00:11.670 ","End":"00:13.080","Text":"so let\u0027s write the set."},{"Start":"00:13.080 ","End":"00:15.960","Text":"We\u0027re looking for the probability. Now, what\u0027s given?"},{"Start":"00:15.960 ","End":"00:20.280","Text":"We\u0027re given that he failed and we\u0027re looking for"},{"Start":"00:20.280 ","End":"00:27.840","Text":"the probability of passing second exam."},{"Start":"00:27.840 ","End":"00:35.680","Text":"In the denominator, we have the probability of failing."},{"Start":"00:36.920 ","End":"00:39.570","Text":"In the numerator,"},{"Start":"00:39.570 ","End":"00:42.705","Text":"we have the probability of the intersect,"},{"Start":"00:42.705 ","End":"00:51.170","Text":"passing second exam and failing."},{"Start":"00:51.170 ","End":"00:54.970","Text":"Now, let\u0027s take a look at the numerator."},{"Start":"00:54.970 ","End":"01:02.900","Text":"If the candidate passed the second exam and he failed than he had to pass the 1 exam,"},{"Start":"01:02.900 ","End":"01:05.425","Text":"he had to pass the second exam,"},{"Start":"01:05.425 ","End":"01:08.780","Text":"but fail on the third exam."},{"Start":"01:09.650 ","End":"01:16.555","Text":"That\u0027s the meaning of passing the second exam and intersect, fail."},{"Start":"01:16.555 ","End":"01:20.650","Text":"Here, that equals to the probability of"},{"Start":"01:20.650 ","End":"01:30.680","Text":"failing 3 exam divided by the probability of failing,"},{"Start":"01:30.680 ","End":"01:37.864","Text":"where the probability of failing is 1 minus the probability of passing."},{"Start":"01:37.864 ","End":"01:40.280","Text":"Now, what does that equal to?"},{"Start":"01:40.280 ","End":"01:45.950","Text":"Well, the probability of passing is 0.14."},{"Start":"01:45.950 ","End":"01:51.670","Text":"1 minus 0.14, that\u0027s 0.86."},{"Start":"01:51.670 ","End":"01:56.195","Text":"What about the probability of failing on the 3 exam?"},{"Start":"01:56.195 ","End":"02:00.587","Text":"Well, that\u0027s calculating the probability of this path right here."},{"Start":"02:00.587 ","End":"02:05.960","Text":"That\u0027s 0.7 times 0.5 times 0.6."},{"Start":"02:05.960 ","End":"02:10.770","Text":"Now, all this comes out to 0.2442."}],"ID":12917},{"Watched":false,"Name":"Exercise 6 - Part a","Duration":"5m 37s","ChapterTopicVideoID":12439,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.271","Text":"In this question, we\u0027re told that the Ministry of Health published the following figures"},{"Start":"00:04.271 ","End":"00:09.507","Text":": 80 percent of children and 60 percent of adults get the flu during winter."},{"Start":"00:09.507 ","End":"00:12.300","Text":"Now, 30 percent of the population are children,"},{"Start":"00:12.300 ","End":"00:13.980","Text":"50 percent are adults,"},{"Start":"00:13.980 ","End":"00:16.095","Text":"and the rest are senior citizens."},{"Start":"00:16.095 ","End":"00:21.580","Text":"It\u0027s also given that 68 percent of the population will get the flu during the winter."},{"Start":"00:21.580 ","End":"00:26.925","Text":"We\u0027re asked, what proportion of those who got the flu are senior citizens?"},{"Start":"00:26.925 ","End":"00:34.310","Text":"Again, we have here a tree diagram that we have to create. Why is that?"},{"Start":"00:34.310 ","End":"00:37.845","Text":"Well, because we have 2 stages here,"},{"Start":"00:37.845 ","End":"00:39.405","Text":"there are 2 levels."},{"Start":"00:39.405 ","End":"00:43.110","Text":"The first level is the population,"},{"Start":"00:43.110 ","End":"00:45.000","Text":"it\u0027s the age group,"},{"Start":"00:45.000 ","End":"00:47.945","Text":"30 percent of the population are children,"},{"Start":"00:47.945 ","End":"00:49.400","Text":"50 percent are adults,"},{"Start":"00:49.400 ","End":"00:51.110","Text":"and the rest are senior citizens."},{"Start":"00:51.110 ","End":"00:53.170","Text":"This would be the first level."},{"Start":"00:53.170 ","End":"01:00.200","Text":"The second level would be whether each of the categories,"},{"Start":"01:00.200 ","End":"01:01.594","Text":"the age categories,"},{"Start":"01:01.594 ","End":"01:04.070","Text":"well, if they have the flu or not."},{"Start":"01:04.070 ","End":"01:06.845","Text":"This would be the second level."},{"Start":"01:06.845 ","End":"01:11.935","Text":"Let\u0027s just draw our tree diagram here."},{"Start":"01:11.935 ","End":"01:16.280","Text":"Here we have the first level right here."},{"Start":"01:16.280 ","End":"01:18.170","Text":"We have children here,"},{"Start":"01:18.170 ","End":"01:20.765","Text":"we have adults here, and seniors here."},{"Start":"01:20.765 ","End":"01:25.310","Text":"Now, we\u0027re told that 30 percent of the population are children."},{"Start":"01:25.310 ","End":"01:27.860","Text":"That\u0027s 0.3 for children,"},{"Start":"01:27.860 ","End":"01:30.980","Text":"0.5 or 50 percent are adults."},{"Start":"01:30.980 ","End":"01:34.850","Text":"So that means that 20 percent are senior citizens."},{"Start":"01:34.850 ","End":"01:37.925","Text":"Now, let\u0027s go to the second level."},{"Start":"01:37.925 ","End":"01:44.085","Text":"Second level ask whether each 1 of the age groups has a flu or not."},{"Start":"01:44.085 ","End":"01:47.675","Text":"Well, what\u0027s the probability of a child having the flu?"},{"Start":"01:47.675 ","End":"01:51.030","Text":"Well, that\u0027s 80 percent, 0.8,"},{"Start":"01:51.030 ","End":"01:56.365","Text":"so that means that 20 percent of children do not get the flu. What about adults?"},{"Start":"01:56.365 ","End":"02:00.435","Text":"60 percent of adults get the flow, so that\u0027s right here."},{"Start":"02:00.435 ","End":"02:02.250","Text":"That means that 0.4,"},{"Start":"02:02.250 ","End":"02:05.285","Text":"or 40 percent do not get the flu."},{"Start":"02:05.285 ","End":"02:13.145","Text":"But we\u0027re not told about the probability of seniors getting flu or not."},{"Start":"02:13.145 ","End":"02:17.720","Text":"Let\u0027s write down the letter p here."},{"Start":"02:17.720 ","End":"02:20.390","Text":"That\u0027s the variable p, and that\u0027ll represent"},{"Start":"02:20.390 ","End":"02:25.085","Text":"the probability of a senior citizen getting the flu."},{"Start":"02:25.085 ","End":"02:27.110","Text":"Now, why do I want to do that?"},{"Start":"02:27.110 ","End":"02:33.580","Text":"Because there\u0027s another piece of information that I haven\u0027t used yet, and this is it."},{"Start":"02:33.580 ","End":"02:40.685","Text":"It\u0027s given that 68 percent of the population will get the flu during the winter."},{"Start":"02:40.685 ","End":"02:45.110","Text":"We\u0027re told that the probability of getting the flu,"},{"Start":"02:45.110 ","End":"02:48.665","Text":"that equals to 0.68."},{"Start":"02:48.665 ","End":"02:54.005","Text":"Now, how do we calculate the probability of getting a flu?"},{"Start":"02:54.005 ","End":"02:59.090","Text":"Well, we have to go to each of the branches where we have a flu here."},{"Start":"02:59.090 ","End":"03:00.655","Text":"We have to go here,"},{"Start":"03:00.655 ","End":"03:02.210","Text":"we have to go here,"},{"Start":"03:02.210 ","End":"03:03.590","Text":"and we have to go here."},{"Start":"03:03.590 ","End":"03:06.080","Text":"What would be the paths?"},{"Start":"03:06.080 ","End":"03:09.605","Text":"I can be a child and have the flu,"},{"Start":"03:09.605 ","End":"03:12.500","Text":"I can be an adult and have the flu,"},{"Start":"03:12.500 ","End":"03:16.280","Text":"and I can be a senior citizen and have the flu."},{"Start":"03:16.280 ","End":"03:19.820","Text":"Now, how do I calculate this probability?"},{"Start":"03:19.820 ","End":"03:22.400","Text":"Well, if I\u0027m a child and I have the flu, well,"},{"Start":"03:22.400 ","End":"03:26.860","Text":"that would be 0.3 times 0.8."},{"Start":"03:26.860 ","End":"03:29.015","Text":"That\u0027s this thing right here."},{"Start":"03:29.015 ","End":"03:33.470","Text":"I have to add to that the probability of an adult getting the flu."},{"Start":"03:33.470 ","End":"03:38.710","Text":"So that would be 0.5 times 0.6."},{"Start":"03:38.710 ","End":"03:41.030","Text":"I have to add onto that,"},{"Start":"03:41.030 ","End":"03:43.415","Text":"this probability right here."},{"Start":"03:43.415 ","End":"03:47.195","Text":"I have to be a senior citizen, that\u0027s 0.2,"},{"Start":"03:47.195 ","End":"03:49.640","Text":"times the probability of getting a flu,"},{"Start":"03:49.640 ","End":"03:52.715","Text":"but we\u0027ve called that p. Now,"},{"Start":"03:52.715 ","End":"03:58.900","Text":"I know that all this has to be equal to 0.68,"},{"Start":"03:58.900 ","End":"04:04.505","Text":"because that\u0027s the total probability of getting the flu."},{"Start":"04:04.505 ","End":"04:08.120","Text":"Well then this is an equation with 1 variable,"},{"Start":"04:08.120 ","End":"04:09.980","Text":"so let\u0027s just solve this."},{"Start":"04:09.980 ","End":"04:15.030","Text":"0.3 times 0.8, that\u0027s 0.24,"},{"Start":"04:15.030 ","End":"04:20.085","Text":"plus 0.5 times 0.6, that\u0027s 0.3,"},{"Start":"04:20.085 ","End":"04:22.718","Text":"plus 0.2 times p,"},{"Start":"04:22.718 ","End":"04:26.990","Text":"and that has to equal to 0.68."},{"Start":"04:26.990 ","End":"04:32.480","Text":"Let\u0027s just move things around, 0.2p,"},{"Start":"04:32.480 ","End":"04:37.475","Text":"that equals to 0.68"},{"Start":"04:37.475 ","End":"04:46.235","Text":"minus 0.24 plus 0.3."},{"Start":"04:46.235 ","End":"04:48.305","Text":"Now, what is that equal to?"},{"Start":"04:48.305 ","End":"04:54.575","Text":"Well, that equals to 0.14."},{"Start":"04:54.575 ","End":"04:59.450","Text":"So 0.2p equals to 0.14."},{"Start":"04:59.450 ","End":"05:00.815","Text":"That means that p,"},{"Start":"05:00.815 ","End":"05:02.645","Text":"let\u0027s divide by 0.2,"},{"Start":"05:02.645 ","End":"05:05.090","Text":"that equals to 0.7."},{"Start":"05:05.090 ","End":"05:12.830","Text":"That\u0027s the probability of a senior citizen getting the flu."},{"Start":"05:12.830 ","End":"05:19.460","Text":"Now, this is a probability and we wanted this in percentage,"},{"Start":"05:19.460 ","End":"05:22.795","Text":"so that\u0027ll be 70 percent."},{"Start":"05:22.795 ","End":"05:26.150","Text":"Now again, that equals this thing right here,"},{"Start":"05:26.150 ","End":"05:29.960","Text":"that equals to the probability of getting"},{"Start":"05:29.960 ","End":"05:37.290","Text":"the flu given that you\u0027re a senior citizen."}],"ID":12918},{"Watched":false,"Name":"Exercise 6 - Part b","Duration":"2m 9s","ChapterTopicVideoID":12440,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.190","Text":"In this section we\u0027re told that a person without flu is selected."},{"Start":"00:04.190 ","End":"00:09.345","Text":"We\u0027re asked what\u0027s the probability that this person is not a senior citizen?"},{"Start":"00:09.345 ","End":"00:12.750","Text":"Again, we have here a conditional probability."},{"Start":"00:12.750 ","End":"00:14.880","Text":"We\u0027re looking for the probability of what?"},{"Start":"00:14.880 ","End":"00:17.040","Text":"Well, first of all, what\u0027s given."},{"Start":"00:17.040 ","End":"00:19.640","Text":"A person without flu is selected,"},{"Start":"00:19.640 ","End":"00:22.415","Text":"so we have no flu."},{"Start":"00:22.415 ","End":"00:28.249","Text":"We\u0027re asked what\u0027s the probability that this person is not a senior citizen?"},{"Start":"00:28.249 ","End":"00:32.415","Text":"Not a senior citizen."},{"Start":"00:32.415 ","End":"00:37.625","Text":"This then is the conditional probability that equals to the following."},{"Start":"00:37.625 ","End":"00:39.275","Text":"In the denominator,"},{"Start":"00:39.275 ","End":"00:45.550","Text":"we have here the probability of not having the flu,"},{"Start":"00:45.680 ","End":"00:48.860","Text":"and in the numerator,"},{"Start":"00:48.860 ","End":"00:59.500","Text":"we have the probability of not being a senior and not having the flu."},{"Start":"00:59.720 ","End":"01:03.575","Text":"Well, let\u0027s take a look at what\u0027s going on here."},{"Start":"01:03.575 ","End":"01:06.920","Text":"In the denominator, what do we have?"},{"Start":"01:06.920 ","End":"01:12.200","Text":"Well, we know what the probability of having a flu is, we\u0027re given that."},{"Start":"01:12.200 ","End":"01:15.485","Text":"We know that the probability of a flu,"},{"Start":"01:15.485 ","End":"01:19.190","Text":"but that equals to 0.68 that was given to us."},{"Start":"01:19.190 ","End":"01:22.355","Text":"The probability of not having a flu,"},{"Start":"01:22.355 ","End":"01:26.180","Text":"that\u0027s 1 minus 0.68."},{"Start":"01:26.180 ","End":"01:33.440","Text":"Now, what about the probability of not being a senior and not having the flu?"},{"Start":"01:33.440 ","End":"01:34.835","Text":"Well, if you\u0027re not a senior,"},{"Start":"01:34.835 ","End":"01:40.850","Text":"then you\u0027re an adult and you don\u0027t have the flu or you\u0027re child,"},{"Start":"01:40.850 ","End":"01:43.160","Text":"and you don\u0027t have the flu."},{"Start":"01:43.160 ","End":"01:48.050","Text":"Let\u0027s calculate these probabilities right here."},{"Start":"01:48.050 ","End":"01:51.620","Text":"Well, being a child and not having the flu,"},{"Start":"01:51.620 ","End":"01:54.935","Text":"that\u0027s 0.3 times 0.2,"},{"Start":"01:54.935 ","End":"01:58.430","Text":"plus being an adult and not having the flu,"},{"Start":"01:58.430 ","End":"02:02.495","Text":"that\u0027s 0.5 times 0.4."},{"Start":"02:02.495 ","End":"02:09.450","Text":"Now, that comes out to 0.8125."}],"ID":12919},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"4m 52s","ChapterTopicVideoID":12441,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this question, we\u0027ll be doing a little bit of ship detection."},{"Start":"00:03.450 ","End":"00:09.930","Text":"A radar station on the beach must detect the ship located in one of the 4 regions, A,"},{"Start":"00:09.930 ","End":"00:14.160","Text":"B, C, and D. If the ship is in region A,"},{"Start":"00:14.160 ","End":"00:17.580","Text":"the chances that the radar would detect it are 0.8."},{"Start":"00:17.580 ","End":"00:22.595","Text":"This chance drops by 0.1 every time the ship advances to a new region,"},{"Start":"00:22.595 ","End":"00:31.145","Text":"from A to B to C to D. It\u0027s given that the ship has a 0.5 chance of being in region D,"},{"Start":"00:31.145 ","End":"00:34.115","Text":"a 0.3 chance of being in region C,"},{"Start":"00:34.115 ","End":"00:36.515","Text":"a 0.2 chance of being in region B,"},{"Start":"00:36.515 ","End":"00:39.480","Text":"and is otherwise in region A."},{"Start":"00:40.180 ","End":"00:42.620","Text":"Now for these types of questions,"},{"Start":"00:42.620 ","End":"00:47.510","Text":"a probability tree is the best technique to display the data. Why is that?"},{"Start":"00:47.510 ","End":"00:50.330","Text":"Well, the data is displayed in 2 levels,"},{"Start":"00:50.330 ","End":"00:52.490","Text":"region, and detection."},{"Start":"00:52.490 ","End":"00:55.040","Text":"Also, the probabilities that"},{"Start":"00:55.040 ","End":"01:00.140","Text":"the detection level are conditional on the region where the ship is in."},{"Start":"01:00.140 ","End":"01:04.230","Text":"So let\u0027s draw our probability tree."},{"Start":"01:04.870 ","End":"01:10.025","Text":"Here it is. Now, what do we see?"},{"Start":"01:10.025 ","End":"01:13.625","Text":"Well, the first thing that we see, are the regions."},{"Start":"01:13.625 ","End":"01:15.455","Text":"That\u0027s region A right here,"},{"Start":"01:15.455 ","End":"01:19.235","Text":"that\u0027s region B, that\u0027s region C,"},{"Start":"01:19.235 ","End":"01:26.885","Text":"and that\u0027s D. We also see what the probabilities are for each region."},{"Start":"01:26.885 ","End":"01:30.890","Text":"The ship has a 50 percent chance, excuse me,"},{"Start":"01:30.890 ","End":"01:32.645","Text":"of being in region D,"},{"Start":"01:32.645 ","End":"01:35.405","Text":"it\u0027s given right here, and there it is."},{"Start":"01:35.405 ","End":"01:38.630","Text":"A 30 percent chance of being in region C,"},{"Start":"01:38.630 ","End":"01:42.180","Text":"a 20 percent chance of being in region B."},{"Start":"01:42.820 ","End":"01:49.130","Text":"Now, if we sum the probabilities of a ship being in region B, C,"},{"Start":"01:49.130 ","End":"01:52.475","Text":"and D. We see that it comes to 1,"},{"Start":"01:52.475 ","End":"01:59.835","Text":"the implication is that the ship will never be in region A."},{"Start":"01:59.835 ","End":"02:04.740","Text":"Another thing that we can see or that\u0027s given to us is"},{"Start":"02:04.740 ","End":"02:09.180","Text":"that the chances that the radar wouldn\u0027t detect it,"},{"Start":"02:09.180 ","End":"02:14.170","Text":"are 0.8 if the ship is in region A."},{"Start":"02:14.300 ","End":"02:22.610","Text":"This chance drops by 0.1 every time the ship advances to a new region."},{"Start":"02:22.610 ","End":"02:29.720","Text":"That means that if the detection is 0.8 in region A,"},{"Start":"02:29.720 ","End":"02:35.670","Text":"that means that the detection in B is 0.7,"},{"Start":"02:35.980 ","End":"02:39.920","Text":"and it\u0027s 0.6 in C,"},{"Start":"02:39.920 ","End":"02:45.440","Text":"and it\u0027s 0.5 in D. Also,"},{"Start":"02:45.440 ","End":"02:48.785","Text":"because we know the probability of detection,"},{"Start":"02:48.785 ","End":"02:53.015","Text":"then we know what the probability of not detection."},{"Start":"02:53.015 ","End":"02:59.510","Text":"Well, let\u0027s look at region D. If the probability of detection is 0.5,"},{"Start":"02:59.510 ","End":"03:04.040","Text":"then the probability of not being detected is also 0.5."},{"Start":"03:04.040 ","End":"03:08.780","Text":"In region C, if the probability of detection is 0.6,"},{"Start":"03:08.780 ","End":"03:12.630","Text":"then the probability of not detection is 0.4."},{"Start":"03:12.630 ","End":"03:16.320","Text":"Likewise, it\u0027ll be 0.3 in region B."},{"Start":"03:16.320 ","End":"03:18.745","Text":"It would be 0.2 in region A."},{"Start":"03:18.745 ","End":"03:20.360","Text":"But as we said,"},{"Start":"03:20.360 ","End":"03:26.240","Text":"we\u0027ll never get to it because the probability of a ship being in region A is 0."},{"Start":"03:26.240 ","End":"03:31.355","Text":"Well, now that we\u0027ve mapped out all the probabilities and the probability tree,"},{"Start":"03:31.355 ","End":"03:33.370","Text":"we can start answering questions."},{"Start":"03:33.370 ","End":"03:36.725","Text":"The first question in section A is,"},{"Start":"03:36.725 ","End":"03:40.100","Text":"what are the chances that the radar will detect the ship?"},{"Start":"03:40.100 ","End":"03:42.110","Text":"We\u0027re looking for the chances,"},{"Start":"03:42.110 ","End":"03:45.690","Text":"the probability of detection."},{"Start":"03:46.750 ","End":"03:50.975","Text":"Well, that means that we have to add up"},{"Start":"03:50.975 ","End":"03:56.120","Text":"all the probabilities of detection for each region that the ship is in."},{"Start":"03:56.120 ","End":"04:00.380","Text":"Now, we know that the ship will never be in section A,"},{"Start":"04:00.380 ","End":"04:04.940","Text":"so we can ignore that the probability of ship being in section A is 0."},{"Start":"04:04.940 ","End":"04:07.100","Text":"So let\u0027s start."},{"Start":"04:07.100 ","End":"04:11.915","Text":"The probability that the ship will be detected in section B, well,"},{"Start":"04:11.915 ","End":"04:18.830","Text":"that\u0027s 0.2 times 0.7, 0.2 times 0.7."},{"Start":"04:18.830 ","End":"04:24.320","Text":"The probability that the ship would be"},{"Start":"04:24.320 ","End":"04:32.850","Text":"detected in area C is 0.3 times 0.6, 0.3 times 0.6."},{"Start":"04:32.850 ","End":"04:41.280","Text":"The probability that the ship will be detected in area D is 0.5 times 0.5."},{"Start":"04:43.090 ","End":"04:47.525","Text":"That equals to 0.57."},{"Start":"04:47.525 ","End":"04:52.650","Text":"That\u0027s the probability that the ship will be detected by the radar."}],"ID":12920},{"Watched":false,"Name":"Exercise 7 - Parts b-c","Duration":"3m 42s","ChapterTopicVideoID":12442,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Section B asks, if the radar detects the ship,"},{"Start":"00:03.600 ","End":"00:07.410","Text":"what\u0027s the probability that it\u0027s in region C?"},{"Start":"00:07.410 ","End":"00:14.400","Text":"Well, here we have to deal with the conditional probability. How do we know that?"},{"Start":"00:14.400 ","End":"00:16.140","Text":"The question starts out,"},{"Start":"00:16.140 ","End":"00:18.090","Text":"if the radar detects the ship,"},{"Start":"00:18.090 ","End":"00:20.280","Text":"given that the radar detects the ship,"},{"Start":"00:20.280 ","End":"00:23.355","Text":"then what\u0027s the probability that it\u0027s in region C?"},{"Start":"00:23.355 ","End":"00:28.710","Text":"But in essence, we\u0027re looking for the probability that the ship is in"},{"Start":"00:28.710 ","End":"00:35.260","Text":"C given that the radar detects it."},{"Start":"00:38.720 ","End":"00:47.240","Text":"That equals to the probability that the ship is in region C"},{"Start":"00:47.240 ","End":"00:49.820","Text":"and that it was"},{"Start":"00:49.820 ","End":"00:59.490","Text":"detected divided by the probability that the ship was detected."},{"Start":"00:59.620 ","End":"01:04.765","Text":"Now we know from section A,"},{"Start":"01:04.765 ","End":"01:10.400","Text":"what\u0027s the probability of the ship being detected? That\u0027s 0.57."},{"Start":"01:10.400 ","End":"01:12.515","Text":"We\u0027ve calculated that."},{"Start":"01:12.515 ","End":"01:14.675","Text":"Let\u0027s look at the numerator."},{"Start":"01:14.675 ","End":"01:21.830","Text":"What\u0027s the probability that the ship is in region C and that it was detected."},{"Start":"01:21.830 ","End":"01:23.915","Text":"That\u0027s this branch right here,"},{"Start":"01:23.915 ","End":"01:27.350","Text":"the ship in C and that it was detected,"},{"Start":"01:27.350 ","End":"01:32.530","Text":"that\u0027s 0.3 times 0.6."},{"Start":"01:32.530 ","End":"01:38.540","Text":"That\u0027s 0.18 divided by 0.57,"},{"Start":"01:38.540 ","End":"01:47.099","Text":"and that equals to 0.3158."},{"Start":"01:47.099 ","End":"01:51.455","Text":"Section C asks, if the radar detects the ship,"},{"Start":"01:51.455 ","End":"01:55.010","Text":"what are the chances that it\u0027s not in region B?"},{"Start":"01:55.010 ","End":"01:58.675","Text":"Well, again, here we have a conditional probability."},{"Start":"01:58.675 ","End":"02:03.110","Text":"Basically, it asks if the radar detects the ship,"},{"Start":"02:03.110 ","End":"02:06.240","Text":"given that the radar detects the ship."},{"Start":"02:06.340 ","End":"02:12.620","Text":"We have the probability of a ship"},{"Start":"02:12.620 ","End":"02:20.529","Text":"not in B given that it\u0027s detected."},{"Start":"02:20.680 ","End":"02:23.165","Text":"This is what we have to find out."},{"Start":"02:23.165 ","End":"02:31.645","Text":"Well, that\u0027s the probability of ship not being in B"},{"Start":"02:31.645 ","End":"02:34.500","Text":"intersect the fact that it was"},{"Start":"02:34.500 ","End":"02:43.500","Text":"detected divided by the probability of a ship being detected."},{"Start":"02:43.810 ","End":"02:51.920","Text":"Now again, we\u0027ve calculated the probability of a ship being detected, that\u0027s 0.57."},{"Start":"02:51.920 ","End":"02:53.600","Text":"That was in section A."},{"Start":"02:53.600 ","End":"03:00.005","Text":"Now what\u0027s the probability of a ship not being in B and being detected?"},{"Start":"03:00.005 ","End":"03:03.950","Text":"What that means is that the ship was either in C and"},{"Start":"03:03.950 ","End":"03:08.645","Text":"was detected or in D and it was detected."},{"Start":"03:08.645 ","End":"03:19.160","Text":"That means that the probability of the numerator is 0.3 times 0.6,"},{"Start":"03:19.160 ","End":"03:22.660","Text":"that\u0027s the probability of a ship being detected in C,"},{"Start":"03:22.660 ","End":"03:26.120","Text":"plus the probability of a ship being detected in D,"},{"Start":"03:26.120 ","End":"03:30.930","Text":"that\u0027s 0.5 times 0.5,"},{"Start":"03:31.480 ","End":"03:43.570","Text":"and that turns out to be 0.7543."}],"ID":12921},{"Watched":false,"Name":"Exercise 8 - Part a","Duration":"6m 50s","ChapterTopicVideoID":12443,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.655","Text":"This question deals with diseases,"},{"Start":"00:02.655 ","End":"00:06.300","Text":"their symptom, and their detection."},{"Start":"00:06.300 ","End":"00:10.155","Text":"We are given that 40 percent of those with disease A,"},{"Start":"00:10.155 ","End":"00:12.420","Text":"60 percent of those with disease B,"},{"Start":"00:12.420 ","End":"00:16.905","Text":"and 50 percent of those with disease C develop symptom X."},{"Start":"00:16.905 ","End":"00:20.250","Text":"Symptom X appears only with these diseases."},{"Start":"00:20.250 ","End":"00:23.565","Text":"A person can\u0027t have more than 1 of these diseases."},{"Start":"00:23.565 ","End":"00:27.350","Text":"The people come into the clinic as follows,"},{"Start":"00:27.350 ","End":"00:30.465","Text":"8 percent of those that come into the clinic have disease A,"},{"Start":"00:30.465 ","End":"00:32.535","Text":"10 percent have disease B,"},{"Start":"00:32.535 ","End":"00:34.575","Text":"and 2 percent have disease C,"},{"Start":"00:34.575 ","End":"00:36.495","Text":"and the rest are healthy."},{"Start":"00:36.495 ","End":"00:43.580","Text":"The chances that symptom X would be detected is 80 percent in a person with disease A and"},{"Start":"00:43.580 ","End":"00:50.330","Text":"90 percent with people with diseases B and C. When approaching this problem,"},{"Start":"00:50.330 ","End":"00:53.469","Text":"how do I know to use the probability tree?"},{"Start":"00:53.469 ","End":"00:57.270","Text":"There are 2 things that I have to look out for."},{"Start":"00:57.270 ","End":"01:01.510","Text":"1 is I see that there are levels and 2,"},{"Start":"01:01.510 ","End":"01:07.505","Text":"I see that the information at each level is conditional on the level above it,"},{"Start":"01:07.505 ","End":"01:09.695","Text":"except for the first level, obviously."},{"Start":"01:09.695 ","End":"01:11.570","Text":"What is the first level?"},{"Start":"01:11.570 ","End":"01:20.435","Text":"The first level is the level that is not conditional on any of the information here."},{"Start":"01:20.435 ","End":"01:25.550","Text":"That would be the people that are entering the clinic."},{"Start":"01:25.550 ","End":"01:29.295","Text":"What\u0027s the next level?"},{"Start":"01:29.295 ","End":"01:32.630","Text":"If we say that the people who have"},{"Start":"01:32.630 ","End":"01:37.685","Text":"the disease is the first level, what\u0027s the next level?"},{"Start":"01:37.685 ","End":"01:41.120","Text":"What\u0027s conditional on the disease?"},{"Start":"01:41.120 ","End":"01:42.995","Text":"That\u0027s the symptoms."},{"Start":"01:42.995 ","End":"01:47.960","Text":"We know that 40 percent of the people with disease A have symptom X."},{"Start":"01:47.960 ","End":"01:50.750","Text":"Symptom X is conditional on the disease,"},{"Start":"01:50.750 ","End":"01:53.275","Text":"therefore, it\u0027s the next level down."},{"Start":"01:53.275 ","End":"01:58.355","Text":"What\u0027s the third level? The third level is the level of detection."},{"Start":"01:58.355 ","End":"02:02.265","Text":"That\u0027s conditional on the symptoms."},{"Start":"02:02.265 ","End":"02:05.000","Text":"A person having symptom X,"},{"Start":"02:05.000 ","End":"02:08.690","Text":"we want to know whether the symptom is detected or not."},{"Start":"02:08.690 ","End":"02:11.495","Text":"If that\u0027s the case,"},{"Start":"02:11.495 ","End":"02:14.970","Text":"then let\u0027s draw the probability tree."},{"Start":"02:15.040 ","End":"02:18.350","Text":"Here we have the probability tree with"},{"Start":"02:18.350 ","End":"02:22.115","Text":"the information that\u0027s been given to us in the problem."},{"Start":"02:22.115 ","End":"02:24.305","Text":"Let\u0027s look at the structure of the tree."},{"Start":"02:24.305 ","End":"02:27.155","Text":"First of all, we see that it has 3 levels."},{"Start":"02:27.155 ","End":"02:30.590","Text":"The first level is that of the disease."},{"Start":"02:30.590 ","End":"02:32.810","Text":"The second level that of the symptom,"},{"Start":"02:32.810 ","End":"02:36.365","Text":"and the third level that of the detection."},{"Start":"02:36.365 ","End":"02:44.130","Text":"We\u0027re also given that 8 percent of the people have disease A. That\u0027s right here."},{"Start":"02:44.260 ","End":"02:46.730","Text":"10 percent have disease B,"},{"Start":"02:46.730 ","End":"02:51.455","Text":"that\u0027s right here and 2 percent have disease C, right here."},{"Start":"02:51.455 ","End":"02:56.030","Text":"What\u0027s the probability of healthy people coming into the clinic?"},{"Start":"02:56.030 ","End":"03:00.680","Text":"Well, If the probability of people coming to"},{"Start":"03:00.680 ","End":"03:05.695","Text":"the clinic is 8 percent plus 1 percent plus 2 percent,"},{"Start":"03:05.695 ","End":"03:08.250","Text":"that comes out to 20 percent,"},{"Start":"03:08.250 ","End":"03:13.430","Text":"so that means probability of healthy people coming into the clinic is 80 percent."},{"Start":"03:13.430 ","End":"03:15.995","Text":"Let\u0027s look at the next level."},{"Start":"03:15.995 ","End":"03:21.230","Text":"Next level is that of the symptoms of the people being sick with"},{"Start":"03:21.230 ","End":"03:27.785","Text":"disease A 40 percent of them have contracted symptom X."},{"Start":"03:27.785 ","End":"03:32.030","Text":"That\u0027s right here. We know that because it\u0027s right here."},{"Start":"03:32.030 ","End":"03:36.710","Text":"60 percent have symptoms X if they\u0027re sick with disease B,"},{"Start":"03:36.710 ","End":"03:40.740","Text":"and 50 percent have symptom X if they\u0027re sick with"},{"Start":"03:40.740 ","End":"03:46.050","Text":"disease C. Let\u0027s just complete the probabilities here."},{"Start":"03:46.720 ","End":"03:50.480","Text":"That\u0027s 60 percent,"},{"Start":"03:50.480 ","End":"03:53.150","Text":"that\u0027s 40 percent,"},{"Start":"03:53.150 ","End":"03:56.645","Text":"and that\u0027s 50 percent."},{"Start":"03:56.645 ","End":"04:00.350","Text":"Let\u0027s go down then to the next level."},{"Start":"04:00.350 ","End":"04:05.405","Text":"Next level deals with the symptom being detected or not."},{"Start":"04:05.405 ","End":"04:08.990","Text":"Out of the 40 percent of the people that"},{"Start":"04:08.990 ","End":"04:13.670","Text":"have symptom X when they\u0027re sick with disease A,"},{"Start":"04:13.670 ","End":"04:16.775","Text":"then 80 percent of them have been detected."},{"Start":"04:16.775 ","End":"04:25.115","Text":"Likewise with people with disease B and of those that have contracted symptom X,"},{"Start":"04:25.115 ","End":"04:27.680","Text":"90 percent of those have been detected."},{"Start":"04:27.680 ","End":"04:32.285","Text":"Likewise for C, 90 percent of those also have been detected."},{"Start":"04:32.285 ","End":"04:33.725","Text":"We can see that here,"},{"Start":"04:33.725 ","End":"04:38.855","Text":"80 percent of people with disease A and 90 percent of"},{"Start":"04:38.855 ","End":"04:44.000","Text":"people with disease B and C. Again,"},{"Start":"04:44.000 ","End":"04:46.430","Text":"let\u0027s just complete the probabilities here."},{"Start":"04:46.430 ","End":"04:52.400","Text":"That\u0027s 0.2, 0.1, and 0.1."},{"Start":"04:52.400 ","End":"04:57.695","Text":"Just to make things clear D means that the symptom has been"},{"Start":"04:57.695 ","End":"05:05.680","Text":"detected and ND means that it has been not detected."},{"Start":"05:06.520 ","End":"05:13.220","Text":"Great. Now that we\u0027ve calculated all the probabilities in our probability tree here,"},{"Start":"05:13.220 ","End":"05:15.880","Text":"we can start answering questions."},{"Start":"05:15.880 ","End":"05:18.335","Text":"Section a asks us,"},{"Start":"05:18.335 ","End":"05:24.065","Text":"what\u0027s the probability of symptom X being detected in a person coming to the clinic?"},{"Start":"05:24.065 ","End":"05:32.132","Text":"What\u0027s the probability of x being detected?"},{"Start":"05:32.132 ","End":"05:36.140","Text":"That equals to the sum of"},{"Start":"05:36.140 ","End":"05:41.435","Text":"the probabilities of X being detected when a person is sick with disease A,"},{"Start":"05:41.435 ","End":"05:48.035","Text":"with disease B, and with disease C. Let\u0027s calculate what\u0027s the probability"},{"Start":"05:48.035 ","End":"05:55.520","Text":"of each 1 of the branches where X has been detected."},{"Start":"05:55.520 ","End":"05:59.105","Text":"For branch A,"},{"Start":"05:59.105 ","End":"06:06.670","Text":"that\u0027s 0.08 times 0.4 times 0.8."},{"Start":"06:06.670 ","End":"06:11.960","Text":"We have to add the probability for branch B. That\u0027s right here."},{"Start":"06:11.960 ","End":"06:18.995","Text":"That\u0027s 0.1 times 0.6 times 0.9."},{"Start":"06:18.995 ","End":"06:24.890","Text":"We have to add the probabilities for branch C. That\u0027s"},{"Start":"06:24.890 ","End":"06:32.170","Text":"0.02 times 0.5 times 0.9."},{"Start":"06:32.620 ","End":"06:37.020","Text":"That equals 0.0886."},{"Start":"06:38.930 ","End":"06:43.170","Text":"That\u0027s the probability of symptom X being"},{"Start":"06:43.170 ","End":"06:49.620","Text":"detected by people that have diseases that come into the clinic."}],"ID":12922},{"Watched":false,"Name":"Exercise 8 - Part b","Duration":"2m 1s","ChapterTopicVideoID":12444,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In section B, we\u0027re asked,"},{"Start":"00:01.980 ","End":"00:04.650","Text":"if symptom X is detected in a person,"},{"Start":"00:04.650 ","End":"00:08.100","Text":"what\u0027s the probability that he has disease A?"},{"Start":"00:08.100 ","End":"00:13.710","Text":"Well, here we\u0027re dealing with a conditional probability. Why is that?"},{"Start":"00:13.710 ","End":"00:15.945","Text":"Because the question reads,"},{"Start":"00:15.945 ","End":"00:18.405","Text":"if symptom X is detected,"},{"Start":"00:18.405 ","End":"00:21.645","Text":"that means that given symptom X,"},{"Start":"00:21.645 ","End":"00:24.210","Text":"what\u0027s the probability that he has disease A?"},{"Start":"00:24.210 ","End":"00:33.970","Text":"That means we\u0027re dealing with the probability of X being detected."},{"Start":"00:36.140 ","End":"00:40.460","Text":"We know that X is detected and what do we want to find out?"},{"Start":"00:40.460 ","End":"00:48.750","Text":"That X is being detected in a person that\u0027s sick with disease A."},{"Start":"00:48.750 ","End":"00:59.310","Text":"Now, that equals to the probability of a person being sick with A,"},{"Start":"00:59.470 ","End":"01:05.250","Text":"and that the symptom has been detected,"},{"Start":"01:05.830 ","End":"01:13.139","Text":"divided by the probability of X being detected."},{"Start":"01:13.660 ","End":"01:18.110","Text":"Now, what\u0027s that equal to?"},{"Start":"01:18.110 ","End":"01:20.330","Text":"Well, in the denominator,"},{"Start":"01:20.330 ","End":"01:24.320","Text":"we know that that\u0027s 0.0886."},{"Start":"01:24.320 ","End":"01:29.470","Text":"Why is that? We\u0027ve calculated that in section A above."},{"Start":"01:29.470 ","End":"01:34.280","Text":"What\u0027s the probability of a person being sick with"},{"Start":"01:34.280 ","End":"01:38.480","Text":"disease A and that the symptom X has been detected?"},{"Start":"01:38.480 ","End":"01:40.535","Text":"Well, on the tree,"},{"Start":"01:40.535 ","End":"01:42.230","Text":"that\u0027s this branch right here,"},{"Start":"01:42.230 ","End":"01:44.375","Text":"that\u0027s this path right here."},{"Start":"01:44.375 ","End":"01:53.285","Text":"That equals to 0.08 times 0.4 times 0.8."},{"Start":"01:53.285 ","End":"01:57.750","Text":"That equals to 0.2889."}],"ID":12923},{"Watched":false,"Name":"Exercise 8 - Part c","Duration":"2m 51s","ChapterTopicVideoID":12445,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Section C asks, if a person has symptom X,"},{"Start":"00:03.570 ","End":"00:06.795","Text":"what\u0027s the probability that he has disease A?"},{"Start":"00:06.795 ","End":"00:13.170","Text":"Well, this is very similar to what we encountered in section B above."},{"Start":"00:13.170 ","End":"00:19.020","Text":"In section B, we were asked if symptom X is detected in a person,"},{"Start":"00:19.020 ","End":"00:21.480","Text":"what\u0027s the probability that he has diseases A?"},{"Start":"00:21.480 ","End":"00:25.960","Text":"Here, we\u0027re not looking at X being detected,"},{"Start":"00:27.410 ","End":"00:33.135","Text":"we\u0027re looking whether somebody has symptom X."},{"Start":"00:33.135 ","End":"00:35.190","Text":"That\u0027s what we\u0027re looking for."},{"Start":"00:35.190 ","End":"00:38.330","Text":"Now, once we understand that,"},{"Start":"00:38.330 ","End":"00:40.955","Text":"then let\u0027s get back to our question."},{"Start":"00:40.955 ","End":"00:44.615","Text":"We see that this is a conditional probability."},{"Start":"00:44.615 ","End":"00:46.940","Text":"If a person has symptom X,"},{"Start":"00:46.940 ","End":"00:49.805","Text":"given that a person has symptom X."},{"Start":"00:49.805 ","End":"00:52.040","Text":"That means that we can write this down."},{"Start":"00:52.040 ","End":"00:58.890","Text":"The probability given that somebody has X,"},{"Start":"00:58.890 ","End":"01:03.210","Text":"what\u0027s the probability that he also has disease A?"},{"Start":"01:04.360 ","End":"01:10.250","Text":"Now, this is equal to the probability of a person"},{"Start":"01:10.250 ","End":"01:17.410","Text":"having A and contracting the symptom X."},{"Start":"01:17.410 ","End":"01:20.165","Text":"What\u0027s the denominator?"},{"Start":"01:20.165 ","End":"01:24.395","Text":"The denominator is what was given to us,"},{"Start":"01:24.395 ","End":"01:29.045","Text":"that the person has the symptom X,"},{"Start":"01:29.045 ","End":"01:32.795","Text":"the probability of a person having symptoms X."},{"Start":"01:32.795 ","End":"01:36.335","Text":"Now, let\u0027s try to calculate the denominator,"},{"Start":"01:36.335 ","End":"01:40.560","Text":"a person contracting or has symptom X."},{"Start":"01:40.560 ","End":"01:44.240","Text":"Well, we\u0027re going to have to sum up all the people"},{"Start":"01:44.240 ","End":"01:48.335","Text":"that have a disease and that have the symptoms."},{"Start":"01:48.335 ","End":"01:53.105","Text":"That means we have to go to this branch,"},{"Start":"01:53.105 ","End":"01:55.685","Text":"and to this branch, and to this branch."},{"Start":"01:55.685 ","End":"02:01.415","Text":"That means that we have 0.08 times 0.4"},{"Start":"02:01.415 ","End":"02:09.750","Text":"plus 0.1 times 0.6 plus 0.02 times 0.5."},{"Start":"02:09.750 ","End":"02:19.140","Text":"Now, this is a person contracting symptom X if he was sick with disease A,"},{"Start":"02:19.140 ","End":"02:22.270","Text":"and this is if he was sick with disease B,"},{"Start":"02:22.270 ","End":"02:27.145","Text":"and this is if he was sick with disease C. Now,"},{"Start":"02:27.145 ","End":"02:30.920","Text":"what\u0027s the probability of a person having disease A,"},{"Start":"02:30.920 ","End":"02:32.990","Text":"he\u0027s sick with disease A,"},{"Start":"02:32.990 ","End":"02:36.350","Text":"and that he has symptom X?"},{"Start":"02:36.350 ","End":"02:39.020","Text":"Well, that\u0027s just this branch right here."},{"Start":"02:39.020 ","End":"02:44.000","Text":"That\u0027s 0.08 times 0.4,"},{"Start":"02:44.000 ","End":"02:48.600","Text":"and that comes out to 0.3137."}],"ID":12924},{"Watched":false,"Name":"Exercise 8 - Part d","Duration":"2m 25s","ChapterTopicVideoID":12446,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"Section D asks, if symptom X is not detected in a person,"},{"Start":"00:04.680 ","End":"00:08.100","Text":"what\u0027s the probability that he\u0027s healthy?"},{"Start":"00:08.100 ","End":"00:12.540","Text":"This is a conditional probability. Why is that?"},{"Start":"00:12.540 ","End":"00:16.545","Text":"We see here, if symptom X is not detected,"},{"Start":"00:16.545 ","End":"00:20.070","Text":"given that symptom X is not detected."},{"Start":"00:20.070 ","End":"00:21.810","Text":"So we can start writing the probability."},{"Start":"00:21.810 ","End":"00:28.165","Text":"What do we want to find out?"},{"Start":"00:28.165 ","End":"00:32.190","Text":"What\u0027s the probability that a person is healthy?"},{"Start":"00:35.110 ","End":"00:40.340","Text":"What\u0027s given to us? That symptom X is not detected."},{"Start":"00:40.340 ","End":"00:44.820","Text":"X is not detected."},{"Start":"00:45.400 ","End":"00:51.200","Text":"What\u0027s the probability of a person being healthy,"},{"Start":"00:51.200 ","End":"01:01.600","Text":"and that symptom X is not detected,"},{"Start":"01:02.870 ","End":"01:11.620","Text":"divided by the probability that X has not being detected."},{"Start":"01:13.930 ","End":"01:21.375","Text":"In section A, we\u0027ve calculated the probability of X being detected,"},{"Start":"01:21.375 ","End":"01:25.185","Text":"and that was 0.0886."},{"Start":"01:25.185 ","End":"01:28.545","Text":"The probability of X not being detected,"},{"Start":"01:28.545 ","End":"01:30.375","Text":"that\u0027s 1 minus that,"},{"Start":"01:30.375 ","End":"01:35.910","Text":"so we can write in the denominator, 1 minus 0.0886."},{"Start":"01:38.650 ","End":"01:41.330","Text":"What about the numerator?"},{"Start":"01:41.330 ","End":"01:45.650","Text":"The numerator is the probability of a person"},{"Start":"01:45.650 ","End":"01:49.955","Text":"being healthy and that X hasn\u0027t been detected."},{"Start":"01:49.955 ","End":"01:53.555","Text":"Now if he\u0027s healthy, he doesn\u0027t have symptom X at all."},{"Start":"01:53.555 ","End":"02:03.180","Text":"So that means that the numerator is basically the probability of a person being healthy."},{"Start":"02:03.490 ","End":"02:08.410","Text":"So that equals to 0.8,"},{"Start":"02:08.410 ","End":"02:10.590","Text":"that\u0027s this guy right here,"},{"Start":"02:10.590 ","End":"02:12.435","Text":"the person being healthy,"},{"Start":"02:12.435 ","End":"02:18.450","Text":"divided by 1 minus 0.0886,"},{"Start":"02:18.450 ","End":"02:26.890","Text":"and that equals 0.8778."}],"ID":12925},{"Watched":false,"Name":"Exercise 9","Duration":"4m 15s","ChapterTopicVideoID":12447,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.785","Text":"This question talks about guessing at a multiple choice exam."},{"Start":"00:04.785 ","End":"00:07.875","Text":"A student takes a multiple choice exam."},{"Start":"00:07.875 ","End":"00:10.110","Text":"The chances that he knows the answer to"},{"Start":"00:10.110 ","End":"00:13.440","Text":"a given question are p. If he doesn\u0027t know the answer,"},{"Start":"00:13.440 ","End":"00:14.760","Text":"he guesses, so,"},{"Start":"00:14.760 ","End":"00:16.670","Text":"he answers the question in any case."},{"Start":"00:16.670 ","End":"00:22.100","Text":"Now, it\u0027s given that each question has k possible answers, and we\u0027re asked,"},{"Start":"00:22.100 ","End":"00:24.935","Text":"if the student answers the question correctly,"},{"Start":"00:24.935 ","End":"00:28.280","Text":"what\u0027s the probability that he knew the answer?"},{"Start":"00:28.280 ","End":"00:35.690","Text":"Well, first, let\u0027s draw the probability tree. Here it is."},{"Start":"00:35.690 ","End":"00:40.140","Text":"Now, let\u0027s look at the information that\u0027s given to us."},{"Start":"00:41.590 ","End":"00:47.780","Text":"We know that the chance is that he knows the answer to a given question are p. Well,"},{"Start":"00:47.780 ","End":"00:50.735","Text":"if he knows that\u0027s right here. That\u0027s this branch."},{"Start":"00:50.735 ","End":"00:54.380","Text":"Now, if he doesn\u0027t know then obviously it\u0027s 1 minus p,"},{"Start":"00:54.380 ","End":"00:56.465","Text":"and if he doesn\u0027t know, he guesses."},{"Start":"00:56.465 ","End":"00:58.415","Text":"Now if he guesses,"},{"Start":"00:58.415 ","End":"01:03.610","Text":"then he has to guess 1 out of k possible answers, so,"},{"Start":"01:03.610 ","End":"01:09.140","Text":"the probability of him guessing correctly is 1 over k,"},{"Start":"01:09.140 ","End":"01:11.930","Text":"and if he guesses incorrectly,"},{"Start":"01:11.930 ","End":"01:17.615","Text":"then it\u0027s 1 minus 1 over k. Let\u0027s get back to our question."},{"Start":"01:17.615 ","End":"01:20.930","Text":"We\u0027re asked, if the student answers the question correctly,"},{"Start":"01:20.930 ","End":"01:23.435","Text":"what\u0027s the probability that he knew the answer?"},{"Start":"01:23.435 ","End":"01:26.240","Text":"But this is a conditional probability question."},{"Start":"01:26.240 ","End":"01:27.950","Text":"Again, how do we know that?"},{"Start":"01:27.950 ","End":"01:31.760","Text":"Again, the wording. If the student answers the question correctly,"},{"Start":"01:31.760 ","End":"01:34.520","Text":"given that the student answers the question correctly."},{"Start":"01:34.520 ","End":"01:36.305","Text":"So we can write this down."},{"Start":"01:36.305 ","End":"01:43.115","Text":"The probability, and what are we given that the student answers correctly?"},{"Start":"01:43.115 ","End":"01:49.370","Text":"Now, what do we want to know?"},{"Start":"01:49.370 ","End":"01:54.193","Text":"What is the probability that he knew the answer?"},{"Start":"01:54.193 ","End":"01:58.800","Text":"The probability that he knew the answer."},{"Start":"01:58.940 ","End":"02:03.630","Text":"Now, we can write this down as this,"},{"Start":"02:03.630 ","End":"02:07.430","Text":"the probability that the student knew"},{"Start":"02:07.430 ","End":"02:17.460","Text":"the answer and that he answered correctly,"},{"Start":"02:18.820 ","End":"02:22.715","Text":"divided by the probability"},{"Start":"02:22.715 ","End":"02:31.560","Text":"that he answers correctly."},{"Start":"02:31.940 ","End":"02:35.045","Text":"Now, what is that?"},{"Start":"02:35.045 ","End":"02:39.905","Text":"Well, what\u0027s the probability of the student answering correctly?"},{"Start":"02:39.905 ","End":"02:46.985","Text":"Well, it\u0027s either this branch right here or it\u0027s this branch right here."},{"Start":"02:46.985 ","End":"02:51.770","Text":"Either he knows or he doesn\u0027t know but answers correctly."},{"Start":"02:51.770 ","End":"02:55.790","Text":"So it\u0027s p, the branch that he knows,"},{"Start":"02:55.790 ","End":"03:04.775","Text":"plus 1 minus p times 1 over k. He doesn\u0027t know,"},{"Start":"03:04.775 ","End":"03:10.415","Text":"the probability that he doesn\u0027t know times the probability that he guessed correctly."},{"Start":"03:10.415 ","End":"03:14.104","Text":"Now, what\u0027s the probability of the numerator"},{"Start":"03:14.104 ","End":"03:18.350","Text":"that he knew the answer and also answered correctly?"},{"Start":"03:18.350 ","End":"03:21.605","Text":"Well, if he knows,"},{"Start":"03:21.605 ","End":"03:24.515","Text":"then the probability is p,"},{"Start":"03:24.515 ","End":"03:28.580","Text":"because the intersection between knowing the answer and"},{"Start":"03:28.580 ","End":"03:33.720","Text":"answering correctly is the probability of knowing the answer."},{"Start":"03:34.010 ","End":"03:38.285","Text":"We can rewrite this this way."},{"Start":"03:38.285 ","End":"03:43.130","Text":"We can multiply both the numerator and the denominator by k, so,"},{"Start":"03:43.130 ","End":"03:54.000","Text":"we\u0027ll have kp divided by kp plus 1 minus p,"},{"Start":"03:54.650 ","End":"04:01.425","Text":"or we can even make this a little bit simpler where we say,"},{"Start":"04:01.425 ","End":"04:09.510","Text":"kp divided by 1 plus p,"},{"Start":"04:09.510 ","End":"04:12.040","Text":"k minus 1,"},{"Start":"04:12.080 ","End":"04:15.340","Text":"and that\u0027s the probability."}],"ID":12926},{"Watched":false,"Name":"Exercise 10 - Part a","Duration":"5m 29s","ChapterTopicVideoID":12448,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.925","Text":"In this question, we\u0027ll be trying to decide what\u0027s the best strategy to win a tournament."},{"Start":"00:05.925 ","End":"00:10.995","Text":"Now, a person is playing against 2 competitors, Sara and Mike."},{"Start":"00:10.995 ","End":"00:16.665","Text":"The person must play 3 games and can select what order of games he prefers."},{"Start":"00:16.665 ","End":"00:18.705","Text":"You can prefer to play."},{"Start":"00:18.705 ","End":"00:22.080","Text":"First Mike and then Sara, and then Mike again."},{"Start":"00:22.080 ","End":"00:24.615","Text":"That\u0027s Strategy 1, Option 1."},{"Start":"00:24.615 ","End":"00:27.255","Text":"Or you can choose Strategy B,"},{"Start":"00:27.255 ","End":"00:29.775","Text":"Option B, first-play Sara,"},{"Start":"00:29.775 ","End":"00:32.505","Text":"and then play Mike, and then Sara again."},{"Start":"00:32.505 ","End":"00:37.680","Text":"Now, someone has to win each game, there aren\u0027t any draws."},{"Start":"00:37.680 ","End":"00:42.040","Text":"A person wins the tournament only if he wins 2 games in a row."},{"Start":"00:42.040 ","End":"00:43.850","Text":"Now, this is important."},{"Start":"00:43.850 ","End":"00:45.050","Text":"We have to remember this."},{"Start":"00:45.050 ","End":"00:48.960","Text":"He has to win 2 games in a row."},{"Start":"00:49.570 ","End":"00:54.065","Text":"It is given that Mike is a better player than Sara,"},{"Start":"00:54.065 ","End":"00:58.520","Text":"and the question is, what\u0027s the best way for a person to win the tournament?"},{"Start":"00:58.520 ","End":"01:00.185","Text":"What option should he choose?"},{"Start":"01:00.185 ","End":"01:03.700","Text":"What\u0027s the best strategy, A or B?"},{"Start":"01:03.700 ","End":"01:06.260","Text":"Well, let\u0027s get started."},{"Start":"01:06.260 ","End":"01:10.640","Text":"The first thing that we need to do is we need to define our variables."},{"Start":"01:10.640 ","End":"01:19.190","Text":"Let\u0027s define P as the chances of winning"},{"Start":"01:19.190 ","End":"01:25.310","Text":"Mike and let\u0027s define"},{"Start":"01:25.310 ","End":"01:30.610","Text":"t as the chances of winning Sara."},{"Start":"01:32.180 ","End":"01:35.600","Text":"Once we define these variables,"},{"Start":"01:35.600 ","End":"01:40.920","Text":"let\u0027s draw the probability tree that fits the question."},{"Start":"01:40.920 ","End":"01:46.425","Text":"This is a probability tree for Strategy A,"},{"Start":"01:46.425 ","End":"01:48.740","Text":"Option A. What does that mean?"},{"Start":"01:48.740 ","End":"01:52.690","Text":"That means that I first play Mike then Sara,"},{"Start":"01:52.690 ","End":"01:57.810","Text":"then Mike again so the tree goes like this. There are 3 levels."},{"Start":"01:57.810 ","End":"02:01.080","Text":"First, I play Mike and then I play Sara,"},{"Start":"02:01.080 ","End":"02:03.105","Text":"then I play Mike again."},{"Start":"02:03.105 ","End":"02:09.570","Text":"Now, the probability of winning Mike is p we\u0027ve defined that."},{"Start":"02:09.570 ","End":"02:14.840","Text":"So the probability of winning is"},{"Start":"02:14.840 ","End":"02:22.610","Text":"P right here for Mike probability of losing is 1 minus p. That\u0027s right here."},{"Start":"02:22.610 ","End":"02:26.435","Text":"Now, if I went here,"},{"Start":"02:26.435 ","End":"02:28.220","Text":"I have to win another game,"},{"Start":"02:28.220 ","End":"02:30.905","Text":"I have to win 2 games in a row."},{"Start":"02:30.905 ","End":"02:33.665","Text":"Now, if I win the other game,"},{"Start":"02:33.665 ","End":"02:35.450","Text":"then I stop here."},{"Start":"02:35.450 ","End":"02:37.340","Text":"I don\u0027t need to go any further because I\u0027ve won"},{"Start":"02:37.340 ","End":"02:43.640","Text":"the tournament and if I won here and lost here then again,"},{"Start":"02:43.640 ","End":"02:48.755","Text":"I\u0027ve lost the tournament because I don\u0027t have 2 wins in a row."},{"Start":"02:48.755 ","End":"02:53.225","Text":"Let\u0027s look at this branch."},{"Start":"02:53.225 ","End":"02:57.995","Text":"This branch means the first game I\u0027ve lost to Mike."},{"Start":"02:57.995 ","End":"03:02.165","Text":"Now means that in the second game I have to play Sara."},{"Start":"03:02.165 ","End":"03:05.560","Text":"What are the chances of reading Sara?"},{"Start":"03:05.560 ","End":"03:13.610","Text":"That\u0027s right here, that\u0027s t. If I lost to Mike in first game,"},{"Start":"03:13.610 ","End":"03:17.360","Text":"now, I can either win with the probability of"},{"Start":"03:17.360 ","End":"03:22.190","Text":"t or lose with a probability of 1 minus t to Sara."},{"Start":"03:22.190 ","End":"03:31.560","Text":"That\u0027s winning with the probability of t and losing with probability of 1 minus t. Now,"},{"Start":"03:32.800 ","End":"03:36.395","Text":"if I lose to Sara,"},{"Start":"03:36.395 ","End":"03:40.040","Text":"then I stop because I\u0027ve lost the tournament."},{"Start":"03:40.040 ","End":"03:43.655","Text":"I don\u0027t have 2 wins in a row."},{"Start":"03:43.655 ","End":"03:48.380","Text":"Now, if I won the game with Sara,"},{"Start":"03:48.380 ","End":"03:50.870","Text":"I need to win with Mike again,"},{"Start":"03:50.870 ","End":"03:52.925","Text":"the third game, is was Mike."},{"Start":"03:52.925 ","End":"03:57.335","Text":"Again, if I went with Mike, what are the chances?"},{"Start":"03:57.335 ","End":"04:02.290","Text":"The chance is p to win,"},{"Start":"04:02.290 ","End":"04:04.635","Text":"and if I lose,"},{"Start":"04:04.635 ","End":"04:11.750","Text":"that\u0027s 1 minus p. In order to win the tournament,"},{"Start":"04:11.750 ","End":"04:16.816","Text":"then I need to either win like this,"},{"Start":"04:16.816 ","End":"04:18.920","Text":"this is a winning branch,"},{"Start":"04:18.920 ","End":"04:26.525","Text":"or to lose the first game and then when the second and when the third game,"},{"Start":"04:26.525 ","End":"04:30.540","Text":"this is the only way that I can win the tournament."},{"Start":"04:32.830 ","End":"04:35.990","Text":"Once I\u0027ve identified the winning branches,"},{"Start":"04:35.990 ","End":"04:38.850","Text":"this branch right here and this branch right here,"},{"Start":"04:39.010 ","End":"04:44.540","Text":"then I want to calculate the probability of winning the tournament."},{"Start":"04:44.540 ","End":"04:51.065","Text":"Well, that\u0027s the probability of winning the tournament."},{"Start":"04:51.065 ","End":"04:59.480","Text":"That equals to p times t plus this branch right here,"},{"Start":"04:59.480 ","End":"05:08.405","Text":"that\u0027s 1 minus p times t times p. Now, let\u0027s simplify that."},{"Start":"05:08.405 ","End":"05:16.805","Text":"That\u0027s pt times 1 plus 1 minus p and equals"},{"Start":"05:16.805 ","End":"05:26.820","Text":"pt times 2 minus p and that\u0027s the probability for Option A."}],"ID":12927},{"Watched":false,"Name":"Exercise 10 - Part b","Duration":"4m 49s","ChapterTopicVideoID":12449,"CourseChapterTopicPlaylistID":245033,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"Now this is the tree for option B or strategy B."},{"Start":"00:04.290 ","End":"00:10.635","Text":"Now strategy B said that I have to first play Sara then I have to play Mike,"},{"Start":"00:10.635 ","End":"00:13.170","Text":"then I have to play Sara again."},{"Start":"00:13.170 ","End":"00:19.650","Text":"If we remember, the probability of beating Sara is t and"},{"Start":"00:19.650 ","End":"00:25.275","Text":"the probability of beating Mike is p. Contrary,"},{"Start":"00:25.275 ","End":"00:27.135","Text":"if I lose to Sara,"},{"Start":"00:27.135 ","End":"00:30.120","Text":"and the probability of losing to her is 1 minus t,"},{"Start":"00:30.120 ","End":"00:35.670","Text":"and the probability of losing to Mike is 1 minus p. Now the first time I"},{"Start":"00:35.670 ","End":"00:41.890","Text":"can either win or lose to Sara with the appropriate probabilities."},{"Start":"00:41.890 ","End":"00:45.695","Text":"Now, in order to win the tournament,"},{"Start":"00:45.695 ","End":"00:54.630","Text":"I have to first beat Sara on the first game and then beat Mike on the second game."},{"Start":"00:54.630 ","End":"00:56.490","Text":"Once I\u0027ve done that,"},{"Start":"00:56.490 ","End":"00:58.680","Text":"I basically won the tournament."},{"Start":"00:58.680 ","End":"01:03.570","Text":"If I beat her in the first name and lost to Mike on the second game,"},{"Start":"01:03.570 ","End":"01:05.730","Text":"well, then I\u0027d lost the tournament."},{"Start":"01:05.730 ","End":"01:08.715","Text":"I don\u0027t have 2 games in a row."},{"Start":"01:08.715 ","End":"01:12.590","Text":"Now, if I\u0027ve lost to Sara on the first game,"},{"Start":"01:12.590 ","End":"01:15.980","Text":"I can still win the tournament by beating Mike on"},{"Start":"01:15.980 ","End":"01:20.315","Text":"the second game and then beating Sara on the third game."},{"Start":"01:20.315 ","End":"01:25.640","Text":"Here I have won. Now if I\u0027ve lost to Sara on"},{"Start":"01:25.640 ","End":"01:30.770","Text":"the first game and lost to Mike on the second game, well,"},{"Start":"01:30.770 ","End":"01:32.510","Text":"I\u0027ve lost the tournament right here,"},{"Start":"01:32.510 ","End":"01:36.800","Text":"and if I\u0027ve lost to Sara on the first game,"},{"Start":"01:36.800 ","End":"01:40.635","Text":"won Mike on the second game,"},{"Start":"01:40.635 ","End":"01:44.745","Text":"and then lost to Sara again on the third game, well I\u0027ve lost."},{"Start":"01:44.745 ","End":"01:49.660","Text":"That means that the winning path"},{"Start":"01:49.660 ","End":"01:56.300","Text":"here is to first lose to Sara and then win Mike,"},{"Start":"01:56.300 ","End":"01:58.895","Text":"then win Sara again."},{"Start":"01:58.895 ","End":"02:02.710","Text":"So I\u0027ll have 2 consecutive wins."},{"Start":"02:02.710 ","End":"02:11.060","Text":"Great. Now that we know again what the paths are for winning in option B,"},{"Start":"02:11.060 ","End":"02:13.595","Text":"let\u0027s calculate the probabilities."},{"Start":"02:13.595 ","End":"02:18.665","Text":"Well, the probability of win is 1."},{"Start":"02:18.665 ","End":"02:20.765","Text":"Well, let\u0027s take this branch,"},{"Start":"02:20.765 ","End":"02:25.685","Text":"that\u0027s tp plus this branch,"},{"Start":"02:25.685 ","End":"02:31.400","Text":"which is 1 minus t times"},{"Start":"02:31.400 ","End":"02:39.640","Text":"p times t. Let\u0027s simplify that out."},{"Start":"02:39.640 ","End":"02:47.660","Text":"That\u0027s pt 1 plus 1 minus t. That equals to"},{"Start":"02:47.660 ","End":"02:52.545","Text":"pt 2 minus t."},{"Start":"02:52.545 ","End":"02:59.380","Text":"That\u0027s the probability of winning using option B."},{"Start":"02:59.870 ","End":"03:04.670","Text":"No we\u0027re not done yet. All we\u0027ve done up to now is"},{"Start":"03:04.670 ","End":"03:09.719","Text":"to calculate the probability of winning the tournament using option A,"},{"Start":"03:09.719 ","End":"03:11.630","Text":"that\u0027s is covered here."},{"Start":"03:11.630 ","End":"03:15.995","Text":"Or the probability of winning the tournament using option B."},{"Start":"03:15.995 ","End":"03:19.805","Text":"That\u0027s right here. We still don\u0027t know which option is better."},{"Start":"03:19.805 ","End":"03:22.160","Text":"Option A or option B."},{"Start":"03:22.160 ","End":"03:27.830","Text":"Well, we\u0027re given that Mike is"},{"Start":"03:27.830 ","End":"03:34.050","Text":"a better player than Sara."},{"Start":"03:34.870 ","End":"03:37.910","Text":"What does that mean? That means that"},{"Start":"03:37.910 ","End":"03:44.780","Text":"the probability or the chances of winning Sara is greater,"},{"Start":"03:44.780 ","End":"03:49.115","Text":"must be greater than the chances of beating Mike."},{"Start":"03:49.115 ","End":"03:57.020","Text":"Having said that, let\u0027s compare the 2 probabilities for option A and for option B."},{"Start":"03:57.020 ","End":"04:05.450","Text":"Well, for option A, the probability of winning the tournament is pt times 2 minus p,"},{"Start":"04:05.450 ","End":"04:11.270","Text":"that\u0027s for A and the probability of winning the tournament for"},{"Start":"04:11.270 ","End":"04:18.600","Text":"option B is pt 2 minus t, that\u0027s for B."},{"Start":"04:18.820 ","End":"04:23.105","Text":"Since t is greater than p,"},{"Start":"04:23.105 ","End":"04:25.121","Text":"well, we can get rid of these,"},{"Start":"04:25.121 ","End":"04:27.680","Text":"pt on both sides."},{"Start":"04:27.680 ","End":"04:30.290","Text":"Since t is greater than p,"},{"Start":"04:30.290 ","End":"04:39.650","Text":"then 2 minus p must be greater than 2 minus t. That means that"},{"Start":"04:39.650 ","End":"04:50.050","Text":"the best strategy or the best option to use is option A."}],"ID":12928}],"Thumbnail":null,"ID":245033},{"Name":"Dependent and Independent Events","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"5m 20s","ChapterTopicVideoID":12450,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.320 ","End":"00:07.665","Text":"In this section, we\u0027ll be talking about dependent and independent events."},{"Start":"00:07.665 ","End":"00:13.710","Text":"Now, when we look at the probability of B given A,"},{"Start":"00:13.710 ","End":"00:16.710","Text":"and if that equals to the probability of B."},{"Start":"00:16.710 ","End":"00:18.270","Text":"Now if that\u0027s true,"},{"Start":"00:18.270 ","End":"00:20.490","Text":"we say that B is independent of A,"},{"Start":"00:20.490 ","End":"00:22.545","Text":"and the opposite is true as well."},{"Start":"00:22.545 ","End":"00:26.610","Text":"If the probability of A given B equals the probability of A,"},{"Start":"00:26.610 ","End":"00:30.390","Text":"that means that A is independent of B."},{"Start":"00:30.390 ","End":"00:34.330","Text":"Let\u0027s take a quick example to illustrate this."},{"Start":"00:35.060 ","End":"00:41.670","Text":"Assume that A is getting a 4 when you throw a die,"},{"Start":"00:41.670 ","End":"00:44.370","Text":"and B is getting heads,"},{"Start":"00:44.370 ","End":"00:46.290","Text":"when you toss a coin."},{"Start":"00:46.290 ","End":"00:48.825","Text":"Now, the probability of A,"},{"Start":"00:48.825 ","End":"00:51.900","Text":"given a fair die is 1/6,"},{"Start":"00:51.900 ","End":"00:58.175","Text":"and the probability of B given a fair coin is a half."},{"Start":"00:58.175 ","End":"01:02.975","Text":"Now, what is the probability of A given B?"},{"Start":"01:02.975 ","End":"01:09.620","Text":"Well, it really doesn\u0027t matter what result you had when you tossed a coin."},{"Start":"01:09.620 ","End":"01:14.360","Text":"The probability of getting any number on a die will be the same,"},{"Start":"01:14.360 ","End":"01:23.970","Text":"so that means that the probability of A given B is equal to the probability of A."},{"Start":"01:24.310 ","End":"01:27.695","Text":"Now, this is true,"},{"Start":"01:27.695 ","End":"01:34.155","Text":"as we said, when A and B are independent."},{"Start":"01:34.155 ","End":"01:37.685","Text":"Now, we also know that the probability of"},{"Start":"01:37.685 ","End":"01:42.665","Text":"a given B equals the probability of A intersect B,"},{"Start":"01:42.665 ","End":"01:45.865","Text":"divided by the probability of B,"},{"Start":"01:45.865 ","End":"01:49.480","Text":"and that\u0027s true always."},{"Start":"01:50.720 ","End":"01:56.650","Text":"We said here that\u0027s true when A and B are independent."},{"Start":"01:56.650 ","End":"01:59.770","Text":"Now since both these equations are equal,"},{"Start":"01:59.770 ","End":"02:01.750","Text":"we can rewrite them and say,"},{"Start":"02:01.750 ","End":"02:05.765","Text":"it\u0027s the probability of A intersect B,"},{"Start":"02:05.765 ","End":"02:08.950","Text":"divided by the probability of B,"},{"Start":"02:08.950 ","End":"02:11.975","Text":"that equals to the probability of A."},{"Start":"02:11.975 ","End":"02:14.415","Text":"Or in other words,"},{"Start":"02:14.415 ","End":"02:18.335","Text":"the probability of A intersect B,"},{"Start":"02:18.335 ","End":"02:22.105","Text":"equals the multiplication of the probabilities."},{"Start":"02:22.105 ","End":"02:27.940","Text":"Probability of A times the probability of B."},{"Start":"02:27.940 ","End":"02:36.380","Text":"Now, this equation is true only when the 2 events are independent."},{"Start":"02:37.150 ","End":"02:41.274","Text":"We use these formulas for independent events"},{"Start":"02:41.274 ","End":"02:46.080","Text":"only if it\u0027s explicitly stated in the exercise that events are independent,"},{"Start":"02:46.080 ","End":"02:49.400","Text":"or if the events are in a context which"},{"Start":"02:49.400 ","End":"02:53.180","Text":"makes it clear that there is no doubt that they are independent."},{"Start":"02:53.180 ","End":"02:54.850","Text":"Let\u0027s take an example."},{"Start":"02:54.850 ","End":"02:59.180","Text":"A researcher conducts 2 independent trials."},{"Start":"02:59.180 ","End":"03:03.830","Text":"The chances of success in the first trial are 0.7,"},{"Start":"03:03.830 ","End":"03:08.390","Text":"and the chances of success in the second trial are 0.4."},{"Start":"03:08.390 ","End":"03:16.110","Text":"So again, we have 2 independent trials and a success in the first trial,"},{"Start":"03:16.110 ","End":"03:19.370","Text":"the probability is 0.7,"},{"Start":"03:19.370 ","End":"03:24.535","Text":"and the probability of success in the second trial is 0.4."},{"Start":"03:24.535 ","End":"03:32.065","Text":"Now we\u0027re asked, what are the chances of the researchers succeeding in both trials?"},{"Start":"03:32.065 ","End":"03:35.440","Text":"Well, since the events are independent,"},{"Start":"03:35.440 ","End":"03:45.155","Text":"then we\u0027re looking at the probability of getting success in A and getting success in B."},{"Start":"03:45.155 ","End":"03:47.950","Text":"Again, since they are independent,"},{"Start":"03:47.950 ","End":"03:51.250","Text":"we can use the formula for independence,"},{"Start":"03:51.250 ","End":"03:57.830","Text":"and that\u0027s the multiplication of success of the 2 trials."},{"Start":"03:57.870 ","End":"04:01.160","Text":"In our scenario,"},{"Start":"04:01.160 ","End":"04:05.630","Text":"the probability of success in the first trial is 0.7."},{"Start":"04:05.630 ","End":"04:09.380","Text":"The success in the second trial is 0.4,"},{"Start":"04:09.380 ","End":"04:13.290","Text":"and that equals to 0.28,"},{"Start":"04:13.370 ","End":"04:18.785","Text":"and that is the probability of success in both trials,"},{"Start":"04:18.785 ","End":"04:23.435","Text":"and what are the chances of the researcher failing in both trials?"},{"Start":"04:23.435 ","End":"04:29.540","Text":"Well, if we have independence in A and B,"},{"Start":"04:29.540 ","End":"04:35.375","Text":"then that carries over into events that are naught A and naught B."},{"Start":"04:35.375 ","End":"04:43.210","Text":"That means that the probability of naught A and naught B,"},{"Start":"04:43.210 ","End":"04:50.150","Text":"that equals to the probability of naught A times the probability of naught B."},{"Start":"04:50.150 ","End":"04:56.570","Text":"Now, the probability of naught A is 1 minus 0.7."},{"Start":"04:56.570 ","End":"05:03.000","Text":"And the probability of naught B is 1 minus 0.4,"},{"Start":"05:03.080 ","End":"05:09.510","Text":"and that equals to 0.3 times 0.6,"},{"Start":"05:09.510 ","End":"05:13.046","Text":"and that equals to 0.18,"},{"Start":"05:13.046 ","End":"05:20.699","Text":"and that\u0027s the probability of naught succeeding in both trials."}],"ID":12929},{"Watched":false,"Name":"Tutorial 2","Duration":"2m 12s","ChapterTopicVideoID":12451,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.590 ","End":"00:06.959","Text":"Now what\u0027s true of 2 events is also true of many events."},{"Start":"00:06.959 ","End":"00:10.275","Text":"So if we have any events,"},{"Start":"00:10.275 ","End":"00:13.225","Text":"let\u0027s say A_1 to A_N,"},{"Start":"00:13.225 ","End":"00:17.390","Text":"and we say that they\u0027re independent only if,"},{"Start":"00:17.390 ","End":"00:26.380","Text":"if and only if the probability of the intersection of all the events from 1 to N,"},{"Start":"00:26.380 ","End":"00:36.320","Text":"if that equals to the multiplication of the probability of each event from 1 to N. Now,"},{"Start":"00:36.320 ","End":"00:38.165","Text":"let\u0027s take an example."},{"Start":"00:38.165 ","End":"00:41.825","Text":"Let\u0027s say i equals 1, 2,"},{"Start":"00:41.825 ","End":"00:43.310","Text":"3, and 4;"},{"Start":"00:43.310 ","End":"00:46.380","Text":"and we have 4 events,"},{"Start":"00:46.380 ","End":"00:49.635","Text":"A_1, A_2,"},{"Start":"00:49.635 ","End":"00:56.060","Text":"A_3, and A_4."},{"Start":"00:56.060 ","End":"01:04.005","Text":"Now, what is the probability of A_1 intersect A_2,"},{"Start":"01:04.005 ","End":"01:10.000","Text":"intersect A_3, intersect A_4?"},{"Start":"01:10.060 ","End":"01:20.710","Text":"Well, if the events A_1 to A_4 are independent that would equal to the multiplication of"},{"Start":"01:20.710 ","End":"01:24.695","Text":"the probabilities of A_1 times the probability of"},{"Start":"01:24.695 ","End":"01:31.695","Text":"A_2 times the probability of A_3 times the probability of A_4."},{"Start":"01:31.695 ","End":"01:36.595","Text":"For those who are mathematically inclined,"},{"Start":"01:36.595 ","End":"01:44.360","Text":"when we have the intersection of many events such as we have here,"},{"Start":"01:44.360 ","End":"01:46.250","Text":"this is the intersection,"},{"Start":"01:46.250 ","End":"01:56.205","Text":"and i runs from 1-4 in our case of all events, from 1-4."},{"Start":"01:56.205 ","End":"02:00.890","Text":"We say that that equals to the multiplication."},{"Start":"02:00.890 ","End":"02:02.450","Text":"This is the sign for multiplication."},{"Start":"02:02.450 ","End":"02:10.590","Text":"Again, I goes from 1-4 of the probabilities of each event."}],"ID":12930},{"Watched":false,"Name":"Exercise 1","Duration":"2m 31s","ChapterTopicVideoID":12452,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:05.880","Text":"In this question, we are given that the probability of A is 0.2,"},{"Start":"00:05.880 ","End":"00:09.855","Text":"the probability of B is 0.5,"},{"Start":"00:09.855 ","End":"00:13.275","Text":"and the probability of A union B is 0.6."},{"Start":"00:13.275 ","End":"00:18.660","Text":"We\u0027re asked the events A and B independent of each other or not."},{"Start":"00:18.660 ","End":"00:22.335","Text":"Now, in order for them to be independent,"},{"Start":"00:22.335 ","End":"00:27.725","Text":"then the probability of A intersect B must"},{"Start":"00:27.725 ","End":"00:34.565","Text":"equal to the probability of A times the probability of B."},{"Start":"00:34.565 ","End":"00:38.315","Text":"Now we do have the probability of A."},{"Start":"00:38.315 ","End":"00:40.790","Text":"We do have the probability of B,"},{"Start":"00:40.790 ","End":"00:46.220","Text":"and we do not have the probability of A intersect B."},{"Start":"00:46.220 ","End":"00:48.035","Text":"In order to calculate that,"},{"Start":"00:48.035 ","End":"00:50.695","Text":"let\u0027s use this formula."},{"Start":"00:50.695 ","End":"00:58.355","Text":"Probability of A union B equals to the probability of"},{"Start":"00:58.355 ","End":"01:08.125","Text":"A plus the probability of B minus the probability of A intersect B."},{"Start":"01:08.125 ","End":"01:11.675","Text":"Now, let\u0027s plug in the numbers."},{"Start":"01:11.675 ","End":"01:17.065","Text":"The probability of A union B is 0.6. Let\u0027s put that in."},{"Start":"01:17.065 ","End":"01:21.180","Text":"The probability of A is 0.2,"},{"Start":"01:21.180 ","End":"01:25.140","Text":"the probability of B is 0.5."},{"Start":"01:25.140 ","End":"01:32.320","Text":"What we are trying to find out is the probability of A intersect B."},{"Start":"01:33.070 ","End":"01:36.725","Text":"Let\u0027s switch sides."},{"Start":"01:36.725 ","End":"01:46.700","Text":"Probability of A intersect B equals 0.2 plus 0.5 minus 0.6,"},{"Start":"01:46.700 ","End":"01:50.555","Text":"and that does equal 0.1."},{"Start":"01:50.555 ","End":"01:58.055","Text":"Let\u0027s get back to our original equation of independence."},{"Start":"01:58.055 ","End":"02:05.550","Text":"Now, we know that now that the probability of A intersect B is 0.1."},{"Start":"02:06.470 ","End":"02:11.180","Text":"We do have the probabilities of A and B,"},{"Start":"02:11.180 ","End":"02:15.062","Text":"which are 0.2 and 0.5,"},{"Start":"02:15.062 ","End":"02:17.430","Text":"so we multiply them."},{"Start":"02:18.050 ","End":"02:21.945","Text":"Does that really equal 0.1?"},{"Start":"02:21.945 ","End":"02:25.060","Text":"Well, yes it does."},{"Start":"02:25.880 ","End":"02:31.650","Text":"This means that A and B are independent of each other."}],"ID":12931},{"Watched":false,"Name":"Exercise 2","Duration":"5m 18s","ChapterTopicVideoID":12453,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.645","Text":"Let\u0027s take a look at the following problem."},{"Start":"00:03.645 ","End":"00:11.055","Text":"Student takes 2 exams where success in 1 exam is independent of the success in the other."},{"Start":"00:11.055 ","End":"00:18.465","Text":"His chances of success are 0.7 in the first exam and 0.4 in the second exam."},{"Start":"00:18.465 ","End":"00:22.950","Text":"We\u0027re asked, what are the student\u0027s chances of succeeding"},{"Start":"00:22.950 ","End":"00:27.795","Text":"in both exams and what are the student\u0027s chances of failing in both exams?"},{"Start":"00:27.795 ","End":"00:31.425","Text":"Now, in order to answer these questions,"},{"Start":"00:31.425 ","End":"00:34.380","Text":"let\u0027s first define our variables."},{"Start":"00:34.380 ","End":"00:39.135","Text":"Let\u0027s define the variable A as"},{"Start":"00:39.135 ","End":"00:42.870","Text":"success in"},{"Start":"00:42.870 ","End":"00:49.680","Text":"the first exam and B,"},{"Start":"00:49.680 ","End":"00:54.810","Text":"let\u0027s define that as success in the second exam."},{"Start":"00:54.810 ","End":"00:59.075","Text":"Now, the probability of"},{"Start":"00:59.075 ","End":"01:06.980","Text":"A then his success on the first exam is equal to 0.7 right here."},{"Start":"01:06.980 ","End":"01:09.170","Text":"Let\u0027s write this down,"},{"Start":"01:09.170 ","End":"01:13.850","Text":"that\u0027s 0.7 and the probability of"},{"Start":"01:13.850 ","End":"01:20.860","Text":"success in the second exam is 0.4."},{"Start":"01:20.860 ","End":"01:22.635","Text":"Let\u0027s point it out here."},{"Start":"01:22.635 ","End":"01:26.830","Text":"Let\u0027s just write this down 0.4."},{"Start":"01:26.990 ","End":"01:29.240","Text":"What do we also know?"},{"Start":"01:29.240 ","End":"01:35.340","Text":"We also know that both events are independent of each other."},{"Start":"01:36.110 ","End":"01:39.905","Text":"Now, if they\u0027re independent of each other,"},{"Start":"01:39.905 ","End":"01:42.275","Text":"then we know"},{"Start":"01:42.275 ","End":"01:50.270","Text":"that probability of the intersection,"},{"Start":"01:50.270 ","End":"01:55.670","Text":"that means that the probability of success in the first exam and success in"},{"Start":"01:55.670 ","End":"02:05.020","Text":"the second exam is equal to the probability of A times the probability of B."},{"Start":"02:05.020 ","End":"02:10.030","Text":"That\u0027s what we\u0027re asked in Section A."},{"Start":"02:10.370 ","End":"02:17.535","Text":"What are the chances of succeeding in both exams?"},{"Start":"02:17.535 ","End":"02:20.570","Text":"Excellent. What does that mean?"},{"Start":"02:20.570 ","End":"02:25.700","Text":"All I have to do is multiply the 2 probabilities,"},{"Start":"02:25.700 ","End":"02:29.090","Text":"the probability of A and the probability of B."},{"Start":"02:29.090 ","End":"02:37.490","Text":"That means that 0.7 times 0.4 and that equals to 0.28."},{"Start":"02:37.490 ","End":"02:46.986","Text":"That is the probability of the student\u0027s chances of succeeding in both exams."},{"Start":"02:46.986 ","End":"02:49.685","Text":"Let\u0027s put that here."},{"Start":"02:49.685 ","End":"02:53.370","Text":"That\u0027s the answer to Section A."},{"Start":"02:53.370 ","End":"02:56.895","Text":"Let\u0027s look now at Section B."},{"Start":"02:56.895 ","End":"03:04.940","Text":"Section B asks, what are the students chances of failing in both exams?"},{"Start":"03:04.940 ","End":"03:07.474","Text":"Now, to failing both exams,"},{"Start":"03:07.474 ","End":"03:13.394","Text":"that\u0027s basically asking for the probabilities of the complimentary events."},{"Start":"03:13.394 ","End":"03:18.140","Text":"The complementary events are written as A-bar and B-bar."},{"Start":"03:18.140 ","End":"03:21.665","Text":"That means that what we want to find out,"},{"Start":"03:21.665 ","End":"03:26.285","Text":"what are the probabilities as"},{"Start":"03:26.285 ","End":"03:32.977","Text":"A-bar and the probability of B-bar."},{"Start":"03:32.977 ","End":"03:37.370","Text":"The probability of the complimentary events"},{"Start":"03:37.370 ","End":"03:42.000","Text":"are 1 minus the probability of the event itself."},{"Start":"03:44.660 ","End":"03:52.710","Text":"That equals to 1 minus 0.7 in Event A,"},{"Start":"03:52.710 ","End":"03:58.440","Text":"and that equals to 1 minus 0.4 in Event B."},{"Start":"03:58.440 ","End":"04:02.165","Text":"Since both events A and B are independent,"},{"Start":"04:02.165 ","End":"04:07.605","Text":"then that means that their complimentary events are also independent."},{"Start":"04:07.605 ","End":"04:15.035","Text":"That means that probability of failing both exams, A-bar,"},{"Start":"04:15.035 ","End":"04:20.900","Text":"intersect B-bar equals the probability"},{"Start":"04:20.900 ","End":"04:28.660","Text":"of A-bar times the probability of B-bar."},{"Start":"04:28.660 ","End":"04:38.620","Text":"We know from here that the complimentary event of A"},{"Start":"04:38.620 ","End":"04:44.030","Text":"is 1 minus 0.7 and the probability of"},{"Start":"04:44.030 ","End":"04:50.620","Text":"the complimentary event B is 1 minus 0.4."},{"Start":"04:52.250 ","End":"04:57.980","Text":"That means that the probability of A-bar is"},{"Start":"04:57.980 ","End":"05:04.435","Text":"0.3 times the probability of the complimentary Event B,"},{"Start":"05:04.435 ","End":"05:10.220","Text":"which is 0.6 and that equals to 0.18."},{"Start":"05:11.600 ","End":"05:19.290","Text":"That is the probability of the student\u0027s chances of failing in both exams."}],"ID":12932},{"Watched":false,"Name":"Exercise 3","Duration":"5m 24s","ChapterTopicVideoID":12454,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.295","Text":"Let\u0027s look at the following problem."},{"Start":"00:02.295 ","End":"00:06.390","Text":"The unemployment rate in a certain country is 8 percent."},{"Start":"00:06.390 ","End":"00:10.215","Text":"2 people in the country are randomly chosen."},{"Start":"00:10.215 ","End":"00:12.780","Text":"We are asked, a,"},{"Start":"00:12.780 ","End":"00:16.305","Text":"what is the probability that both people chosen are unemployed?"},{"Start":"00:16.305 ","End":"00:20.460","Text":"B, what is the probability that at least 1 of"},{"Start":"00:20.460 ","End":"00:25.350","Text":"the people chosen is unemployed? Let\u0027s begin."},{"Start":"00:25.350 ","End":"00:28.530","Text":"First, let\u0027s define our variables."},{"Start":"00:28.530 ","End":"00:31.770","Text":"Let A be the first person chosen,"},{"Start":"00:31.770 ","End":"00:40.770","Text":"and B be the second person chosen."},{"Start":"00:40.770 ","End":"00:43.190","Text":"But before we can continue,"},{"Start":"00:43.190 ","End":"00:47.225","Text":"we need to discuss the issue of independence."},{"Start":"00:47.225 ","End":"00:51.005","Text":"Is the first person independent of the second person?"},{"Start":"00:51.005 ","End":"00:55.145","Text":"Well, it\u0027s not explicitly stated in the question."},{"Start":"00:55.145 ","End":"01:01.505","Text":"But whenever we have a small sample size taken from a large population,"},{"Start":"01:01.505 ","End":"01:06.110","Text":"we can assume that the sample elements are independent of each other."},{"Start":"01:06.110 ","End":"01:13.760","Text":"In our case, we\u0027ve only taken 2 people from a very large population size,"},{"Start":"01:13.760 ","End":"01:16.170","Text":"the population of the whole country."},{"Start":"01:16.210 ","End":"01:18.650","Text":"The minute we encounter this thing,"},{"Start":"01:18.650 ","End":"01:24.710","Text":"we can assume that the first person is independent of the second person."},{"Start":"01:24.710 ","End":"01:26.885","Text":"Well, having said that,"},{"Start":"01:26.885 ","End":"01:29.790","Text":"let\u0027s go on and solve for a."},{"Start":"01:29.800 ","End":"01:36.290","Text":"A asks us, what is the probability that both people chosen are unemployed?"},{"Start":"01:36.290 ","End":"01:39.095","Text":"Well, that\u0027s basically saying,"},{"Start":"01:39.095 ","End":"01:44.500","Text":"what is the probability of A intersect B?"},{"Start":"01:44.500 ","End":"01:47.240","Text":"Now because we\u0027re assuming independence,"},{"Start":"01:47.240 ","End":"01:52.520","Text":"we know that that equals to the probability of A times the probability of B,"},{"Start":"01:52.520 ","End":"01:56.735","Text":"and we even know what the probability of A is."},{"Start":"01:56.735 ","End":"01:59.195","Text":"Well, that\u0027s 8 percent."},{"Start":"01:59.195 ","End":"02:01.430","Text":"It\u0027s given right here."},{"Start":"02:01.430 ","End":"02:05.045","Text":"Let\u0027s plug that in. That\u0027s 0.08."},{"Start":"02:05.045 ","End":"02:08.060","Text":"What\u0027s the probability that B is unemployed?"},{"Start":"02:08.060 ","End":"02:09.830","Text":"Well, it\u0027s the same probability."},{"Start":"02:09.830 ","End":"02:12.390","Text":"Again, that\u0027s 0.08."},{"Start":"02:12.390 ","End":"02:17.170","Text":"That comes out to 0.0064."},{"Start":"02:17.170 ","End":"02:22.370","Text":"That is the probability that both people"},{"Start":"02:22.370 ","End":"02:27.990","Text":"that have been chosen from a very large population are unemployed."},{"Start":"02:30.010 ","End":"02:34.320","Text":"Let\u0027s take a look at section B."},{"Start":"02:34.420 ","End":"02:37.700","Text":"Section B asks us,"},{"Start":"02:37.700 ","End":"02:42.785","Text":"what is the probability that at least 1 of the people chosen is unemployed?"},{"Start":"02:42.785 ","End":"02:45.830","Text":"Well, that\u0027s basically asking,"},{"Start":"02:45.830 ","End":"02:49.745","Text":"what is the probability of A union B?"},{"Start":"02:49.745 ","End":"02:53.090","Text":"Now, if we use De Morgan\u0027s rule,"},{"Start":"02:53.090 ","End":"03:00.440","Text":"we know that that equals to 1 minus the probability of the intersection"},{"Start":"03:00.440 ","End":"03:07.560","Text":"of the complementary events A bar intersect B bar."},{"Start":"03:07.560 ","End":"03:09.960","Text":"Now, what\u0027s A bar and B bar?"},{"Start":"03:09.960 ","End":"03:13.745","Text":"Now, if A is a person that is unemployed,"},{"Start":"03:13.745 ","End":"03:17.255","Text":"then A bar is a person that\u0027s employed."},{"Start":"03:17.255 ","End":"03:19.595","Text":"The same goes for B."},{"Start":"03:19.595 ","End":"03:23.975","Text":"Now because both A and B are independent,"},{"Start":"03:23.975 ","End":"03:29.795","Text":"we also know that A bar and B bar are independent as well."},{"Start":"03:29.795 ","End":"03:35.030","Text":"That means that we can write this equation in this form,"},{"Start":"03:35.030 ","End":"03:42.810","Text":"1 minus the probability of A bar times the probability of B bar."},{"Start":"03:43.150 ","End":"03:47.855","Text":"Now, what is the probability of A bar?"},{"Start":"03:47.855 ","End":"03:53.630","Text":"Well if the probability of a person being unemployed is 8 percent,"},{"Start":"03:53.630 ","End":"04:00.180","Text":"that means that the probability of a person being employed is 92 percent."},{"Start":"04:00.250 ","End":"04:10.490","Text":"That means that equals to 1 minus 0.92 times 0.92."},{"Start":"04:10.490 ","End":"04:15.900","Text":"That comes out to 0.1536."},{"Start":"04:17.050 ","End":"04:22.070","Text":"We can solve this in a different way."},{"Start":"04:22.070 ","End":"04:26.280","Text":"We know also that the probability of"},{"Start":"04:26.280 ","End":"04:35.870","Text":"A union B is equals to probability of A plus the probability of B,"},{"Start":"04:35.870 ","End":"04:40.720","Text":"minus the probability of A intersect B."},{"Start":"04:41.060 ","End":"04:44.430","Text":"Now, the probability of A,"},{"Start":"04:44.430 ","End":"04:47.890","Text":"we know that to be 0.08."},{"Start":"04:49.130 ","End":"04:55.110","Text":"We know that that\u0027s the same probability as B."},{"Start":"04:55.270 ","End":"05:06.735","Text":"What we also know is the probability of A intersect B is this 0.0064."},{"Start":"05:06.735 ","End":"05:09.700","Text":"Let\u0027s plug that in."},{"Start":"05:10.970 ","End":"05:15.490","Text":"Again that comes out to 0.1536."},{"Start":"05:16.010 ","End":"05:22.800","Text":"As we can see, we get the same results using both methods."}],"ID":12933},{"Watched":false,"Name":"Exercise 4","Duration":"4m 56s","ChapterTopicVideoID":12455,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"This problem deals with quality control."},{"Start":"00:02.415 ","End":"00:07.455","Text":"A product must pass 4 independent tests before it\u0027s marketed,"},{"Start":"00:07.455 ","End":"00:10.815","Text":"otherwise it is rejected for marketing."},{"Start":"00:10.815 ","End":"00:16.005","Text":"The product\u0027s chances of success on each test are 80 percent."},{"Start":"00:16.005 ","End":"00:19.215","Text":"All 4 tests must be performed."},{"Start":"00:19.215 ","End":"00:24.825","Text":"We\u0027re asked, what are the chances that the product will be rejected?"},{"Start":"00:24.825 ","End":"00:31.630","Text":"What is the probability that the product will successfully pass at least 1 test?"},{"Start":"00:32.000 ","End":"00:38.550","Text":"Here, we have to take into consideration several things."},{"Start":"00:38.550 ","End":"00:44.875","Text":"The first 1 is that each test is independent of each other."},{"Start":"00:44.875 ","End":"00:54.515","Text":"That means that a test is not dependent on the success or failure of another test."},{"Start":"00:54.515 ","End":"00:58.850","Text":"Another thing is that the probability of success is 80 percent."},{"Start":"00:58.850 ","End":"01:03.665","Text":"The third thing, that a product must pass all 4 tests."},{"Start":"01:03.665 ","End":"01:06.795","Text":"It can pass or be rejected by 3."},{"Start":"01:06.795 ","End":"01:09.610","Text":"All 4 tests must be executed."},{"Start":"01:09.610 ","End":"01:14.600","Text":"Misconceptions concerning the rejection of"},{"Start":"01:14.600 ","End":"01:20.110","Text":"a product is that the product has to be rejected by all 4 tests."},{"Start":"01:20.110 ","End":"01:26.795","Text":"This is not true. A product can be rejected if it doesn\u0027t pass 1 test."},{"Start":"01:26.795 ","End":"01:30.545","Text":"Having said that, the question is,"},{"Start":"01:30.545 ","End":"01:35.203","Text":"what are the chances that the product will be rejected?"},{"Start":"01:35.203 ","End":"01:39.785","Text":"Well, let\u0027s take a look."},{"Start":"01:39.785 ","End":"01:46.910","Text":"Probability that the product will be rejected."},{"Start":"01:46.910 ","End":"01:50.750","Text":"Let\u0027s take a look at it from the other side."},{"Start":"01:50.750 ","End":"01:58.560","Text":"What is the probability that the product will not be rejected?"},{"Start":"02:04.030 ","End":"02:08.210","Text":"That\u0027s probability of being rejected is 1 minus"},{"Start":"02:08.210 ","End":"02:12.515","Text":"the probability of a product not being rejected."},{"Start":"02:12.515 ","End":"02:15.545","Text":"What does that mean?"},{"Start":"02:15.545 ","End":"02:18.499","Text":"The probability of a product not being rejected."},{"Start":"02:18.499 ","End":"02:25.440","Text":"That means that it\u0027s the probability of passing all tests."},{"Start":"02:33.170 ","End":"02:35.580","Text":"That\u0027s 1 minus."},{"Start":"02:35.580 ","End":"02:38.510","Text":"What\u0027s the probability of passing all the tests?"},{"Start":"02:38.510 ","End":"02:42.620","Text":"Well, we know that the probability of passing 1 test is 0.8,"},{"Start":"02:42.620 ","End":"02:46.670","Text":"but we also know that the tests are independent."},{"Start":"02:46.670 ","End":"02:49.580","Text":"In order to pass all 4 tests,"},{"Start":"02:49.580 ","End":"02:56.395","Text":"we have to multiply the probability of passing 1 test 4 times,"},{"Start":"02:56.395 ","End":"03:02.900","Text":"0.8,"},{"Start":"03:02.900 ","End":"03:08.430","Text":"which is 1 minus 0.8^4."},{"Start":"03:08.430 ","End":"03:12.965","Text":"That comes out to 0.5904,"},{"Start":"03:12.965 ","End":"03:19.252","Text":"which is the probability that a product will be rejected."},{"Start":"03:19.252 ","End":"03:23.345","Text":"Now let\u0027s take a look at Section B."},{"Start":"03:23.345 ","End":"03:28.535","Text":"What is the probability that the product will successfully pass at least 1 test?"},{"Start":"03:28.535 ","End":"03:32.240","Text":"Well, probability that it will"},{"Start":"03:32.240 ","End":"03:40.140","Text":"pass at least 1"},{"Start":"03:40.140 ","End":"03:43.065","Text":"test."},{"Start":"03:43.065 ","End":"03:47.045","Text":"Again, let\u0027s look at the complimentary probability,"},{"Start":"03:47.045 ","End":"03:57.840","Text":"1 minus the probability that it will be rejected by all tests."},{"Start":"03:59.500 ","End":"04:08.620","Text":"The probability of being rejected by 1 test is 1 minus 0.8 or 0.2."},{"Start":"04:08.980 ","End":"04:14.270","Text":"If a product is being rejected by all tests,"},{"Start":"04:14.270 ","End":"04:18.410","Text":"then we have to multiply 0.2,"},{"Start":"04:18.410 ","End":"04:21.970","Text":"the probability of rejecting 4 times."},{"Start":"04:21.970 ","End":"04:31.435","Text":"That\u0027ll be 1 minus 0.2 times 0.2 times 0.2 times 0.2,"},{"Start":"04:31.435 ","End":"04:35.030","Text":"which is equal to 1 minus 0.2^4."},{"Start":"04:35.750 ","End":"04:41.840","Text":"Because it has to be rejected by all tests or 4 tests."},{"Start":"04:41.840 ","End":"04:48.590","Text":"That comes out to 0.9984."},{"Start":"04:48.590 ","End":"04:56.150","Text":"That is a probability that the product will successfully pass at least 1 test."}],"ID":12934},{"Watched":false,"Name":"Exercise 5","Duration":"3m 38s","ChapterTopicVideoID":12456,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this question, we\u0027re given that"},{"Start":"00:02.700 ","End":"00:07.035","Text":"the unemployment rate in a certain country is 8 percent."},{"Start":"00:07.035 ","End":"00:12.015","Text":"5 people in the country are randomly chosen and we\u0027re asked: a,"},{"Start":"00:12.015 ","End":"00:15.765","Text":"what is the probability that everyone in the sample is unemployed?"},{"Start":"00:15.765 ","End":"00:21.345","Text":"b, what is the probability that at least 1 person in the sample is unemployed?"},{"Start":"00:21.345 ","End":"00:26.490","Text":"In these types of questions we have to understand that when selecting"},{"Start":"00:26.490 ","End":"00:31.725","Text":"a very small sample from a very large population,"},{"Start":"00:31.725 ","End":"00:38.055","Text":"we can assume that the sample elements are basically independent."},{"Start":"00:38.055 ","End":"00:45.005","Text":"In our case, that the 5 people that have been chosen are independent of each other."},{"Start":"00:45.005 ","End":"00:47.315","Text":"Now, having said that,"},{"Start":"00:47.315 ","End":"00:50.135","Text":"let\u0027s solve question a."},{"Start":"00:50.135 ","End":"00:51.845","Text":"Note a asks us,"},{"Start":"00:51.845 ","End":"00:57.300","Text":"what is the probability of everyone being unemployed?"},{"Start":"01:02.620 ","End":"01:09.365","Text":"Now, because the people are randomly selected,"},{"Start":"01:09.365 ","End":"01:12.365","Text":"and because they\u0027re independent of each other,"},{"Start":"01:12.365 ","End":"01:17.095","Text":"we have to multiply the probability of there being unemployed."},{"Start":"01:17.095 ","End":"01:20.150","Text":"What is the probability of a person being unemployed?"},{"Start":"01:20.150 ","End":"01:22.255","Text":"Well, that\u0027s given; that\u0027s 8 percent."},{"Start":"01:22.255 ","End":"01:27.135","Text":"All we have to do right now is multiply 8 percent 5 times."},{"Start":"01:27.135 ","End":"01:31.965","Text":"That\u0027s 0.08 times 0.08"},{"Start":"01:31.965 ","End":"01:38.160","Text":"times 0.08 times 0.08."},{"Start":"01:38.160 ","End":"01:45.795","Text":"That\u0027s 5 times. That\u0027s basically 0.08^5."},{"Start":"01:45.795 ","End":"01:50.835","Text":"That is a very small number."},{"Start":"01:50.835 ","End":"01:53.800","Text":"Let\u0027s just leave it at that."},{"Start":"01:55.310 ","End":"01:57.995","Text":"Question b asks us,"},{"Start":"01:57.995 ","End":"02:02.435","Text":"what is the probability that at least 1 person in the sample is unemployed?"},{"Start":"02:02.435 ","End":"02:06.005","Text":"That\u0027s the person, the probability of"},{"Start":"02:06.005 ","End":"02:13.220","Text":"at least 1 being unemployed."},{"Start":"02:13.220 ","End":"02:18.890","Text":"Well, in order to solve this,"},{"Start":"02:18.890 ","End":"02:22.955","Text":"let\u0027s take the complimentary set, which states,"},{"Start":"02:22.955 ","End":"02:28.830","Text":"that this is equal to 1 minus the probability"},{"Start":"02:28.830 ","End":"02:39.460","Text":"of everyone being employed."},{"Start":"02:41.750 ","End":"02:46.535","Text":"Now, what\u0027s the probability of being employed?"},{"Start":"02:46.535 ","End":"02:52.370","Text":"The probability of being employed is 1 minus the probability of being unemployed."},{"Start":"02:52.370 ","End":"02:56.415","Text":"That is 1 minus 8 percent,"},{"Start":"02:56.415 ","End":"02:58.320","Text":"which is 92 percent."},{"Start":"02:58.320 ","End":"03:03.890","Text":"Now, if we have 5 people that are all employed,"},{"Start":"03:03.890 ","End":"03:07.820","Text":"all we have to do is multiply 92 percent times"},{"Start":"03:07.820 ","End":"03:12.575","Text":"5 because each person is independent of the other."},{"Start":"03:12.575 ","End":"03:15.280","Text":"We get 1 minus"},{"Start":"03:15.280 ","End":"03:24.240","Text":"0.92^5 because we can multiply 0.92 5 times."},{"Start":"03:24.240 ","End":"03:28.740","Text":"That comes out to 0.3409."},{"Start":"03:28.740 ","End":"03:37.770","Text":"That is the probability of at least 1 person in the sample being unemployed."}],"ID":12935},{"Watched":false,"Name":"Exercise 6","Duration":"7m 11s","ChapterTopicVideoID":12457,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this exercise, we have 2 events,"},{"Start":"00:02.565 ","End":"00:06.195","Text":"A and B that are defined on a sample space."},{"Start":"00:06.195 ","End":"00:11.040","Text":"We assume that the probability of A union B is 0.9,"},{"Start":"00:11.040 ","End":"00:16.110","Text":"the probability of A intersect naught B is 0.3,"},{"Start":"00:16.110 ","End":"00:20.625","Text":"and the probability of A given B is 0.6."},{"Start":"00:20.625 ","End":"00:24.645","Text":"We\u0027re asked whether A and B are independent events?"},{"Start":"00:24.645 ","End":"00:28.120","Text":"In order for them to be independent events,"},{"Start":"00:28.340 ","End":"00:33.630","Text":"probability of A intersect B must"},{"Start":"00:33.630 ","End":"00:38.850","Text":"equal to the probability of A times the probability of B."},{"Start":"00:38.850 ","End":"00:41.815","Text":"Now, let\u0027s look at what the question gives us."},{"Start":"00:41.815 ","End":"00:44.420","Text":"Well, first of all,"},{"Start":"00:44.420 ","End":"00:50.000","Text":"let\u0027s look at the probability of A union B."},{"Start":"00:50.000 ","End":"00:56.150","Text":"By definition, that equals probability of A plus"},{"Start":"00:56.150 ","End":"01:03.425","Text":"the probability of B minus the probability of A intersect B,"},{"Start":"01:03.425 ","End":"01:07.230","Text":"and that equals 0.9."},{"Start":"01:07.880 ","End":"01:12.100","Text":"Why is that? Because of here."},{"Start":"01:13.880 ","End":"01:21.095","Text":"Let\u0027s look at the other probabilities."},{"Start":"01:21.095 ","End":"01:28.055","Text":"Probability of A given B, and by definition,"},{"Start":"01:28.055 ","End":"01:36.240","Text":"that equals the probability of A intersect B divided by the probability of B,"},{"Start":"01:36.240 ","End":"01:40.370","Text":"and that equals 0.6."},{"Start":"01:40.370 ","End":"01:44.640","Text":"Again, that\u0027s given to us from here."},{"Start":"01:46.090 ","End":"01:50.225","Text":"The third equation is this,"},{"Start":"01:50.225 ","End":"01:55.580","Text":"probability of A intersect naught B."},{"Start":"01:55.580 ","End":"01:57.485","Text":"That\u0027s a little bit tricky."},{"Start":"01:57.485 ","End":"02:02.430","Text":"So let\u0027s look at this diagram right here."},{"Start":"02:02.630 ","End":"02:10.890","Text":"A intersect naught B is basically this area right here."},{"Start":"02:16.600 ","End":"02:20.060","Text":"So in essence, what we have here,"},{"Start":"02:20.060 ","End":"02:25.709","Text":"the probability of A intersect naught B is the probability"},{"Start":"02:25.709 ","End":"02:32.135","Text":"of A minus the probability of A intersect B."},{"Start":"02:32.135 ","End":"02:36.905","Text":"That equals 0.3. How do I know that?"},{"Start":"02:36.905 ","End":"02:40.220","Text":"Well, again, that\u0027s given to us here."},{"Start":"02:40.220 ","End":"02:44.195","Text":"Now we have 3 equations;"},{"Start":"02:44.195 ","End":"02:45.650","Text":"equation number 1, 2,"},{"Start":"02:45.650 ","End":"02:48.295","Text":"and 3, and 3 variables."},{"Start":"02:48.295 ","End":"02:50.510","Text":"P of A intersect B,"},{"Start":"02:50.510 ","End":"02:52.895","Text":"P of A, and P of B,"},{"Start":"02:52.895 ","End":"02:55.250","Text":"and we have to figure out the values of"},{"Start":"02:55.250 ","End":"03:00.815","Text":"these variables in order to see whether this equation holds true."},{"Start":"03:00.815 ","End":"03:06.210","Text":"If it does, then A and B will be independent and if they\u0027re not equal,"},{"Start":"03:06.210 ","End":"03:11.080","Text":"then A and B will be dependent variables. So let\u0027s get started."},{"Start":"03:11.080 ","End":"03:17.285","Text":"The first thing we want to do is write these equations out a little bit more comfortably."},{"Start":"03:17.285 ","End":"03:19.380","Text":"We\u0027ll write this out."},{"Start":"03:19.380 ","End":"03:27.750","Text":"Probability of A intersect B that would equal 0.6 times the probability of B."},{"Start":"03:28.180 ","End":"03:35.525","Text":"We will write this out as the probability of A being"},{"Start":"03:35.525 ","End":"03:42.920","Text":"equal to 0.3 plus the probability of A intersect B."},{"Start":"03:42.920 ","End":"03:46.580","Text":"Now we know what the probability of A intersect B is."},{"Start":"03:46.580 ","End":"03:48.770","Text":"That\u0027s 0.6 P of B."},{"Start":"03:48.770 ","End":"03:50.450","Text":"We can see that from here."},{"Start":"03:50.450 ","End":"03:53.134","Text":"So let\u0027s write this out."},{"Start":"03:53.134 ","End":"04:03.320","Text":"The probability of A equals 0.3 plus 0.6 times the probability of B."},{"Start":"04:03.320 ","End":"04:09.495","Text":"Great. Now, let\u0027s take these 2 values,"},{"Start":"04:09.495 ","End":"04:15.845","Text":"these 2 results, and plug them back into equation number 1."},{"Start":"04:15.845 ","End":"04:19.315","Text":"When we do that, we get this."},{"Start":"04:19.315 ","End":"04:26.835","Text":"0.9 is equal to P of A. P of A is this,"},{"Start":"04:26.835 ","End":"04:32.740","Text":"that\u0027s 0.3 plus 0.6 P of B,"},{"Start":"04:32.870 ","End":"04:43.055","Text":"plus P of B minus the probability of A intersect B."},{"Start":"04:43.055 ","End":"04:45.560","Text":"Well, that\u0027s this guy right here,"},{"Start":"04:45.560 ","End":"04:51.960","Text":"0.6 P of B minus 0.6 P of B."},{"Start":"04:52.480 ","End":"04:58.020","Text":"Now, we see here that this cancels this out."},{"Start":"04:59.200 ","End":"05:01.550","Text":"Let\u0027s rearrange this,"},{"Start":"05:01.550 ","End":"05:08.070","Text":"and we get that the probability of B equals 0.6."},{"Start":"05:09.650 ","End":"05:15.345","Text":"Excellent. Having calculated P of B,"},{"Start":"05:15.345 ","End":"05:18.435","Text":"let\u0027s try to calculate P of A."},{"Start":"05:18.435 ","End":"05:26.035","Text":"Well, we know that P of A equals 0.3 plus 0.6 P of B from here."},{"Start":"05:26.035 ","End":"05:34.530","Text":"So that would be 0.3 plus 0.6 times P of B, and that\u0027s 0.6."},{"Start":"05:34.720 ","End":"05:44.930","Text":"P of A equals 0.66."},{"Start":"05:44.930 ","End":"05:51.600","Text":"The last thing that we have to calculate is the probability of A intersect B."},{"Start":"05:51.600 ","End":"05:54.100","Text":"We get that from here."},{"Start":"05:54.100 ","End":"06:03.975","Text":"So the probability of A intersect B is equal to 0.6 times P of B."},{"Start":"06:03.975 ","End":"06:06.580","Text":"Now that\u0027s 0.6."},{"Start":"06:09.470 ","End":"06:17.130","Text":"That would equal 0.36."},{"Start":"06:17.130 ","End":"06:20.390","Text":"We\u0027ve calculated all 3 variables."},{"Start":"06:20.390 ","End":"06:24.620","Text":"Now let\u0027s see if they are equal to each other."},{"Start":"06:24.620 ","End":"06:29.830","Text":"The probability of A intersect B,"},{"Start":"06:30.140 ","End":"06:39.610","Text":"is that really equal to the probability of A times the probability of B?"},{"Start":"06:39.610 ","End":"06:44.285","Text":"Well, P of A intersect B is 0.36."},{"Start":"06:44.285 ","End":"06:46.340","Text":"We get that from here."},{"Start":"06:46.340 ","End":"06:51.035","Text":"P of A is 0.66,"},{"Start":"06:51.035 ","End":"06:55.020","Text":"P of B is 0.6."},{"Start":"06:55.240 ","End":"06:59.250","Text":"We see that it\u0027s not."},{"Start":"07:02.320 ","End":"07:08.640","Text":"That means that A and B are dependent."}],"ID":12936},{"Watched":false,"Name":"Exercise 7","Duration":"1m 55s","ChapterTopicVideoID":12458,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.050","Text":"In this exercise, we\u0027re asked to prove that if"},{"Start":"00:04.050 ","End":"00:08.655","Text":"the probability of A given B equals the probability of B given A,"},{"Start":"00:08.655 ","End":"00:13.410","Text":"then the probability of A equals the probability of B."},{"Start":"00:13.410 ","End":"00:17.580","Text":"Let\u0027s get started. By definition,"},{"Start":"00:17.580 ","End":"00:26.535","Text":"we know that the probability of A given B equals the probability of the intersect,"},{"Start":"00:26.535 ","End":"00:28.005","Text":"A intersect B,"},{"Start":"00:28.005 ","End":"00:31.980","Text":"divided by the probability of B."},{"Start":"00:31.980 ","End":"00:40.980","Text":"Likewise, the probability of B given A equals the probability of the intersect again,"},{"Start":"00:40.980 ","End":"00:42.630","Text":"A intersect B,"},{"Start":"00:42.630 ","End":"00:47.900","Text":"but now we divide by the probability of A."},{"Start":"00:47.900 ","End":"00:54.630","Text":"We\u0027re given that the 2 conditional probabilities are equal, right here."},{"Start":"00:56.600 ","End":"00:59.545","Text":"We can write this down."},{"Start":"00:59.545 ","End":"01:06.200","Text":"We can write that the probability of A intersect B divided by"},{"Start":"01:06.200 ","End":"01:14.075","Text":"the probability of B must equal the probability of A intersect B again,"},{"Start":"01:14.075 ","End":"01:17.615","Text":"divided by the probability of A."},{"Start":"01:17.615 ","End":"01:25.805","Text":"Let\u0027s cross multiply probability of A times probability of"},{"Start":"01:25.805 ","End":"01:29.900","Text":"A intersect B equals"},{"Start":"01:29.900 ","End":"01:37.305","Text":"the probability of B times the probability of A intersect B."},{"Start":"01:37.305 ","End":"01:42.285","Text":"Let\u0027s get rid of the common factor."},{"Start":"01:42.285 ","End":"01:48.050","Text":"What we have here is the probability of A being equal to"},{"Start":"01:48.050 ","End":"01:54.600","Text":"the probability of B and this is what we\u0027re asked to prove."}],"ID":12937},{"Watched":false,"Name":"Exercise 8 - Parts a-c","Duration":"5m 21s","ChapterTopicVideoID":12459,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"In this question, we\u0027re asked to determine which of the following assertions are correct."},{"Start":"00:05.535 ","End":"00:08.500","Text":"Well, let\u0027s start with a."},{"Start":"00:08.900 ","End":"00:13.320","Text":"A states that if the probability of the union of A and"},{"Start":"00:13.320 ","End":"00:16.830","Text":"B equal the multiplication of the probabilities,"},{"Start":"00:16.830 ","End":"00:20.100","Text":"the probability of A times the probability of B,"},{"Start":"00:20.100 ","End":"00:22.125","Text":"then the events are independent."},{"Start":"00:22.125 ","End":"00:27.180","Text":"Well, we know that\u0027s obviously not true because the definition of"},{"Start":"00:27.180 ","End":"00:33.720","Text":"independence means that the probability of A intersect B,"},{"Start":"00:33.720 ","End":"00:41.290","Text":"equals the multiplication of A times the multiplication of B."},{"Start":"00:41.720 ","End":"00:50.155","Text":"That is obviously not equal to the probability of the union."},{"Start":"00:50.155 ","End":"00:56.785","Text":"The probability of the intersect does not equal the probability of the union."},{"Start":"00:56.785 ","End":"01:01.250","Text":"Therefore, assertion a is not true."},{"Start":"01:01.380 ","End":"01:07.840","Text":"The next section gives us that event A is a subset of event B."},{"Start":"01:07.840 ","End":"01:11.320","Text":"If the probability of A is greater than 0,"},{"Start":"01:11.320 ","End":"01:15.095","Text":"the probability of B is between 0 and 1."},{"Start":"01:15.095 ","End":"01:19.450","Text":"We want to know whether the assertion that"},{"Start":"01:19.450 ","End":"01:25.520","Text":"the probability of A given B is less than the probability of A."},{"Start":"01:25.800 ","End":"01:29.330","Text":"Well, let\u0027s see."},{"Start":"01:29.330 ","End":"01:34.655","Text":"What\u0027s the definition of the probability of A given B?"},{"Start":"01:34.655 ","End":"01:39.285","Text":"Probability of A given B, by definition,"},{"Start":"01:39.285 ","End":"01:47.550","Text":"is the probability of A intersect B divided by probability of B."},{"Start":"01:48.020 ","End":"01:56.845","Text":"We\u0027re given that A is a subset of B, what does that mean?"},{"Start":"01:56.845 ","End":"02:00.665","Text":"Well, if we look at the Venn diagram right here,"},{"Start":"02:00.665 ","End":"02:03.500","Text":"we can see exactly what that means."},{"Start":"02:03.500 ","End":"02:06.690","Text":"A is totally a subset of B."},{"Start":"02:06.690 ","End":"02:16.460","Text":"That means that the probability of A intersect B is actually just the probability of A."},{"Start":"02:17.280 ","End":"02:23.275","Text":"To complete the conditional probability of A given B,"},{"Start":"02:23.275 ","End":"02:30.400","Text":"that means that that would equal to the probability of A divided by the probability of B."},{"Start":"02:30.400 ","End":"02:37.330","Text":"Now, we want to know whether this is less than the probability of A."},{"Start":"02:37.330 ","End":"02:42.489","Text":"Now, let\u0027s write down the probability of A as a fraction."},{"Start":"02:42.489 ","End":"02:46.790","Text":"Well, if we write it like this,"},{"Start":"02:46.790 ","End":"02:54.700","Text":"that equals to probability of A. Probability of A/1 equals the probability of A."},{"Start":"02:54.700 ","End":"03:01.054","Text":"Now, when we have 2 fractions where they have the same numerator,"},{"Start":"03:01.054 ","End":"03:04.880","Text":"probability of A, but different denominators."},{"Start":"03:04.880 ","End":"03:07.370","Text":"Here we have 1,"},{"Start":"03:07.370 ","End":"03:11.000","Text":"and here we have the probability of B."},{"Start":"03:11.000 ","End":"03:15.620","Text":"Now, 1 is obviously larger than the probability of B,"},{"Start":"03:15.620 ","End":"03:22.200","Text":"which makes this fraction much less than this fraction."},{"Start":"03:22.820 ","End":"03:28.750","Text":"Which means that assertion b, is not true."},{"Start":"03:29.560 ","End":"03:37.220","Text":"The conditional probability is greater than the probability of A, not less."},{"Start":"03:37.220 ","End":"03:39.605","Text":"In section c,"},{"Start":"03:39.605 ","End":"03:45.365","Text":"we\u0027re told that A and B are mutually exclusive events with positive chances of occurring."},{"Start":"03:45.365 ","End":"03:51.475","Text":"The assertion is that they are dependent events. Well, let\u0027s see."},{"Start":"03:51.475 ","End":"04:00.680","Text":"First of all, we\u0027re given that A and B are mutually exclusive. What does that mean?"},{"Start":"04:00.680 ","End":"04:09.080","Text":"That means that the probability of A intersect B must equal 0. What else are we told?"},{"Start":"04:09.080 ","End":"04:14.210","Text":"We\u0027re told that both events have a positive chance of occurring."},{"Start":"04:14.210 ","End":"04:19.130","Text":"That means that the probability of A is greater than"},{"Start":"04:19.130 ","End":"04:24.215","Text":"0 and also the probability of B is greater than 0."},{"Start":"04:24.215 ","End":"04:30.005","Text":"We want to find out whether they\u0027re dependent or independent events."},{"Start":"04:30.005 ","End":"04:34.290","Text":"Well, if they\u0027re independent events,"},{"Start":"04:34.870 ","End":"04:40.295","Text":"then the probability of A intersect B"},{"Start":"04:40.295 ","End":"04:46.620","Text":"must equal the probability of A times the probability of B."},{"Start":"04:46.690 ","End":"04:52.415","Text":"We\u0027re given that the probability of A intersect B is 0."},{"Start":"04:52.415 ","End":"04:58.490","Text":"We\u0027re also given that the probabilities of both A and B are greater than 0,"},{"Start":"04:58.490 ","End":"05:05.480","Text":"that means that the multiplication of those probabilities are greater than 0."},{"Start":"05:05.480 ","End":"05:12.210","Text":"That means that the equation for independence doesn\u0027t hold up."},{"Start":"05:12.520 ","End":"05:20.370","Text":"If that\u0027s the case, then A and B are dependent."}],"ID":12938},{"Watched":false,"Name":"Exercise 8 - Parts d-e","Duration":"4m 41s","ChapterTopicVideoID":12460,"CourseChapterTopicPlaylistID":245034,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"Section D, we\u0027re given that A and B are dependent events,"},{"Start":"00:03.990 ","End":"00:06.210","Text":"with a positive chances of occurring."},{"Start":"00:06.210 ","End":"00:11.370","Text":"The assertion is that A and B are mutually exclusive events."},{"Start":"00:11.370 ","End":"00:14.190","Text":"Well, let\u0027s see if that\u0027s true."},{"Start":"00:14.190 ","End":"00:19.890","Text":"The first thing that we\u0027re given is that A and B are dependent events."},{"Start":"00:19.890 ","End":"00:25.500","Text":"All that means, is that probability of"},{"Start":"00:25.500 ","End":"00:32.250","Text":"A given B does not equal the probability of A."},{"Start":"00:32.250 ","End":"00:40.290","Text":"We\u0027re also given that both A and B have positive chances of occurring."},{"Start":"00:40.290 ","End":"00:44.900","Text":"All that means is that the probability of A is greater than"},{"Start":"00:44.900 ","End":"00:50.030","Text":"0 and the probability of B is also greater than 0."},{"Start":"00:50.030 ","End":"00:54.035","Text":"Now, let\u0027s get back to our conditional probability."},{"Start":"00:54.035 ","End":"01:00.530","Text":"If the probability of A given B does not equal to the probability of A only,"},{"Start":"01:00.530 ","End":"01:08.655","Text":"does that imply that the probability of A intersect B equals 0?"},{"Start":"01:08.655 ","End":"01:10.725","Text":"No, it doesn\u0027t apply."},{"Start":"01:10.725 ","End":"01:12.970","Text":"Let\u0027s give an example."},{"Start":"01:14.030 ","End":"01:16.440","Text":"The example is this,"},{"Start":"01:16.440 ","End":"01:21.330","Text":"assume that the probability of A is 0.1,"},{"Start":"01:21.330 ","End":"01:26.280","Text":"the probability of B is 0.3."},{"Start":"01:26.280 ","End":"01:35.035","Text":"Let\u0027s assume probability of A intersect B equals 0.05."},{"Start":"01:35.035 ","End":"01:39.245","Text":"If A and B would be mutually exclusive,"},{"Start":"01:39.245 ","End":"01:41.435","Text":"then the intercept would be 0."},{"Start":"01:41.435 ","End":"01:42.950","Text":"But we\u0027re assuming the opposite."},{"Start":"01:42.950 ","End":"01:49.430","Text":"We\u0027re assuming that the probability is a positive number and it\u0027s 0.05."},{"Start":"01:49.430 ","End":"01:53.420","Text":"Therefore, let\u0027s multiply this out,"},{"Start":"01:53.420 ","End":"02:01.085","Text":"0.05, does that equal to 0.1 times 0.3?"},{"Start":"02:01.085 ","End":"02:03.565","Text":"Well, obviously it doesn\u0027t,"},{"Start":"02:03.565 ","End":"02:08.285","Text":"that equals 0.03. But what does this show?"},{"Start":"02:08.285 ","End":"02:11.360","Text":"This shows that on 1 hand,"},{"Start":"02:11.360 ","End":"02:17.154","Text":"A and B are dependent."},{"Start":"02:17.154 ","End":"02:19.310","Text":"B, what does it mean?"},{"Start":"02:19.310 ","End":"02:24.094","Text":"It means that they are not mutually exclusive."},{"Start":"02:24.094 ","End":"02:27.845","Text":"Therefore, the assertion is not true."},{"Start":"02:27.845 ","End":"02:32.465","Text":"In Section E, we\u0027re given that the probability of not"},{"Start":"02:32.465 ","End":"02:38.330","Text":"A intersect not B equals 1 minus the probability of A minus the probability of B."},{"Start":"02:38.330 ","End":"02:40.190","Text":"If that\u0027s the case,"},{"Start":"02:40.190 ","End":"02:42.800","Text":"then the assertion is that A and B are"},{"Start":"02:42.800 ","End":"02:46.850","Text":"mutually exclusive events. Well, let\u0027s check this out."},{"Start":"02:46.850 ","End":"02:49.525","Text":"According to De Morgan\u0027s rule,"},{"Start":"02:49.525 ","End":"02:55.340","Text":"we know the probability of A union B equals"},{"Start":"02:55.340 ","End":"03:01.610","Text":"1 minus the probability of not A intersect not B."},{"Start":"03:01.610 ","End":"03:08.300","Text":"We also know that the probability of A union B equals the probability of"},{"Start":"03:08.300 ","End":"03:16.805","Text":"A plus the probability of B minus the probability of A intersect B."},{"Start":"03:16.805 ","End":"03:20.435","Text":"Now since these 2 equations are equal,"},{"Start":"03:20.435 ","End":"03:27.965","Text":"let\u0027s write this out, 1 minus the probability of not A intersect not B."},{"Start":"03:27.965 ","End":"03:32.375","Text":"That would equal to probability of A plus"},{"Start":"03:32.375 ","End":"03:38.820","Text":"the probability of B minus the probability of A intersect B."},{"Start":"03:39.140 ","End":"03:42.910","Text":"Now, let\u0027s rewrite this."},{"Start":"03:42.910 ","End":"03:48.070","Text":"Probability of not A intersect not B."},{"Start":"03:48.070 ","End":"03:53.870","Text":"That would equal to 1 minus probability of A minus"},{"Start":"03:53.870 ","End":"04:00.890","Text":"the probability of B plus probability of A intersect B."},{"Start":"04:00.890 ","End":"04:04.350","Text":"Now, in order for"},{"Start":"04:04.350 ","End":"04:12.180","Text":"this equation to be equal to this equation because that\u0027s what\u0027s given,"},{"Start":"04:12.180 ","End":"04:19.570","Text":"then this factor right here needs to equal to 0."},{"Start":"04:19.570 ","End":"04:26.205","Text":"Now, if the probability of A intersect B equals 0,"},{"Start":"04:26.205 ","End":"04:34.390","Text":"that means that A and B are mutually exclusive."},{"Start":"04:34.990 ","End":"04:40.410","Text":"That means that the assertion is true."}],"ID":12939}],"Thumbnail":null,"ID":245034},{"Name":"Summary Questions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"37s","ChapterTopicVideoID":12461,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.790","Text":"In this chapter, we have a number of problems to summarize the subject of probability."},{"Start":"00:05.790 ","End":"00:09.210","Text":"Now I want to give you a number of these problems that really"},{"Start":"00:09.210 ","End":"00:13.080","Text":"summarize the subjects that we\u0027ve been learning these past chapters,"},{"Start":"00:13.080 ","End":"00:15.285","Text":"such as operations and events,"},{"Start":"00:15.285 ","End":"00:18.330","Text":"conditional probabilities, probability trees,"},{"Start":"00:18.330 ","End":"00:21.084","Text":"dependencies between events, and more."},{"Start":"00:21.084 ","End":"00:24.900","Text":"That\u0027s important for you to deal with these problems that have"},{"Start":"00:24.900 ","End":"00:29.790","Text":"a mix of these subjects so you\u0027ll be able to do well on the exams."},{"Start":"00:29.790 ","End":"00:33.180","Text":"Let\u0027s go. Please try to solve the problems by yourself"},{"Start":"00:33.180 ","End":"00:37.660","Text":"before you see the solution videos. Good luck."}],"ID":12940},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"3m 57s","ChapterTopicVideoID":12462,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"In this question, we\u0027re given a sample space that consists of families with 2 cars."},{"Start":"00:05.550 ","End":"00:07.725","Text":"In 30 percent of these families,"},{"Start":"00:07.725 ","End":"00:09.930","Text":"the older car is manufactured in Europe,"},{"Start":"00:09.930 ","End":"00:12.135","Text":"and in 60 percent of these families,"},{"Start":"00:12.135 ","End":"00:15.030","Text":"the newer car is manufactured in Europe."},{"Start":"00:15.030 ","End":"00:17.160","Text":"In 15 percent of these families,"},{"Start":"00:17.160 ","End":"00:19.590","Text":"both cards are manufactured in Europe."},{"Start":"00:19.590 ","End":"00:24.000","Text":"We\u0027re asked, what\u0027s the probability that a family chosen at"},{"Start":"00:24.000 ","End":"00:28.545","Text":"random from the sample space will not have any car manufactured in Europe."},{"Start":"00:28.545 ","End":"00:31.395","Text":"Now before we can solve this problem,"},{"Start":"00:31.395 ","End":"00:38.680","Text":"let\u0027s define some variables and write down the information that\u0027s provided for us."},{"Start":"00:38.960 ","End":"00:46.015","Text":"Let\u0027s define A as the families that have an old European car,"},{"Start":"00:46.015 ","End":"00:52.890","Text":"and B as families having a new European car."},{"Start":"00:52.890 ","End":"01:00.080","Text":"Let\u0027s also define the probability of A that\u0027s given to us,"},{"Start":"01:00.080 ","End":"01:02.300","Text":"and it\u0027s 30 percent."},{"Start":"01:02.300 ","End":"01:08.410","Text":"The probability of B is 60 percent,"},{"Start":"01:08.410 ","End":"01:13.250","Text":"and the probability of A intersect B,"},{"Start":"01:13.250 ","End":"01:18.695","Text":"that means the probability of families having both an old and a new car,"},{"Start":"01:18.695 ","End":"01:22.650","Text":"European car is 15 percent."},{"Start":"01:25.990 ","End":"01:32.570","Text":"I know these probabilities because they\u0027ve been given to me here,"},{"Start":"01:32.570 ","End":"01:36.355","Text":"here, and here."},{"Start":"01:36.355 ","End":"01:42.980","Text":"Now, what I want to do is I want to identify"},{"Start":"01:42.980 ","End":"01:50.075","Text":"not only these probabilities but all other probabilities that I may or may not use."},{"Start":"01:50.075 ","End":"01:55.115","Text":"For that, I want to define a probability matrix."},{"Start":"01:55.115 ","End":"01:58.925","Text":"Here it is. What do I see?"},{"Start":"01:58.925 ","End":"02:05.865","Text":"I see here the events A and not A and B and not B."},{"Start":"02:05.865 ","End":"02:11.240","Text":"A being families that have an old European car,"},{"Start":"02:11.240 ","End":"02:16.160","Text":"not A are families that do not have an old European car."},{"Start":"02:16.160 ","End":"02:21.890","Text":"Likewise, B families that have a new European car here,"},{"Start":"02:21.890 ","End":"02:27.200","Text":"and not B as families that do not have a new European car."},{"Start":"02:27.200 ","End":"02:33.570","Text":"Now I know that all the probabilities here have to add up to 1."},{"Start":"02:33.580 ","End":"02:37.520","Text":"I\u0027ve also been given these probabilities,"},{"Start":"02:37.520 ","End":"02:41.200","Text":"30 percent, 60 percent, and 15 percent."},{"Start":"02:41.200 ","End":"02:44.970","Text":"You can see them here in blue."},{"Start":"02:44.970 ","End":"02:48.950","Text":"Now, once I have these probabilities,"},{"Start":"02:48.950 ","End":"02:53.450","Text":"it\u0027s very easy for me to fill out all the matrix."},{"Start":"02:53.450 ","End":"02:59.660","Text":"Here you can see the probabilities that had been written in red."},{"Start":"02:59.660 ","End":"03:02.665","Text":"Now, let\u0027s get back to our question."},{"Start":"03:02.665 ","End":"03:05.920","Text":"What were we asked?"},{"Start":"03:05.920 ","End":"03:09.980","Text":"We were asked, what is the probability that a family chosen at"},{"Start":"03:09.980 ","End":"03:15.350","Text":"random from a sample space will not have any car manufactured in Europe?"},{"Start":"03:15.350 ","End":"03:20.300","Text":"Basically, what they\u0027re saying is they want to find out what is"},{"Start":"03:20.300 ","End":"03:27.335","Text":"the probability of not A and not B."},{"Start":"03:27.335 ","End":"03:32.450","Text":"Well, let\u0027s go to our probability matrix."},{"Start":"03:32.450 ","End":"03:34.355","Text":"Let\u0027s go to not A."},{"Start":"03:34.355 ","End":"03:36.565","Text":"This is the column not A,"},{"Start":"03:36.565 ","End":"03:40.290","Text":"and this is the row not B."},{"Start":"03:40.290 ","End":"03:47.890","Text":"The intersection of not A and not B is 0.25."},{"Start":"03:48.800 ","End":"03:57.450","Text":"That is the probability of a family not having any car manufactured in Europe."}],"ID":12941},{"Watched":false,"Name":"Exercise 1 - Parts b-c","Duration":"5m 54s","ChapterTopicVideoID":12464,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"In this question, we\u0027re asked,"},{"Start":"00:01.890 ","End":"00:06.960","Text":"what\u0027s the probability that at least 1 car will be manufactured in Europe?"},{"Start":"00:06.960 ","End":"00:15.675","Text":"Well, we can translate that into the probability of A union B."},{"Start":"00:15.675 ","End":"00:19.770","Text":"Remember A are the families that have"},{"Start":"00:19.770 ","End":"00:25.320","Text":"an old European car and B are the families that have a new European car."},{"Start":"00:25.320 ","End":"00:29.640","Text":"Now, if we use the De Morgan\u0027s rule,"},{"Start":"00:29.640 ","End":"00:31.170","Text":"it will be much easier."},{"Start":"00:31.170 ","End":"00:35.850","Text":"Because to calculate the probability that at least 1 car would be"},{"Start":"00:35.850 ","End":"00:40.780","Text":"manufactured in Europe is the same as saying let\u0027s"},{"Start":"00:40.780 ","End":"00:45.620","Text":"calculate the probability that no car will be"},{"Start":"00:45.620 ","End":"00:52.520","Text":"manufactured in Europe and subtract that from 1."},{"Start":"00:52.520 ","End":"00:55.460","Text":"Now this is De Morgan\u0027s rule."},{"Start":"00:55.460 ","End":"01:00.215","Text":"If we go back to our probability matrix,"},{"Start":"01:00.215 ","End":"01:09.145","Text":"we can see that the probability of not A intersect not B is again 25 percent."},{"Start":"01:09.145 ","End":"01:13.830","Text":"That will be equal to 1 minus 0.25,"},{"Start":"01:14.030 ","End":"01:18.505","Text":"which is equal to 0.75."},{"Start":"01:18.505 ","End":"01:21.950","Text":"That means that 75 percent of"},{"Start":"01:21.950 ","End":"01:28.055","Text":"the families will have at least 1 car that will be manufactured in Europe."},{"Start":"01:28.055 ","End":"01:34.220","Text":"In Section c, we\u0027re given that a family has a car that was manufactured in Europe."},{"Start":"01:34.220 ","End":"01:39.500","Text":"We\u0027re asked, what\u0027s the probability that it would be the newer one?"},{"Start":"01:39.500 ","End":"01:42.665","Text":"Well, when looking at this question,"},{"Start":"01:42.665 ","End":"01:45.905","Text":"we see that we have the phrase, given that."},{"Start":"01:45.905 ","End":"01:52.195","Text":"That automatically suggests that we\u0027re dealing with conditional probabilities."},{"Start":"01:52.195 ","End":"01:54.695","Text":"We\u0027re dealing with conditional probabilities,"},{"Start":"01:54.695 ","End":"01:57.440","Text":"we have to identify the events."},{"Start":"01:57.440 ","End":"02:02.770","Text":"The first event is that a family has a car that was manufactured in Europe."},{"Start":"02:02.770 ","End":"02:08.715","Text":"That means it\u0027s either A or it\u0027s B."},{"Start":"02:08.715 ","End":"02:14.290","Text":"We want to know what is the probability of this event A union B."},{"Start":"02:14.290 ","End":"02:19.320","Text":"Now, we\u0027ve already encountered this probability in section b."},{"Start":"02:19.320 ","End":"02:25.750","Text":"It equals to 1 minus the probability of not A intersect not B."},{"Start":"02:25.750 ","End":"02:29.590","Text":"We calculated that to be 75 percent."},{"Start":"02:29.590 ","End":"02:33.400","Text":"The second event is the fact that we want"},{"Start":"02:33.400 ","End":"02:36.940","Text":"the car to be the newer one, not the older one."},{"Start":"02:36.940 ","End":"02:39.925","Text":"That means that the event is not A,"},{"Start":"02:39.925 ","End":"02:41.515","Text":"not the old car."},{"Start":"02:41.515 ","End":"02:44.155","Text":"We do want this to be the new car."},{"Start":"02:44.155 ","End":"02:49.655","Text":"What\u0027s the probability of not A intersect B?"},{"Start":"02:49.655 ","End":"02:52.880","Text":"Now we have the 2 events,"},{"Start":"02:52.880 ","End":"02:55.235","Text":"this event and this event,"},{"Start":"02:55.235 ","End":"02:58.310","Text":"that comprise the conditional probability."},{"Start":"02:58.310 ","End":"03:04.850","Text":"Now, the numerator of the conditional probability is the intersect of"},{"Start":"03:04.850 ","End":"03:12.485","Text":"these 2 events and the denominator is the probability of the known event."},{"Start":"03:12.485 ","End":"03:17.375","Text":"Let\u0027s write this out. The conditional probability"},{"Start":"03:17.375 ","End":"03:21.285","Text":"is the intersect of these 2 events as we said."},{"Start":"03:21.285 ","End":"03:30.640","Text":"That\u0027s not A intersect B. Intersect A union B."},{"Start":"03:32.060 ","End":"03:35.585","Text":"The denominator, as we said,"},{"Start":"03:35.585 ","End":"03:38.100","Text":"is A union B."},{"Start":"03:38.100 ","End":"03:41.540","Text":"Now, as we said earlier,"},{"Start":"03:41.540 ","End":"03:45.065","Text":"we\u0027ve already calculated this to be 75 percent."},{"Start":"03:45.065 ","End":"03:47.965","Text":"Now we have to deal with the numerator."},{"Start":"03:47.965 ","End":"03:50.870","Text":"How do we calculate this?"},{"Start":"03:50.870 ","End":"03:56.675","Text":"Well, let\u0027s draw out a Venn diagram and see what we can find out."},{"Start":"03:56.675 ","End":"03:59.050","Text":"Here\u0027s our Venn diagram."},{"Start":"03:59.050 ","End":"04:05.190","Text":"Now, let\u0027s look at A union B."},{"Start":"04:05.190 ","End":"04:13.545","Text":"A union B, in essence is all the area in yellow."},{"Start":"04:13.545 ","End":"04:17.260","Text":"That\u0027s A and that\u0027s B."},{"Start":"04:19.820 ","End":"04:22.560","Text":"That\u0027s this guy right here."},{"Start":"04:22.560 ","End":"04:24.480","Text":"Now what about this guy?"},{"Start":"04:24.480 ","End":"04:28.520","Text":"Not A intersect B?"},{"Start":"04:28.520 ","End":"04:35.010","Text":"Not A intersect B is this guy right here."},{"Start":"04:35.010 ","End":"04:37.900","Text":"The area in green."},{"Start":"04:40.460 ","End":"04:45.740","Text":"We can write out the numerator in this way."},{"Start":"04:45.740 ","End":"04:51.395","Text":"The probability of not A intersect B. Intersect"},{"Start":"04:51.395 ","End":"04:58.705","Text":"A union B is basically just the probability of not A intersect B."},{"Start":"04:58.705 ","End":"05:04.325","Text":"We divide that by the probability of A union B."},{"Start":"05:04.325 ","End":"05:09.200","Text":"Now the denominator is 75 percent."},{"Start":"05:09.200 ","End":"05:11.525","Text":"What about the numerator?"},{"Start":"05:11.525 ","End":"05:15.830","Text":"Well, let\u0027s get back to a probability matrix and see if we"},{"Start":"05:15.830 ","End":"05:20.600","Text":"can identify the probability of not A intersect B."},{"Start":"05:20.600 ","End":"05:27.600","Text":"Well, here\u0027s the not A column and here\u0027s the B column."},{"Start":"05:27.650 ","End":"05:37.830","Text":"That means that the intersect of not A and B is right here, and that\u0027s 0.45."},{"Start":"05:37.830 ","End":"05:41.375","Text":"In this ratio, the conditional probability,"},{"Start":"05:41.375 ","End":"05:44.675","Text":"turns out to be 60 percent."},{"Start":"05:44.675 ","End":"05:51.620","Text":"That\u0027s the probability of a family having a European car."},{"Start":"05:51.620 ","End":"05:54.000","Text":"That\u0027s a new car."}],"ID":12942},{"Watched":false,"Name":"Exercise 1 - Part d","Duration":"1m 3s","ChapterTopicVideoID":12463,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In Section D, we\u0027re asked,"},{"Start":"00:02.580 ","End":"00:05.580","Text":"if a family\u0027s older car is manufactured in Europe,"},{"Start":"00:05.580 ","End":"00:10.200","Text":"what\u0027s the probability that their newer car is also manufactured in Europe?"},{"Start":"00:10.200 ","End":"00:12.285","Text":"Well, just like in Section C,"},{"Start":"00:12.285 ","End":"00:15.240","Text":"we\u0027re dealing here with conditional probabilities."},{"Start":"00:15.240 ","End":"00:22.500","Text":"Basically, what we\u0027re being asked is what is the probability of B given A?"},{"Start":"00:22.500 ","End":"00:27.510","Text":"By definition, that\u0027s the probability of"},{"Start":"00:27.510 ","End":"00:34.380","Text":"A intersect B divided by the probability of A."},{"Start":"00:34.380 ","End":"00:39.150","Text":"Now, if we look at the probability matrix,"},{"Start":"00:39.150 ","End":"00:43.020","Text":"we can see that the probability of A is"},{"Start":"00:43.020 ","End":"00:52.810","Text":"0.3 and the probability of A intersect B is 0.15."},{"Start":"00:53.660 ","End":"01:03.060","Text":"That means that we have a conditional probability of 1/2 or 50 percent."}],"ID":12943},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"5m 11s","ChapterTopicVideoID":12465,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"In this question, we\u0027ll be dealing with sour milk."},{"Start":"00:03.675 ","End":"00:10.290","Text":"In Never-Never-Land, 50% of the milk in supermarkets is produced at company A,"},{"Start":"00:10.290 ","End":"00:12.780","Text":"40% at company B,"},{"Start":"00:12.780 ","End":"00:17.610","Text":"and the rest at company C. 3% of the milk produced at company A and"},{"Start":"00:17.610 ","End":"00:23.270","Text":"10% of the milk produced at company B is sour when it reaches the supermarkets."},{"Start":"00:23.270 ","End":"00:30.020","Text":"It\u0027s also known that 7.5% of all milk at Never-Never-Land supermarkets is sour."},{"Start":"00:30.020 ","End":"00:35.345","Text":"Now, before we try to solve any questions,"},{"Start":"00:35.345 ","End":"00:41.465","Text":"let\u0027s see if we can present the data in such a way that it\u0027ll help us."},{"Start":"00:41.465 ","End":"00:45.980","Text":"Now, how can we best present the data that\u0027s given to us?"},{"Start":"00:45.980 ","End":"00:51.617","Text":"Well, we can see that we have basically 2 levels,"},{"Start":"00:51.617 ","End":"00:56.495","Text":"the level of the companies and the level of the milk they produce,"},{"Start":"00:56.495 ","End":"01:01.730","Text":"and this suggests that we\u0027re dealing with conditional probabilities."},{"Start":"01:01.730 ","End":"01:09.765","Text":"As such, the best way to display this type of information is on a probability tree."},{"Start":"01:09.765 ","End":"01:16.485","Text":"Well, let\u0027s draw out a probability tree. Here we go."},{"Start":"01:16.485 ","End":"01:22.070","Text":"Here we can see basically all the information that\u0027s provided to us."},{"Start":"01:22.070 ","End":"01:28.375","Text":"We can see company A that produces 50% of the milk,"},{"Start":"01:28.375 ","End":"01:35.705","Text":"company B and company C producing 40% and 10% respectively."},{"Start":"01:35.705 ","End":"01:44.015","Text":"Also, we\u0027re given that company A produces 3% sour milk,"},{"Start":"01:44.015 ","End":"01:47.780","Text":"the company produces 10% sour milk and there\u0027s"},{"Start":"01:47.780 ","End":"01:51.780","Text":"no information about the quality of the milk at"},{"Start":"01:51.780 ","End":"01:55.730","Text":"company C. But I do believe we have"},{"Start":"01:55.730 ","End":"02:01.860","Text":"enough information to try to solve or complete the probabilities on this tree."},{"Start":"02:01.910 ","End":"02:05.445","Text":"That\u0027s the first thing that we need to do."},{"Start":"02:05.445 ","End":"02:08.080","Text":"Well, let\u0027s look at company A."},{"Start":"02:08.080 ","End":"02:10.955","Text":"If they produce 3% of the sour milk,"},{"Start":"02:10.955 ","End":"02:17.130","Text":"that means that they produce 97% fresh milk."},{"Start":"02:17.380 ","End":"02:26.700","Text":"Company B produces 10% sour milk that means that they produce 90% fresh milk."},{"Start":"02:26.700 ","End":"02:32.560","Text":"Again, we have no knowledge of the quality of the milk at company C,"},{"Start":"02:32.560 ","End":"02:36.590","Text":"but we do have something else."},{"Start":"02:36.800 ","End":"02:44.940","Text":"We do know that 7.5% of all the milk at Never-Never-Land supermarkets is sour."},{"Start":"02:44.940 ","End":"02:48.460","Text":"Let\u0027s see if we can calculate the quality of"},{"Start":"02:48.460 ","End":"02:51.265","Text":"the milk with the probability of the sour milk,"},{"Start":"02:51.265 ","End":"02:57.790","Text":"more precisely, at company C. Well, let\u0027s do this."},{"Start":"02:58.890 ","End":"03:07.900","Text":"The proportion of sour milk at Never-Never-Land from company A is 50% times"},{"Start":"03:07.900 ","End":"03:15.540","Text":"3% and the proportion of sour milk at company B is"},{"Start":"03:15.540 ","End":"03:23.570","Text":"40% times 10% and"},{"Start":"03:23.570 ","End":"03:27.500","Text":"the proportion of sour milk at company C,"},{"Start":"03:27.500 ","End":"03:31.940","Text":"well we don\u0027t know that but we do know that company C produces"},{"Start":"03:31.940 ","End":"03:37.910","Text":"10% of the milk and let\u0027s write down P as the probability of sour milk."},{"Start":"03:37.910 ","End":"03:43.460","Text":"Now, we know that this equals 7.5%."},{"Start":"03:43.460 ","End":"03:46.130","Text":"It\u0027s given to us right here."},{"Start":"03:46.130 ","End":"03:49.260","Text":"Let\u0027s try to solve this."},{"Start":"03:50.140 ","End":"03:55.310","Text":"That equals to 0.015 plus"},{"Start":"03:55.310 ","End":"04:05.400","Text":"0.04 plus 0.1P and that all equals 0.075."},{"Start":"04:05.680 ","End":"04:13.760","Text":"Let\u0027s solve for P, 0.1P equals 0.02."},{"Start":"04:13.760 ","End":"04:20.640","Text":"Therefore, P equals 0.2 or 20%."},{"Start":"04:21.080 ","End":"04:24.160","Text":"We can write this down."},{"Start":"04:24.230 ","End":"04:33.360","Text":"Sour milk here is 20% so that means 80% is"},{"Start":"04:33.360 ","End":"04:38.220","Text":"fresh and now we\u0027ve"},{"Start":"04:38.220 ","End":"04:44.640","Text":"solved or calculated all the probabilities in the probability tree."},{"Start":"04:44.860 ","End":"04:48.470","Text":"We have all the information that we need right now."},{"Start":"04:48.470 ","End":"04:52.400","Text":"Let\u0027s try and solve question a."},{"Start":"04:52.400 ","End":"05:00.075","Text":"Question a asks, what\u0027s the proportion of milk from company C is sour? We\u0027ve solved that."},{"Start":"05:00.075 ","End":"05:10.320","Text":"It\u0027s right here. That means that 20% of the milk at company C is sour."}],"ID":12944},{"Watched":false,"Name":"Exercise 2 - Parts b-d","Duration":"5m 32s","ChapterTopicVideoID":12466,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.725","Text":"Section B asks,"},{"Start":"00:01.725 ","End":"00:04.965","Text":"if sour milk is purchased from a supermarket,"},{"Start":"00:04.965 ","End":"00:09.495","Text":"what is the chances that it was produced at company C?"},{"Start":"00:09.495 ","End":"00:13.710","Text":"Well, this looks like a conditional probability question."},{"Start":"00:13.710 ","End":"00:16.425","Text":"What do we have? What are we asked?"},{"Start":"00:16.425 ","End":"00:22.935","Text":"We\u0027re asked, what is the probability that given that the milk is sour,"},{"Start":"00:22.935 ","End":"00:27.720","Text":"what is the probability that it was produced at company C?"},{"Start":"00:27.720 ","End":"00:35.100","Text":"Well, that\u0027s the probability of milk being produced in company C and that the milk is"},{"Start":"00:35.100 ","End":"00:42.880","Text":"sour divided by the probability that the milk is sour."},{"Start":"00:43.090 ","End":"00:46.130","Text":"Now we have these probabilities."},{"Start":"00:46.130 ","End":"00:51.575","Text":"The probability that all milk is sour is 0.075."},{"Start":"00:51.575 ","End":"00:58.835","Text":"The probability of milk being sour and coming from a company C,"},{"Start":"00:58.835 ","End":"01:03.065","Text":"is this, 10 percent times 20 percent,"},{"Start":"01:03.065 ","End":"01:06.650","Text":"10 percent of the milk comes from company C. Out of that,"},{"Start":"01:06.650 ","End":"01:09.020","Text":"20 percent of the milk is sour."},{"Start":"01:09.020 ","End":"01:13.650","Text":"That\u0027s 0.1 times 0.2."},{"Start":"01:13.820 ","End":"01:21.000","Text":"That comes out to 0.02 over 0.075."},{"Start":"01:21.000 ","End":"01:27.210","Text":"That equals 0.267."},{"Start":"01:27.210 ","End":"01:34.335","Text":"The probability of the milk coming from company C,"},{"Start":"01:34.335 ","End":"01:39.405","Text":"given that it\u0027s sour, is 26.7 percent."},{"Start":"01:39.405 ","End":"01:45.200","Text":"In Section C, we were given that milk was purchased and was not sour."},{"Start":"01:45.200 ","End":"01:50.435","Text":"We were asked what are the chances that it was produced at company A."},{"Start":"01:50.435 ","End":"01:59.105","Text":"Well, this is a conditional probability stating that if milk was not sour,"},{"Start":"01:59.105 ","End":"02:04.080","Text":"what\u0027s the probability that it was produced at company A?"},{"Start":"02:04.450 ","End":"02:09.920","Text":"That equals to the probability of milk being produced at company A,"},{"Start":"02:09.920 ","End":"02:13.490","Text":"and it not being sour,"},{"Start":"02:13.490 ","End":"02:22.050","Text":"divided by the probability of all milk [inaudible] being not sour."},{"Start":"02:24.160 ","End":"02:33.580","Text":"The probability of all milk in [inaudible] being not sour is 1 minus 7.5 percent."},{"Start":"02:34.060 ","End":"02:37.940","Text":"The probability of milk being produced in"},{"Start":"02:37.940 ","End":"02:41.705","Text":"company A and not being sour, well, that\u0027s right here."},{"Start":"02:41.705 ","End":"02:48.330","Text":"That\u0027s 50 percent of the milk in [inaudible] being produced at company A and out of that,"},{"Start":"02:48.330 ","End":"02:52.425","Text":"97 percent of it being fresh milk or not sour."},{"Start":"02:52.425 ","End":"02:58.390","Text":"That\u0027s 0.5 times 0.97."},{"Start":"02:58.640 ","End":"03:04.890","Text":"That equals to 0.485"},{"Start":"03:04.890 ","End":"03:11.460","Text":"divided by 0.925."},{"Start":"03:11.460 ","End":"03:17.140","Text":"That comes out to 0.524."},{"Start":"03:17.140 ","End":"03:23.240","Text":"This is the probability of the milk being produced at company A if we"},{"Start":"03:23.240 ","End":"03:29.480","Text":"purchased it and it was not sour, 52.4 percent."},{"Start":"03:29.480 ","End":"03:31.670","Text":"Section D, asks,"},{"Start":"03:31.670 ","End":"03:37.520","Text":"whether the events sour milk and produced at company A are independent events."},{"Start":"03:37.520 ","End":"03:41.285","Text":"Well, let\u0027s define a as the milk being sour,"},{"Start":"03:41.285 ","End":"03:47.400","Text":"and b as the milk being produced at"},{"Start":"03:47.400 ","End":"03:54.790","Text":"company A. Probability of the milk being sour is 7.5 percent."},{"Start":"03:55.790 ","End":"04:03.690","Text":"The probability milk coming from company A that\u0027s here, that\u0027s 50 percent."},{"Start":"04:04.250 ","End":"04:08.175","Text":"Now in order for a and b to be independent,"},{"Start":"04:08.175 ","End":"04:13.160","Text":"then the probability of a intersect b"},{"Start":"04:13.160 ","End":"04:19.055","Text":"must equal the probability of a times the probability of b."},{"Start":"04:19.055 ","End":"04:26.005","Text":"That the probability of a intersect b means that we\u0027re looking for the probability"},{"Start":"04:26.005 ","End":"04:33.905","Text":"of milk being produced at company A and the probability of it being sour."},{"Start":"04:33.905 ","End":"04:41.225","Text":"That means that we\u0027re looking at the probability 0.5,"},{"Start":"04:41.225 ","End":"04:43.100","Text":"milk coming from company A,"},{"Start":"04:43.100 ","End":"04:48.785","Text":"times 0.03, the milking sour at company A."},{"Start":"04:48.785 ","End":"04:52.100","Text":"That equals 0.015."},{"Start":"04:52.100 ","End":"04:55.700","Text":"Now, let\u0027s check for independence."},{"Start":"04:55.700 ","End":"05:00.660","Text":"The probability of a intersect b is 0.015."},{"Start":"05:00.660 ","End":"05:11.700","Text":"The probability of a times the probability of b is 0.075 times 0.5,"},{"Start":"05:11.700 ","End":"05:16.125","Text":"which equals to 0.0375."},{"Start":"05:16.125 ","End":"05:20.890","Text":"We see that the 2 sides are not equal,"},{"Start":"05:20.890 ","End":"05:24.020","Text":"right here versus right here."},{"Start":"05:24.020 ","End":"05:29.340","Text":"That means that a and b are dependent."}],"ID":12945},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"5m 16s","ChapterTopicVideoID":12467,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"Here we have a dating problem."},{"Start":"00:03.225 ","End":"00:07.455","Text":"John and Betty want to spend an evening out at the mall."},{"Start":"00:07.455 ","End":"00:11.310","Text":"The probability that they will go bowling is 0.3."},{"Start":"00:11.310 ","End":"00:15.660","Text":"The probability that they will go to a cafe is 0.5."},{"Start":"00:15.660 ","End":"00:19.560","Text":"The probability that they will do at least 1 of these things,"},{"Start":"00:19.560 ","End":"00:24.060","Text":"either go bowling or visiting a cafe is 0.7."},{"Start":"00:24.060 ","End":"00:27.240","Text":"Now, before we get started,"},{"Start":"00:27.240 ","End":"00:29.670","Text":"let\u0027s write down everything that we know."},{"Start":"00:29.670 ","End":"00:35.560","Text":"Let\u0027s first of all, define A as the event of going bowling,"},{"Start":"00:36.230 ","End":"00:41.170","Text":"and B as the event of going to a cafe."},{"Start":"00:41.260 ","End":"00:46.670","Text":"Now, we\u0027re given that the probability of going bowling is"},{"Start":"00:46.670 ","End":"00:53.730","Text":"0.3 and the probability of going to a cafe is 0.5,"},{"Start":"00:54.280 ","End":"01:01.640","Text":"right here, going bowling at 0.3 and going to a cafe at 0.5."},{"Start":"01:01.640 ","End":"01:06.145","Text":"Now, we\u0027re also given that"},{"Start":"01:06.145 ","End":"01:12.215","Text":"the probability that they will do at least 1 of these things is 0.7."},{"Start":"01:12.215 ","End":"01:18.020","Text":"Now that means that they will either go bowling or go to a cafe."},{"Start":"01:18.020 ","End":"01:20.615","Text":"Now that probability, as I said,"},{"Start":"01:20.615 ","End":"01:24.390","Text":"is given to us at 0.7."},{"Start":"01:24.400 ","End":"01:28.550","Text":"Again, we can see that here."},{"Start":"01:28.550 ","End":"01:30.950","Text":"Now, before we continue,"},{"Start":"01:30.950 ","End":"01:34.775","Text":"I want to talk about how we can best describe the data."},{"Start":"01:34.775 ","End":"01:38.705","Text":"Now, we have basically 3 techniques."},{"Start":"01:38.705 ","End":"01:41.194","Text":"We have the Venn diagrams,"},{"Start":"01:41.194 ","End":"01:43.595","Text":"we have a probability matrix,"},{"Start":"01:43.595 ","End":"01:46.250","Text":"and we can use a probability tree."},{"Start":"01:46.250 ","End":"01:52.070","Text":"Now, out of the 3 probability tree is not relevant because we didn\u0027t"},{"Start":"01:52.070 ","End":"01:58.520","Text":"encounter any conditions nor hierarchies in the question, so that\u0027s out."},{"Start":"01:58.520 ","End":"02:01.910","Text":"Out of the 2 remaining techniques,"},{"Start":"02:01.910 ","End":"02:06.890","Text":"either Venn diagrams or probability matrix we can use both of them,"},{"Start":"02:06.890 ","End":"02:10.115","Text":"but I prefer using the probability matrix."},{"Start":"02:10.115 ","End":"02:14.870","Text":"Let\u0027s look at the probability matrix here."},{"Start":"02:14.870 ","End":"02:16.615","Text":"We have 2 events,"},{"Start":"02:16.615 ","End":"02:23.090","Text":"bowling and cafe, and we have various probabilities."},{"Start":"02:23.090 ","End":"02:29.645","Text":"Now, let\u0027s fill out the probability matrix with these probabilities."},{"Start":"02:29.645 ","End":"02:33.545","Text":"Now the probability of A is 0.3."},{"Start":"02:33.545 ","End":"02:36.610","Text":"That\u0027s right here, 0.3."},{"Start":"02:36.610 ","End":"02:44.655","Text":"The probability of B is 0.5, we write that here."},{"Start":"02:44.655 ","End":"02:49.770","Text":"The probability of A union B is 0.7."},{"Start":"02:49.770 ","End":"02:57.410","Text":"But hold on, I don\u0027t have any union signs between A and B in the probability matrix."},{"Start":"02:57.410 ","End":"03:00.290","Text":"They\u0027re all intersect sides."},{"Start":"03:00.290 ","End":"03:06.665","Text":"But we can easily convert this into a function of intersect and how do we do that?"},{"Start":"03:06.665 ","End":"03:10.989","Text":"Well, we use our famous equation."},{"Start":"03:10.989 ","End":"03:17.810","Text":"Probability of A union B equals probability of A plus"},{"Start":"03:17.810 ","End":"03:25.030","Text":"the probability of B minus the probability of A intersect B."},{"Start":"03:25.030 ","End":"03:27.465","Text":"Now let\u0027s plug in the numbers,"},{"Start":"03:27.465 ","End":"03:33.590","Text":"we have the probability of A union B, that\u0027s 0.7."},{"Start":"03:33.590 ","End":"03:37.055","Text":"That equals to the probability of A that\u0027s 0.3,"},{"Start":"03:37.055 ","End":"03:39.054","Text":"plus the probability of B,"},{"Start":"03:39.054 ","End":"03:45.185","Text":"that\u0027s 0.5, minus the probability of A intersect B."},{"Start":"03:45.185 ","End":"03:51.120","Text":"Solving for probability of A intersect B,"},{"Start":"03:52.000 ","End":"03:56.930","Text":"that comes out to 0.1."},{"Start":"03:56.930 ","End":"04:02.810","Text":"Let\u0027s put that in, that\u0027s 0.1.Now once I have this,"},{"Start":"04:02.810 ","End":"04:06.290","Text":"I can easily fill out all the other probabilities,"},{"Start":"04:06.290 ","End":"04:10.595","Text":"because all the probabilities have to add up to 1."},{"Start":"04:10.595 ","End":"04:16.805","Text":"This would be 0.5, this would be 0.2,"},{"Start":"04:16.805 ","End":"04:20.315","Text":"this would be 0.3,"},{"Start":"04:20.315 ","End":"04:25.650","Text":"this would be 0.4 and that\u0027s 0.7."},{"Start":"04:28.270 ","End":"04:33.034","Text":"Now that we have all the probabilities in our matrix,"},{"Start":"04:33.034 ","End":"04:36.365","Text":"let\u0027s try and solve question a."},{"Start":"04:36.365 ","End":"04:42.355","Text":"Question a asks, what\u0027s the probability that they will go only bowling?"},{"Start":"04:42.355 ","End":"04:47.825","Text":"That means they we\u0027re asking for the probability that they will go bowling,"},{"Start":"04:47.825 ","End":"04:52.340","Text":"but they will not go to a cafe."},{"Start":"04:52.340 ","End":"04:57.725","Text":"Now if we look at our matrix we have to go bowling,"},{"Start":"04:57.725 ","End":"05:03.260","Text":"what\u0027s the intersection between going bowling and not going to a cafe?"},{"Start":"05:03.260 ","End":"05:10.410","Text":"That\u0027s right here. That equals 20 percent."},{"Start":"05:10.510 ","End":"05:15.810","Text":"That\u0027s the probability that they will only go bowling."}],"ID":12946},{"Watched":false,"Name":"Exercise 3 - Parts b-d","Duration":"4m 41s","ChapterTopicVideoID":12468,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In section b, we\u0027re asked whether the events going bowling and going to the cafe,"},{"Start":"00:06.030 ","End":"00:08.939","Text":"whether these events are mutually exclusive."},{"Start":"00:08.939 ","End":"00:19.240","Text":"Well, let\u0027s define A as going bowling and B as going to a cafe."},{"Start":"00:19.820 ","End":"00:24.360","Text":"Now, in order for A and B to be mutually exclusive,"},{"Start":"00:24.360 ","End":"00:30.210","Text":"then the probability of A intersect B must equal 0."},{"Start":"00:30.210 ","End":"00:34.110","Text":"But if we look at the probability matrix,"},{"Start":"00:34.110 ","End":"00:41.595","Text":"the probability of the intersection between A and B is 0.1."},{"Start":"00:41.595 ","End":"00:45.060","Text":"That equals to 0.1."},{"Start":"00:45.060 ","End":"00:48.065","Text":"It doesn\u0027t equal 0, it equals 0.1."},{"Start":"00:48.065 ","End":"00:56.280","Text":"That means that A and B are not mutually exclusive."},{"Start":"00:56.650 ","End":"01:02.855","Text":"Section C asks whether the events going bowling and going to a cafe,"},{"Start":"01:02.855 ","End":"01:06.150","Text":"whether these events are dependent events."},{"Start":"01:06.340 ","End":"01:15.125","Text":"Let\u0027s define A as going bowling and B as going to a cafe."},{"Start":"01:15.125 ","End":"01:19.250","Text":"Now in order for A and B to be independent,"},{"Start":"01:19.250 ","End":"01:23.390","Text":"then the probability of A intersect B must equal to"},{"Start":"01:23.390 ","End":"01:28.115","Text":"the probability of A times the probability of B."},{"Start":"01:28.115 ","End":"01:33.875","Text":"Now, let\u0027s go to our probability matrix and plug in a percent."},{"Start":"01:33.875 ","End":"01:37.025","Text":"The probability of A is right here,"},{"Start":"01:37.025 ","End":"01:43.830","Text":"that\u0027s 0.3, and the probability of B is 0.5."},{"Start":"01:45.470 ","End":"01:51.100","Text":"The probability of A intersect B is 0.1."},{"Start":"01:51.320 ","End":"01:57.090","Text":"Now, 0.3 times 0.5 equals 0.15."},{"Start":"01:57.090 ","End":"02:01.260","Text":"That does not equal 0.1."},{"Start":"02:01.260 ","End":"02:06.940","Text":"That means that A and B are dependent."},{"Start":"02:07.120 ","End":"02:11.390","Text":"Section D asks, what is the probability that"},{"Start":"02:11.390 ","End":"02:14.810","Text":"in 1 day they will only go bowling and on the next day,"},{"Start":"02:14.810 ","End":"02:18.365","Text":"they will not go bowling nor go to a cafe."},{"Start":"02:18.365 ","End":"02:22.160","Text":"Let\u0027s define A as going bowling"},{"Start":"02:22.160 ","End":"02:30.390","Text":"and B as going to a cafe."},{"Start":"02:31.090 ","End":"02:33.710","Text":"Before we can continue,"},{"Start":"02:33.710 ","End":"02:39.420","Text":"we have to make an assumption that the days are independent of each other."},{"Start":"02:42.510 ","End":"02:52.185","Text":"The days are independent."},{"Start":"02:52.185 ","End":"02:54.920","Text":"Once we do that, let\u0027s write it out."},{"Start":"02:54.920 ","End":"02:58.235","Text":"What we want to know is what is the probability"},{"Start":"02:58.235 ","End":"03:02.060","Text":"that on the first day they will only go bowling?"},{"Start":"03:02.060 ","End":"03:07.850","Text":"That means that it will be A intersect not B."},{"Start":"03:07.850 ","End":"03:16.950","Text":"That\u0027s 1 day and we want that the next day,"},{"Start":"03:18.130 ","End":"03:25.710","Text":"that they won\u0027t go bowling and they will not go to cafe as well."},{"Start":"03:26.060 ","End":"03:30.110","Text":"Because we assumed that the days are independent,"},{"Start":"03:30.110 ","End":"03:32.810","Text":"then we can write this."},{"Start":"03:32.810 ","End":"03:37.130","Text":"This is actually the multiplication of each 1 of these events."},{"Start":"03:37.130 ","End":"03:42.020","Text":"That on 1 day, they will only go"},{"Start":"03:42.020 ","End":"03:49.920","Text":"bowling times the probability that on the next day,"},{"Start":"03:50.750 ","End":"03:53.860","Text":"they won\u0027t do anything."},{"Start":"03:53.870 ","End":"03:57.835","Text":"Won\u0027t go bowling and won\u0027t go to a cafe."},{"Start":"03:57.835 ","End":"04:01.865","Text":"Now, if we go to our probability matrix,"},{"Start":"04:01.865 ","End":"04:03.800","Text":"we can plug in the numbers."},{"Start":"04:03.800 ","End":"04:07.175","Text":"Now, A intersect not B,"},{"Start":"04:07.175 ","End":"04:12.515","Text":"A intersect not B, that\u0027s right here."},{"Start":"04:12.515 ","End":"04:15.800","Text":"That\u0027s 0.2."},{"Start":"04:15.800 ","End":"04:19.470","Text":"Not A intersect not B,"},{"Start":"04:19.470 ","End":"04:25.890","Text":"that\u0027s right here, 0.3."},{"Start":"04:25.890 ","End":"04:29.970","Text":"That turns out to be 0.06."},{"Start":"04:29.970 ","End":"04:34.310","Text":"That\u0027s the probability that on 1 day they will only go"},{"Start":"04:34.310 ","End":"04:41.340","Text":"bowling and the next day they won\u0027t go bowling and they won\u0027t go to a cafe."}],"ID":12947},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"3m 1s","ChapterTopicVideoID":12469,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this question, we\u0027ll be talking about passing a statistics exam."},{"Start":"00:04.860 ","End":"00:10.905","Text":"We\u0027re given the 70 percent of all people taking a stats exam pass on the first try."},{"Start":"00:10.905 ","End":"00:14.400","Text":"All those who fail will repeat the exam,"},{"Start":"00:14.400 ","End":"00:17.175","Text":"and of those, 80 percent will pass it."},{"Start":"00:17.175 ","End":"00:20.450","Text":"50 percent of those who fail both the first and"},{"Start":"00:20.450 ","End":"00:26.290","Text":"the second exam will register for the course again. All the rest will drop out."},{"Start":"00:26.290 ","End":"00:29.030","Text":"Now, before we go on,"},{"Start":"00:29.030 ","End":"00:33.665","Text":"I want to talk about how we can best represent the data."},{"Start":"00:33.665 ","End":"00:35.765","Text":"Well, we have several techniques,"},{"Start":"00:35.765 ","End":"00:37.040","Text":"as we said before."},{"Start":"00:37.040 ","End":"00:39.184","Text":"We have the Venn diagrams,"},{"Start":"00:39.184 ","End":"00:41.330","Text":"we have probability matrix,"},{"Start":"00:41.330 ","End":"00:44.150","Text":"and we have a probability tree."},{"Start":"00:44.150 ","End":"00:50.420","Text":"Now in this case, the probability tree would be the best choice. Why is that?"},{"Start":"00:50.420 ","End":"00:53.180","Text":"Because the data is presented to us in"},{"Start":"00:53.180 ","End":"00:59.735","Text":"stages and we have conditional probabilities at each stage."},{"Start":"00:59.735 ","End":"01:06.755","Text":"Let\u0027s draw out the probability tree. Here it is."},{"Start":"01:06.755 ","End":"01:12.650","Text":"Again, we can see that on the first pass,"},{"Start":"01:12.650 ","End":"01:18.910","Text":"70 percent of all people taking the statistics and will pass in the first try."},{"Start":"01:18.910 ","End":"01:24.060","Text":"Here we have 70 percent and if 70 percent pass,"},{"Start":"01:24.060 ","End":"01:27.210","Text":"that means that 30 percent will fail."},{"Start":"01:27.210 ","End":"01:30.030","Text":"On the second try,"},{"Start":"01:30.030 ","End":"01:34.120","Text":"80 percent will pass, and here it is."},{"Start":"01:34.310 ","End":"01:37.700","Text":"If 80 percent pass on the second try,"},{"Start":"01:37.700 ","End":"01:40.710","Text":"then 20 percent will fail."},{"Start":"01:40.940 ","End":"01:45.260","Text":"If we fail both the first second try,"},{"Start":"01:45.260 ","End":"01:49.835","Text":"then 50 percent will register."},{"Start":"01:49.835 ","End":"01:54.245","Text":"That means 50 percent will drop out."},{"Start":"01:54.245 ","End":"01:59.675","Text":"Once we\u0027ve calculated all the probabilities on the probability tree,"},{"Start":"01:59.675 ","End":"02:03.400","Text":"we can begin to answer the questions."},{"Start":"02:03.400 ","End":"02:05.230","Text":"Let\u0027s look at Section a."},{"Start":"02:05.230 ","End":"02:10.685","Text":"What are the chances that a randomly selected student pass the course?"},{"Start":"02:10.685 ","End":"02:12.980","Text":"Well, you pass the course"},{"Start":"02:12.980 ","End":"02:19.620","Text":"here on the first try and he passed the course here on the second trial."},{"Start":"02:19.620 ","End":"02:25.650","Text":"That means that the probability of passing"},{"Start":"02:25.650 ","End":"02:32.010","Text":"a course is equal to 0.7 on the first try,"},{"Start":"02:32.800 ","End":"02:38.735","Text":"plus 0.3 times 0.8."},{"Start":"02:38.735 ","End":"02:44.240","Text":"That means that he failed the first try and passed the second try."},{"Start":"02:44.240 ","End":"02:50.150","Text":"Now, that equals to 0.7 plus 0.24,"},{"Start":"02:50.150 ","End":"02:53.390","Text":"that equals to 0.94."},{"Start":"02:53.390 ","End":"03:00.780","Text":"That\u0027s the probability of a randomly selected students having passed the course."}],"ID":12948},{"Watched":false,"Name":"Exercise 4 - Parts b-c","Duration":"3m 18s","ChapterTopicVideoID":12471,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"Section B asks, if a randomly selected student passed the course,"},{"Start":"00:05.340 ","End":"00:10.215","Text":"what are the chances that he passed the repeat exam on his second try?"},{"Start":"00:10.215 ","End":"00:16.077","Text":"Well, this is a conditional probability, and what\u0027s given?"},{"Start":"00:16.077 ","End":"00:19.780","Text":"We\u0027re given that he passed the course."},{"Start":"00:20.930 ","End":"00:24.180","Text":"Now, what do we want to know?"},{"Start":"00:24.180 ","End":"00:27.915","Text":"The probability that he passed on the second try."},{"Start":"00:27.915 ","End":"00:31.960","Text":"Not the first, but the second try, right here."},{"Start":"00:32.390 ","End":"00:37.390","Text":"So pass on second try."},{"Start":"00:37.690 ","End":"00:41.165","Text":"Now, if that\u0027s a conditional probability,"},{"Start":"00:41.165 ","End":"00:50.070","Text":"then that equals to the probability of him passing on the second try,"},{"Start":"00:50.070 ","End":"00:56.730","Text":"intersect, him passing the course,"},{"Start":"00:56.890 ","End":"01:04.530","Text":"divided by the probability of him passing the course."},{"Start":"01:04.570 ","End":"01:14.000","Text":"Now, we know what the probability of a student passing the course, that\u0027s 0.94."},{"Start":"01:14.000 ","End":"01:16.910","Text":"We\u0027ve calculated that previously in section A."},{"Start":"01:16.910 ","End":"01:21.800","Text":"Now, what\u0027s the probability of a person passing on a second try,"},{"Start":"01:21.800 ","End":"01:24.335","Text":"intersect, with him passing the course?"},{"Start":"01:24.335 ","End":"01:32.972","Text":"Well, that\u0027s exactly the probability of him passing on the second try,"},{"Start":"01:32.972 ","End":"01:36.695","Text":"because if he passed on the second try,"},{"Start":"01:36.695 ","End":"01:39.070","Text":"then he obviously pass the course."},{"Start":"01:39.070 ","End":"01:45.060","Text":"Now, that means that we have 0.94 here,"},{"Start":"01:45.060 ","End":"01:51.290","Text":"and the probability of a person passing on the second try is 0.3,"},{"Start":"01:51.290 ","End":"01:54.515","Text":"the probability of him failing on the first try,"},{"Start":"01:54.515 ","End":"02:00.470","Text":"and times 0.8, which is the probability of him passing on the second try."},{"Start":"02:00.470 ","End":"02:09.958","Text":"That\u0027s 0.3 times 0.8 equals 0.24 divided by 0.94,"},{"Start":"02:09.958 ","End":"02:15.060","Text":"that equals to 0.255."},{"Start":"02:16.270 ","End":"02:22.685","Text":"In section C, we\u0027re asked what is the proportion of students that quit?"},{"Start":"02:22.685 ","End":"02:32.190","Text":"Well, that means that we want to find the probability of a person dropping out."},{"Start":"02:35.300 ","End":"02:38.015","Text":"Now, how do we do that?"},{"Start":"02:38.015 ","End":"02:43.040","Text":"Well, let\u0027s look at our probability tree."},{"Start":"02:43.230 ","End":"02:47.971","Text":"We see that we\u0027re on this long branch right here,"},{"Start":"02:47.971 ","End":"02:52.790","Text":"because the person has to fail on the first try,"},{"Start":"02:52.790 ","End":"02:56.060","Text":"he has to fail on the second trial,"},{"Start":"02:56.060 ","End":"02:59.025","Text":"and then he has to drop out."},{"Start":"02:59.025 ","End":"03:02.570","Text":"That means that, basically,"},{"Start":"03:02.570 ","End":"03:10.590","Text":"we\u0027re looking at 0.3 times 0.2 times 0.5."},{"Start":"03:10.660 ","End":"03:16.830","Text":"That equals to 0.03."}],"ID":12949},{"Watched":false,"Name":"Exercise 4 - Part d","Duration":"2m 35s","ChapterTopicVideoID":12470,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In Section D we\u0027re given 2 students Sam and Cathy,"},{"Start":"00:04.065 ","End":"00:07.680","Text":"that they\u0027re randomly selected and we\u0027re asked,"},{"Start":"00:07.680 ","End":"00:10.800","Text":"what is the probability that Sam passed the exam at"},{"Start":"00:10.800 ","End":"00:15.820","Text":"the first try and Cathy passed the exam on the second try."},{"Start":"00:16.400 ","End":"00:27.660","Text":"Let\u0027s see. The probability that Sam passed on the first try."},{"Start":"00:28.970 ","End":"00:32.790","Text":"We have that. That\u0027s 0.7,"},{"Start":"00:32.790 ","End":"00:38.820","Text":"that\u0027s right here and the probability that"},{"Start":"00:38.820 ","End":"00:48.675","Text":"Cathy passed on the second try,"},{"Start":"00:48.675 ","End":"00:58.020","Text":"that equals to 0.3 times 0.8."},{"Start":"00:58.020 ","End":"01:00.210","Text":"Remember it\u0027s this."},{"Start":"01:00.210 ","End":"01:04.245","Text":"She failed on the first try and she passed on the second try."},{"Start":"01:04.245 ","End":"01:10.420","Text":"It\u0027s 0.3 times 0.8, that equals 0.24."},{"Start":"01:12.170 ","End":"01:17.220","Text":"What we\u0027re asked is what is the probability that Sam"},{"Start":"01:17.220 ","End":"01:23.535","Text":"passed the exam on the first try and that Cathy passed on the second try."},{"Start":"01:23.535 ","End":"01:30.660","Text":"Probability Sam passed on"},{"Start":"01:30.660 ","End":"01:40.360","Text":"the first try and that Cathy passed on the second try."},{"Start":"01:46.130 ","End":"01:50.565","Text":"Sam and Cathy are independent of each other."},{"Start":"01:50.565 ","End":"01:53.550","Text":"They\u0027re not dependent on each other."},{"Start":"01:53.550 ","End":"02:04.390","Text":"We can write this out as the probability of Sam passed on the first try,"},{"Start":"02:05.450 ","End":"02:16.420","Text":"times the probability that Cathy passed on the second try,"},{"Start":"02:18.410 ","End":"02:21.600","Text":"that will equal to 0.7,"},{"Start":"02:21.600 ","End":"02:25.545","Text":"and this equals to 0.24,"},{"Start":"02:25.545 ","End":"02:36.430","Text":"and when we multiply these 2 probabilities out, we\u0027ll get 0.168."}],"ID":12950},{"Watched":false,"Name":"Exercise 5","Duration":"6m 47s","ChapterTopicVideoID":12472,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this question, we\u0027ll be dealing with unemployment."},{"Start":"00:03.450 ","End":"00:09.555","Text":"Well, 40 percent of a given population is comprised of men and 60 percent of women."},{"Start":"00:09.555 ","End":"00:12.180","Text":"10 percent of the men are unemployed and"},{"Start":"00:12.180 ","End":"00:16.390","Text":"13 percent of the total population is unemployed."},{"Start":"00:16.400 ","End":"00:20.640","Text":"We\u0027re asked, what\u0027s the unemployment rate for women?"},{"Start":"00:20.640 ","End":"00:26.500","Text":"Well, the first thing that we need to do is to draw our probability tree."},{"Start":"00:26.720 ","End":"00:30.060","Text":"Here it is. Now,"},{"Start":"00:30.060 ","End":"00:31.470","Text":"let\u0027s take a look at it."},{"Start":"00:31.470 ","End":"00:37.845","Text":"We said that 40 percent of a population is comprised of men,"},{"Start":"00:37.845 ","End":"00:43.290","Text":"that\u0027s here and 60 percent of women, that\u0027s here."},{"Start":"00:43.290 ","End":"00:46.050","Text":"Now, 10 percent of the men are unemployed,"},{"Start":"00:46.050 ","End":"00:50.435","Text":"that\u0027s here, men and that are unemployed."},{"Start":"00:50.435 ","End":"00:55.760","Text":"That means that 90 percent of the men are employed."},{"Start":"00:55.760 ","End":"01:05.450","Text":"Now, what we don\u0027t know is what\u0027s the unemployment rate or employment rate of women."},{"Start":"01:05.450 ","End":"01:11.015","Text":"But, what we\u0027re given is that the probability of"},{"Start":"01:11.015 ","End":"01:17.495","Text":"all the unemployed people in the population equals 13 percent."},{"Start":"01:17.495 ","End":"01:21.110","Text":"Now, given that, we can calculate"},{"Start":"01:21.110 ","End":"01:26.000","Text":"the probability of unemployed women. How can we do this?"},{"Start":"01:26.000 ","End":"01:27.740","Text":"Well, we know that"},{"Start":"01:27.740 ","End":"01:32.600","Text":"the overall population that\u0027s unemployed is 13 percent. That\u0027s given right here."},{"Start":"01:32.600 ","End":"01:42.010","Text":"It\u0027s 0.13 and that equals to the probability of unemployed men and of unemployed women."},{"Start":"01:42.010 ","End":"01:45.825","Text":"That\u0027s 0.4 times 0.1."},{"Start":"01:45.825 ","End":"01:48.150","Text":"That\u0027s this branch right here,"},{"Start":"01:48.150 ","End":"01:52.355","Text":"plus 0.6 times p,"},{"Start":"01:52.355 ","End":"01:55.235","Text":"the probability of unemployed women."},{"Start":"01:55.235 ","End":"02:03.690","Text":"Now, that equals to 0.04 plus 0.6p."},{"Start":"02:03.880 ","End":"02:06.830","Text":"This is basic algebra,"},{"Start":"02:06.830 ","End":"02:11.840","Text":"so that\u0027ll be 0.09 equal 0.6p."},{"Start":"02:11.840 ","End":"02:22.655","Text":"That means that p equals to 0.09 divided by 0.6 and that equals to 0.15."},{"Start":"02:22.655 ","End":"02:27.500","Text":"That means that the probability of unemployed women is"},{"Start":"02:27.500 ","End":"02:36.330","Text":"0.15 and the probability of employed women is 0.85."},{"Start":"02:36.730 ","End":"02:39.425","Text":"In section b we\u0027re asked,"},{"Start":"02:39.425 ","End":"02:45.020","Text":"what\u0027s the probability that a randomly selected unemployed person is a woman?"},{"Start":"02:45.020 ","End":"02:49.280","Text":"Well, this is a conditional probability question."},{"Start":"02:49.280 ","End":"02:51.470","Text":"Now, what\u0027s given to us?"},{"Start":"02:51.470 ","End":"02:55.235","Text":"We\u0027re given that a person is unemployed."},{"Start":"02:55.235 ","End":"02:58.520","Text":"What do we have to know? We want to find"},{"Start":"02:58.520 ","End":"03:02.720","Text":"out what\u0027s the probability that this unemployed person is a woman."},{"Start":"03:02.720 ","End":"03:04.790","Text":"We can write this out like this."},{"Start":"03:04.790 ","End":"03:10.165","Text":"What\u0027s the probability of a person being a woman,"},{"Start":"03:10.165 ","End":"03:13.179","Text":"given that she\u0027s unemployed?"},{"Start":"03:13.179 ","End":"03:16.785","Text":"Well, let\u0027s look at it."},{"Start":"03:16.785 ","End":"03:18.720","Text":"In the denominator,"},{"Start":"03:18.720 ","End":"03:22.745","Text":"we\u0027ll put the probability of what\u0027s given to us."},{"Start":"03:22.745 ","End":"03:26.900","Text":"That means that the probability of a person being unemployed."},{"Start":"03:26.900 ","End":"03:29.705","Text":"Then in the numerator,"},{"Start":"03:29.705 ","End":"03:39.735","Text":"we\u0027ll put down the probability that a person is a woman and that she is unemployed."},{"Start":"03:39.735 ","End":"03:42.580","Text":"Now, section a,"},{"Start":"03:42.580 ","End":"03:48.570","Text":"we\u0027ve been given that the probability of an unemployed person is 0.13."},{"Start":"03:48.610 ","End":"03:51.845","Text":"We have to calculate the numerator,"},{"Start":"03:51.845 ","End":"03:57.095","Text":"what\u0027s the probability that there\u0027s a woman and she\u0027s unemployed?"},{"Start":"03:57.095 ","End":"03:59.825","Text":"Well, this is this branch right here."},{"Start":"03:59.825 ","End":"04:10.770","Text":"That\u0027s 0.6 times 0.15 and this comes out to 0.692."},{"Start":"04:12.320 ","End":"04:16.710","Text":"In section c we define the following events, A,"},{"Start":"04:16.710 ","End":"04:19.785","Text":"a randomly selected person is unemployed,"},{"Start":"04:19.785 ","End":"04:23.040","Text":"and B, a randomly selected person is male."},{"Start":"04:23.040 ","End":"04:26.420","Text":"We\u0027re asked, are these events mutually exclusive?"},{"Start":"04:26.420 ","End":"04:29.045","Text":"Are they independent events?"},{"Start":"04:29.045 ","End":"04:31.460","Text":"Well, let\u0027s see. First of all,"},{"Start":"04:31.460 ","End":"04:33.780","Text":"let\u0027s see if they\u0027re mutually exclusive."},{"Start":"04:33.780 ","End":"04:38.555","Text":"In order for A and B to be mutually exclusive,"},{"Start":"04:38.555 ","End":"04:45.065","Text":"that means that the probability of the intersect, has to equal 0."},{"Start":"04:45.065 ","End":"04:48.710","Text":"Well, what\u0027s the probability of A?"},{"Start":"04:48.710 ","End":"04:56.430","Text":"The probability of A is the probability of a randomly selected person being unemployed."},{"Start":"04:56.430 ","End":"04:59.415","Text":"Well, that\u0027s 0.13 that\u0027s been given to us."},{"Start":"04:59.415 ","End":"05:01.965","Text":"What\u0027s the probability of B?"},{"Start":"05:01.965 ","End":"05:04.475","Text":"A randomly selected person is a male."},{"Start":"05:04.475 ","End":"05:10.380","Text":"Well, that\u0027s right here. That\u0027s 0.4."},{"Start":"05:10.380 ","End":"05:13.770","Text":"We say that the probability of"},{"Start":"05:13.770 ","End":"05:21.170","Text":"A intersect B equals the probability of A times the probability of B."},{"Start":"05:21.170 ","End":"05:24.500","Text":"Now, that\u0027s 0.13,"},{"Start":"05:24.500 ","End":"05:32.575","Text":"that\u0027s 0.4, and there is no way that the multiplication of these 2 numbers equals 0."},{"Start":"05:32.575 ","End":"05:41.980","Text":"That means that A and B are not mutually exclusive."},{"Start":"05:43.870 ","End":"05:46.325","Text":"Now, the next question is,"},{"Start":"05:46.325 ","End":"05:48.320","Text":"are they independent events?"},{"Start":"05:48.320 ","End":"05:50.440","Text":"Well, again,"},{"Start":"05:50.440 ","End":"05:52.095","Text":"to be independent,"},{"Start":"05:52.095 ","End":"05:56.240","Text":"the probability of A intersect B must"},{"Start":"05:56.240 ","End":"06:02.030","Text":"equal to the probability of A times the probability of B."},{"Start":"06:02.030 ","End":"06:07.280","Text":"Now, the probability of A is 0.13,"},{"Start":"06:07.280 ","End":"06:11.250","Text":"probability of B is 0.4."},{"Start":"06:11.470 ","End":"06:16.640","Text":"Now, what\u0027s the probability of A intersect B of"},{"Start":"06:16.640 ","End":"06:23.700","Text":"a randomly selected person being unemployed and that he\u0027s a male?"},{"Start":"06:23.700 ","End":"06:26.835","Text":"Well, that\u0027s this,"},{"Start":"06:26.835 ","End":"06:29.955","Text":"0.4 times 0.1."},{"Start":"06:29.955 ","End":"06:33.645","Text":"0.4 times 0.1."},{"Start":"06:33.645 ","End":"06:39.845","Text":"Now, obviously, this side does not equal to this side."},{"Start":"06:39.845 ","End":"06:45.960","Text":"That means that A and B are dependent."}],"ID":12951},{"Watched":false,"Name":"Exercise 6 - Part a","Duration":"4m 33s","ChapterTopicVideoID":12473,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"This question deals with ordinary and non-ordinary coins."},{"Start":"00:04.335 ","End":"00:07.005","Text":"Now, there are 10 coins in a box,"},{"Start":"00:07.005 ","End":"00:10.440","Text":"7 of them are ordinary coins with heads and tails,"},{"Start":"00:10.440 ","End":"00:12.750","Text":"and 3 have heads on both sides."},{"Start":"00:12.750 ","End":"00:16.950","Text":"A person randomly selects a coin and tosses it twice."},{"Start":"00:16.950 ","End":"00:21.450","Text":"Now, let A be the event that the first toss is head,"},{"Start":"00:21.450 ","End":"00:25.620","Text":"and let B be the event that the second toss is heads."},{"Start":"00:25.620 ","End":"00:31.510","Text":"Now, this is obviously best displayed as a probability tree."},{"Start":"00:31.510 ","End":"00:34.205","Text":"Why is that? Well, we have levels."},{"Start":"00:34.205 ","End":"00:36.935","Text":"We have the first level being the coins,"},{"Start":"00:36.935 ","End":"00:39.665","Text":"and then we have a sequence of events."},{"Start":"00:39.665 ","End":"00:42.470","Text":"The first toss and the second toss."},{"Start":"00:42.470 ","End":"00:45.115","Text":"Let\u0027s draw the probability tree."},{"Start":"00:45.115 ","End":"00:47.010","Text":"Let\u0027s see how it looks."},{"Start":"00:47.010 ","End":"00:50.595","Text":"Again, we have the first level,"},{"Start":"00:50.595 ","End":"00:55.000","Text":"the coins, ordinary coin versus not ordinary coin."},{"Start":"00:55.000 ","End":"00:56.930","Text":"In an ordinary coin,"},{"Start":"00:56.930 ","End":"00:59.670","Text":"we have both heads and tails and it"},{"Start":"00:59.670 ","End":"01:02.850","Text":"doesn\u0027t matter for the first toss and the second toss."},{"Start":"01:02.850 ","End":"01:05.205","Text":"Now in a non-ordinary coin,"},{"Start":"01:05.205 ","End":"01:07.365","Text":"we only have heads."},{"Start":"01:07.365 ","End":"01:11.864","Text":"We don\u0027t have the tails branch."},{"Start":"01:11.864 ","End":"01:13.845","Text":"Another thing that we know,"},{"Start":"01:13.845 ","End":"01:15.815","Text":"let\u0027s look at the probabilities."},{"Start":"01:15.815 ","End":"01:20.870","Text":"Well, 7 out of 10 coins are ordinary."},{"Start":"01:20.870 ","End":"01:24.785","Text":"That means that we have a 70 percent chance of choosing"},{"Start":"01:24.785 ","End":"01:30.460","Text":"an ordinary coin and 30 percent chance of choosing a non-ordinary coin."},{"Start":"01:30.460 ","End":"01:33.590","Text":"Now, if we choose a non-ordinary coin,"},{"Start":"01:33.590 ","End":"01:38.960","Text":"we have a probability of 1 to get heads both in the first and second toss."},{"Start":"01:38.960 ","End":"01:40.790","Text":"In an ordinary coin,"},{"Start":"01:40.790 ","End":"01:45.050","Text":"we have a 50 percent chance of getting heads and"},{"Start":"01:45.050 ","End":"01:51.215","Text":"a 50 percent chance of getting tails no matter which toss we have,"},{"Start":"01:51.215 ","End":"01:53.980","Text":"be it the first toss or the second toss."},{"Start":"01:54.050 ","End":"01:57.170","Text":"Let\u0027s now look at the questions."},{"Start":"01:57.170 ","End":"02:03.865","Text":"The first thing that we want to do is we want to calculate the chances of events A and B."},{"Start":"02:03.865 ","End":"02:07.285","Text":"What\u0027s the probability of A?"},{"Start":"02:07.285 ","End":"02:14.335","Text":"That\u0027s the probability of A being the event that the first toss is heads."},{"Start":"02:14.335 ","End":"02:18.090","Text":"Now, where is that?"},{"Start":"02:20.800 ","End":"02:23.563","Text":"If we toss an ordinary coin, then this is the branch,"},{"Start":"02:23.563 ","End":"02:26.190","Text":"and we have to stop here because that\u0027s the first toss."},{"Start":"02:26.190 ","End":"02:29.515","Text":"If we toss a non-ordinary coin,"},{"Start":"02:29.515 ","End":"02:31.915","Text":"then we have to stop here as well."},{"Start":"02:31.915 ","End":"02:40.070","Text":"That\u0027s this branch. Now, the probability then of A being heads in"},{"Start":"02:40.070 ","End":"02:49.075","Text":"the first toss is equal to 0.7 times 0.5 plus 0.3 times 1."},{"Start":"02:49.075 ","End":"02:53.565","Text":"Again, why is that, 0.7 times 0.5?"},{"Start":"02:53.565 ","End":"02:55.544","Text":"That\u0027s this branch right here,"},{"Start":"02:55.544 ","End":"02:57.525","Text":"and 0.3 times 1,"},{"Start":"02:57.525 ","End":"02:59.955","Text":"well, that\u0027s this branch right here."},{"Start":"02:59.955 ","End":"03:05.350","Text":"That equals to 0.65."},{"Start":"03:05.350 ","End":"03:09.965","Text":"Now, what\u0027s the probability of B?"},{"Start":"03:09.965 ","End":"03:12.160","Text":"Well, what\u0027s B?"},{"Start":"03:12.160 ","End":"03:16.445","Text":"B is the event that the second toss is heads."},{"Start":"03:16.445 ","End":"03:21.435","Text":"Well, where is that with the branches?"},{"Start":"03:21.435 ","End":"03:22.875","Text":"Well, it\u0027s here."},{"Start":"03:22.875 ","End":"03:25.865","Text":"That means if we have an ordinary coin,"},{"Start":"03:25.865 ","End":"03:29.427","Text":"we can go here,"},{"Start":"03:29.427 ","End":"03:35.160","Text":"or go here,"},{"Start":"03:35.160 ","End":"03:40.860","Text":"or go here."},{"Start":"03:40.860 ","End":"03:43.925","Text":"Let\u0027s calculate the probabilities."},{"Start":"03:43.925 ","End":"03:50.300","Text":"Well, that\u0027s 0.7 times 0.5 times 0.5,"},{"Start":"03:50.300 ","End":"03:52.975","Text":"that\u0027s this branch right here,"},{"Start":"03:52.975 ","End":"04:01.715","Text":"plus 0.7 times 0.5 times 0.5,"},{"Start":"04:01.715 ","End":"04:04.235","Text":"that\u0027s this branch right here."},{"Start":"04:04.235 ","End":"04:06.955","Text":"Ordinary coin, tails on"},{"Start":"04:06.955 ","End":"04:11.235","Text":"the first toss and the heads on the second, and that\u0027s what we want."},{"Start":"04:11.235 ","End":"04:18.605","Text":"We want 0.3 times 1 times 1."},{"Start":"04:18.605 ","End":"04:21.365","Text":"That\u0027s a non-ordinary coin."},{"Start":"04:21.365 ","End":"04:25.310","Text":"Heads on the first toss and the heads on the second toss."},{"Start":"04:25.310 ","End":"04:32.100","Text":"Now, that equals also to 0.65."}],"ID":12952},{"Watched":false,"Name":"Exercise 6 - Parts b-c","Duration":"4m 33s","ChapterTopicVideoID":12474,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"Now, in section A, we\u0027ve calculated the probabilities of A and B."},{"Start":"00:05.310 ","End":"00:10.515","Text":"The probability of A equals 0.65,"},{"Start":"00:10.515 ","End":"00:15.675","Text":"and the probability of B also equals 0.65."},{"Start":"00:15.675 ","End":"00:21.480","Text":"Now, in section B, we\u0027re asked whether A and B are independent events."},{"Start":"00:21.480 ","End":"00:24.480","Text":"Well, in order for them to be independent events,"},{"Start":"00:24.480 ","End":"00:28.455","Text":"then the probability of A intersect B must"},{"Start":"00:28.455 ","End":"00:33.630","Text":"equal to the probability of A times the probability of B."},{"Start":"00:33.630 ","End":"00:36.930","Text":"Now, we\u0027ve calculated the individual probabilities."},{"Start":"00:36.930 ","End":"00:42.195","Text":"What we need to do is basically calculate the probability of the intersect."},{"Start":"00:42.195 ","End":"00:48.745","Text":"Now, the probability of the intersect, what are we looking for?"},{"Start":"00:48.745 ","End":"00:58.210","Text":"We\u0027re looking for an event that\u0027s heads in the first toss and heads in the second toss."},{"Start":"00:58.760 ","End":"01:01.210","Text":"In an ordinary coin,"},{"Start":"01:01.210 ","End":"01:04.450","Text":"we\u0027re looking at heads in the first toss,"},{"Start":"01:04.450 ","End":"01:06.505","Text":"and heads in the second toss."},{"Start":"01:06.505 ","End":"01:08.440","Text":"In a non ordinary coin,"},{"Start":"01:08.440 ","End":"01:13.845","Text":"we\u0027re looking at heads in the first toss and heads in the second toss."},{"Start":"01:13.845 ","End":"01:17.335","Text":"Now, let\u0027s calculate the probabilities."},{"Start":"01:17.335 ","End":"01:23.020","Text":"That equals to 0.7 times 0.5 times"},{"Start":"01:23.020 ","End":"01:28.815","Text":"0.5 plus 0.3 times 1 times 1."},{"Start":"01:28.815 ","End":"01:32.910","Text":"Again, 0.7 times 0.5 times 0.5,"},{"Start":"01:32.910 ","End":"01:34.613","Text":"that\u0027s this branch,"},{"Start":"01:34.613 ","End":"01:36.945","Text":"and 0.3 times 1 times 1,"},{"Start":"01:36.945 ","End":"01:46.800","Text":"that\u0027s this branch and that turns out to be 0.475."},{"Start":"01:46.800 ","End":"01:52.215","Text":"Now, let\u0027s see about independence,"},{"Start":"01:52.215 ","End":"01:54.360","Text":"are A and B really independent?"},{"Start":"01:54.360 ","End":"01:56.445","Text":"Well, as we said,"},{"Start":"01:56.445 ","End":"02:03.990","Text":"P of A intersect B just as equal to P of A times P of B."},{"Start":"02:03.990 ","End":"02:10.965","Text":"Well, P of A intersect B is 0.475."},{"Start":"02:10.965 ","End":"02:13.805","Text":"Now, P of A times P of B,"},{"Start":"02:13.805 ","End":"02:17.510","Text":"that\u0027s 0.65 times 0.65,"},{"Start":"02:17.510 ","End":"02:23.870","Text":"and that equals to 0.4225."},{"Start":"02:23.870 ","End":"02:28.220","Text":"Now, obviously, the 2 sides are not equal."},{"Start":"02:28.220 ","End":"02:34.350","Text":"That means that A and B are dependent."},{"Start":"02:35.140 ","End":"02:40.610","Text":"In section C, we\u0027re asked if the first toss is heads,"},{"Start":"02:40.610 ","End":"02:45.215","Text":"what\u0027s the probability that the coin tossed has both heads and tails?"},{"Start":"02:45.215 ","End":"02:48.255","Text":"Well, this is a conditional probability,"},{"Start":"02:48.255 ","End":"02:51.080","Text":"and why is that? Well, we\u0027re given."},{"Start":"02:51.080 ","End":"02:53.930","Text":"If the first toss is heads, that means,"},{"Start":"02:53.930 ","End":"02:55.865","Text":"given that the first toss is heads,"},{"Start":"02:55.865 ","End":"03:00.965","Text":"what\u0027s the probability that the coin toss has both heads and tails?"},{"Start":"03:00.965 ","End":"03:03.205","Text":"Well, let\u0027s write this out."},{"Start":"03:03.205 ","End":"03:08.700","Text":"That\u0027s a probability of the first toss being heads."},{"Start":"03:08.700 ","End":"03:12.150","Text":"Well, that\u0027s event A from above,"},{"Start":"03:12.150 ","End":"03:14.055","Text":"and what do we try to find out?"},{"Start":"03:14.055 ","End":"03:20.525","Text":"We\u0027re trying to find out what\u0027s the probability of a coin that has both heads and tails."},{"Start":"03:20.525 ","End":"03:23.520","Text":"That means that it\u0027s an ordinary coin."},{"Start":"03:26.540 ","End":"03:32.348","Text":"Now, we can write the cell like this,"},{"Start":"03:32.348 ","End":"03:38.360","Text":"and the denominator will put the probability of what\u0027s given to us,"},{"Start":"03:38.360 ","End":"03:40.280","Text":"the probability of A,"},{"Start":"03:40.280 ","End":"03:41.510","Text":"and in the numerator,"},{"Start":"03:41.510 ","End":"03:43.750","Text":"that\u0027s the probability of the intersect,"},{"Start":"03:43.750 ","End":"03:50.660","Text":"that\u0027s intersect of an ordinary coin,"},{"Start":"03:50.660 ","End":"03:56.420","Text":"and that we have heads and the first toss."},{"Start":"03:56.420 ","End":"04:02.650","Text":"Well, the probability in the denominator is 0.65,"},{"Start":"04:02.650 ","End":"04:04.790","Text":"we\u0027ve calculated that above."},{"Start":"04:04.790 ","End":"04:08.090","Text":"Now, let\u0027s see about the numerator."},{"Start":"04:08.090 ","End":"04:15.660","Text":"What\u0027s the probability of an ordinary coin and that we get heads on the first toss?"},{"Start":"04:15.660 ","End":"04:18.940","Text":"Well, that\u0027s this branch right here."},{"Start":"04:19.850 ","End":"04:26.820","Text":"That equals to 0.7 times 0.5."},{"Start":"04:26.820 ","End":"04:31.900","Text":"That equals to 0.538."}],"ID":12953},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"3m 20s","ChapterTopicVideoID":12475,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.535","Text":"This question deals in used cars."},{"Start":"00:02.535 ","End":"00:05.010","Text":"Jack wants to sell his car."},{"Start":"00:05.010 ","End":"00:09.720","Text":"He places ads on both the Internet and in the newspaper."},{"Start":"00:09.720 ","End":"00:15.645","Text":"Now, 30 percent of the people interested in buying a used car see the Internet ads,"},{"Start":"00:15.645 ","End":"00:19.409","Text":"and 50 percent see the ads in the newspaper."},{"Start":"00:19.409 ","End":"00:27.375","Text":"Now we\u0027re also given that 72 percent of the people see the ads in at least 1 medium."},{"Start":"00:27.375 ","End":"00:31.980","Text":"We\u0027re asked, what\u0027s the proportion of people interested in buying a used car?"},{"Start":"00:31.980 ","End":"00:35.145","Text":"What\u0027s the proportion of people that see both ads?"},{"Start":"00:35.145 ","End":"00:39.760","Text":"Now, let\u0027s first define our variables."},{"Start":"00:39.770 ","End":"00:45.880","Text":"Let\u0027s define A as the people that see the ads in the Internet,"},{"Start":"00:46.160 ","End":"00:52.709","Text":"and B as the people seeing the ad in the newspaper."},{"Start":"00:54.370 ","End":"00:58.875","Text":"Now, the probability of A, that\u0027s given,"},{"Start":"00:58.875 ","End":"01:01.020","Text":"that\u0027s 30 percent,"},{"Start":"01:01.020 ","End":"01:03.465","Text":"and the probability of B,"},{"Start":"01:03.465 ","End":"01:07.620","Text":"that\u0027s also given, that\u0027s 50 percent."},{"Start":"01:07.620 ","End":"01:10.275","Text":"Now, see that here,"},{"Start":"01:10.275 ","End":"01:16.849","Text":"30 percent of the Internet and 50 percent in the newspaper."},{"Start":"01:16.849 ","End":"01:26.430","Text":"Now, we\u0027re also given that 72 percent of the people see the ad in at least 1 medium."},{"Start":"01:26.430 ","End":"01:31.725","Text":"That means that A or B,"},{"Start":"01:31.725 ","End":"01:42.940","Text":"that\u0027s people who see the ad in the Internet or the newspaper."},{"Start":"01:42.940 ","End":"01:51.500","Text":"Therefore, the probability of A union B equals 0.72."},{"Start":"01:51.500 ","End":"02:00.395","Text":"Now, we know from our studies that the probability of A union B"},{"Start":"02:00.395 ","End":"02:05.405","Text":"equals the probability of A plus the probability"},{"Start":"02:05.405 ","End":"02:10.850","Text":"of B minus the probability of A intersect B."},{"Start":"02:10.850 ","End":"02:16.110","Text":"Now, we have the probability of A union B,"},{"Start":"02:16.110 ","End":"02:21.000","Text":"that\u0027s 0.72, and we have the probability of A,"},{"Start":"02:21.000 ","End":"02:26.920","Text":"that\u0027s 0.3, and the probability of B that\u0027s 0.5,"},{"Start":"02:26.920 ","End":"02:31.655","Text":"and we have to subtract the probability of A intersect B."},{"Start":"02:31.655 ","End":"02:33.890","Text":"Now what\u0027s the meaning of A intersect B?"},{"Start":"02:33.890 ","End":"02:36.740","Text":"That\u0027s exactly what we\u0027re trying to find out."},{"Start":"02:36.740 ","End":"02:38.450","Text":"The people that see the ads,"},{"Start":"02:38.450 ","End":"02:42.560","Text":"both in the Internet and in the newspaper."},{"Start":"02:42.560 ","End":"02:45.035","Text":"Let\u0027s do a little bit of juggling."},{"Start":"02:45.035 ","End":"02:50.285","Text":"We see that the probability of A intersect B equals"},{"Start":"02:50.285 ","End":"02:57.250","Text":"0.3 plus 0.5 minus 0.72."},{"Start":"02:57.250 ","End":"03:00.270","Text":"That equals 0.08,"},{"Start":"03:00.290 ","End":"03:09.245","Text":"which basically means that 8 percent of the people see the ads in"},{"Start":"03:09.245 ","End":"03:18.840","Text":"both the newspaper and the Internet."}],"ID":12954},{"Watched":false,"Name":"Exercise 7 - Parts b-c","Duration":"4m 31s","ChapterTopicVideoID":12477,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In section B, we\u0027re asked,"},{"Start":"00:02.340 ","End":"00:04.545","Text":"if a person sees the Internet ad,"},{"Start":"00:04.545 ","End":"00:08.910","Text":"what\u0027s the probability that he doesn\u0027t see the newspaper ad?"},{"Start":"00:08.910 ","End":"00:12.285","Text":"Let\u0027s recall from section A."},{"Start":"00:12.285 ","End":"00:16.480","Text":"A is a person seeing the ad in the Internet,"},{"Start":"00:16.940 ","End":"00:22.149","Text":"and B is a person seeing the ad in the newspaper."},{"Start":"00:23.060 ","End":"00:28.425","Text":"Now, the probability of A is 30 percent,"},{"Start":"00:28.425 ","End":"00:32.460","Text":"and the probability of B is 50 percent."},{"Start":"00:32.460 ","End":"00:35.220","Text":"The probability of A union B,"},{"Start":"00:35.220 ","End":"00:39.005","Text":"that was 72 percent."},{"Start":"00:39.005 ","End":"00:46.925","Text":"We\u0027ve calculated, the probability of the intersection is 8 percent."},{"Start":"00:46.925 ","End":"00:51.110","Text":"Now, in order to calculate,"},{"Start":"00:51.110 ","End":"00:53.600","Text":"to see all the probabilities,"},{"Start":"00:53.600 ","End":"00:57.155","Text":"it\u0027s best to use the probability matrix."},{"Start":"00:57.155 ","End":"01:00.950","Text":"Let\u0027s look at it. Here it is."},{"Start":"01:00.950 ","End":"01:04.685","Text":"Now, let\u0027s fill in the probabilities."},{"Start":"01:04.685 ","End":"01:09.170","Text":"Well, the probability of A is 30 percent,"},{"Start":"01:09.170 ","End":"01:11.855","Text":"so we\u0027ll put that here 0.3."},{"Start":"01:11.855 ","End":"01:16.565","Text":"Probability of B is 0.5, that\u0027s right here."},{"Start":"01:16.565 ","End":"01:20.450","Text":"The probability of A intersect B,"},{"Start":"01:20.450 ","End":"01:21.890","Text":"that\u0027s B, that\u0027s A,"},{"Start":"01:21.890 ","End":"01:23.240","Text":"so that\u0027s the intercept."},{"Start":"01:23.240 ","End":"01:26.360","Text":"That\u0027s 0.08."},{"Start":"01:26.360 ","End":"01:30.020","Text":"Now, let\u0027s fill in the other probabilities."},{"Start":"01:30.020 ","End":"01:40.260","Text":"That\u0027s 0.5 and that\u0027s 0.42, and that\u0027s 0.7,"},{"Start":"01:40.360 ","End":"01:44.030","Text":"and that will be 0.22. and"},{"Start":"01:44.030 ","End":"01:53.400","Text":"that would be 0.28."},{"Start":"01:53.400 ","End":"01:58.505","Text":"Great. Now, in this question,"},{"Start":"01:58.505 ","End":"02:02.285","Text":"we\u0027re actually looking at a conditional probability."},{"Start":"02:02.285 ","End":"02:06.230","Text":"The conditional probability is this."},{"Start":"02:06.230 ","End":"02:10.865","Text":"We\u0027re asked what\u0027s the probability and what\u0027s given to us."},{"Start":"02:10.865 ","End":"02:13.295","Text":"If a person sees the Internet ad,"},{"Start":"02:13.295 ","End":"02:17.990","Text":"that means that we\u0027re given A."},{"Start":"02:17.990 ","End":"02:20.810","Text":"What\u0027s the probability, that\u0027s what we\u0027re looking for."},{"Start":"02:20.810 ","End":"02:24.995","Text":"What\u0027s the probability that he doesn\u0027t see the newspaper ad?"},{"Start":"02:24.995 ","End":"02:27.400","Text":"That means not B."},{"Start":"02:27.400 ","End":"02:30.215","Text":"This is what we\u0027re looking to calculate."},{"Start":"02:30.215 ","End":"02:35.090","Text":"Well, that equals to the probability of A, that\u0027s the denominator,"},{"Start":"02:35.090 ","End":"02:42.740","Text":"that\u0027s probability of a person seeing the ad in the Internet that\u0027s given,"},{"Start":"02:42.740 ","End":"02:44.300","Text":"and in the numerator,"},{"Start":"02:44.300 ","End":"02:49.380","Text":"that\u0027s the probability of naught B intersect A."},{"Start":"02:49.380 ","End":"02:56.505","Text":"Well, the probability of A is right here, it\u0027s 0.3."},{"Start":"02:56.505 ","End":"03:02.150","Text":"The probability of naught B intersect A, here it is."},{"Start":"03:02.150 ","End":"03:08.240","Text":"That\u0027s A, that\u0027s naught B so the intercept is right here, 0.22."},{"Start":"03:08.240 ","End":"03:11.850","Text":"That equal to 0.733."},{"Start":"03:15.170 ","End":"03:17.985","Text":"In section C we\u0027re asked,"},{"Start":"03:17.985 ","End":"03:23.025","Text":"are seeing the Internet ad and seeing the newspaper ad, independent events?"},{"Start":"03:23.025 ","End":"03:32.950","Text":"Well, basically, we\u0027re being asked if events A and events B are they independent?"},{"Start":"03:35.160 ","End":"03:39.310","Text":"Well, in order for them to be independent,"},{"Start":"03:39.310 ","End":"03:45.010","Text":"then the probability of A intersect B must be equal"},{"Start":"03:45.010 ","End":"03:51.205","Text":"to the probability of A times the probability of B."},{"Start":"03:51.205 ","End":"03:53.485","Text":"Let\u0027s look at our matrix."},{"Start":"03:53.485 ","End":"03:57.280","Text":"The probability of A is 0.3."},{"Start":"03:57.280 ","End":"04:01.490","Text":"The probability of B is 0.5."},{"Start":"04:01.940 ","End":"04:08.469","Text":"The probability of A intersect B, that\u0027s 0.08."},{"Start":"04:08.690 ","End":"04:13.575","Text":"Now here, that equals to 0.15."},{"Start":"04:13.575 ","End":"04:19.240","Text":"Obviously, that doesn\u0027t equal to 0.08."},{"Start":"04:20.200 ","End":"04:22.610","Text":"To answer the question,"},{"Start":"04:22.610 ","End":"04:24.440","Text":"are A and B independent?"},{"Start":"04:24.440 ","End":"04:30.030","Text":"No, A and B are dependent."}],"ID":12955},{"Watched":false,"Name":"Exercise 7 - Part d","Duration":"9m ","ChapterTopicVideoID":12476,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.090","Text":"In section D, we\u0027re told that the probability of a person calling Jack when he only saw"},{"Start":"00:06.090 ","End":"00:09.090","Text":"the Internet ad is 0.7 and"},{"Start":"00:09.090 ","End":"00:14.430","Text":"the probability of him calling Jack when he only saw the newspaper ad,"},{"Start":"00:14.430 ","End":"00:16.470","Text":"well, that\u0027s 0.6."},{"Start":"00:16.470 ","End":"00:21.390","Text":"If you sum both ads in the Internet and in the newspaper,"},{"Start":"00:21.390 ","End":"00:24.855","Text":"what\u0027s the probability of him calling Jack? Well, that\u0027s 0.9."},{"Start":"00:24.855 ","End":"00:27.780","Text":"Now, we\u0027re asked, what\u0027s the probability of"},{"Start":"00:27.780 ","End":"00:30.330","Text":"a person that\u0027s interested in buying a used car?"},{"Start":"00:30.330 ","End":"00:33.190","Text":"What\u0027s the probability of him calling Jack?"},{"Start":"00:33.190 ","End":"00:35.540","Text":"Well, before we can answer that,"},{"Start":"00:35.540 ","End":"00:39.140","Text":"let\u0027s look at the question a little bit more closely."},{"Start":"00:39.140 ","End":"00:41.360","Text":"Let\u0027s look at the events."},{"Start":"00:41.360 ","End":"00:45.605","Text":"The probability of a person seeing only the Internet ad,"},{"Start":"00:45.605 ","End":"00:51.310","Text":"well, that\u0027s the event a and not b."},{"Start":"00:51.310 ","End":"00:56.190","Text":"Only Internet."},{"Start":"00:57.680 ","End":"01:01.740","Text":"Now, let\u0027s look at the next event."},{"Start":"01:01.740 ","End":"01:09.120","Text":"The probability that a person saw the ad only in the newspaper."},{"Start":"01:09.120 ","End":"01:17.470","Text":"Well, that\u0027s not a and b, that\u0027s only newspaper."},{"Start":"01:19.100 ","End":"01:23.790","Text":"What\u0027s the third one, the probability that a person saw"},{"Start":"01:23.790 ","End":"01:27.720","Text":"the ad both in the Internet and in the newspaper."},{"Start":"01:27.720 ","End":"01:31.870","Text":"Well, that\u0027s a and b."},{"Start":"01:32.060 ","End":"01:41.770","Text":"That\u0027s both Internet and newspaper."},{"Start":"01:44.390 ","End":"01:50.720","Text":"Now, when we try to find out what the probability of these events,"},{"Start":"01:50.720 ","End":"01:54.965","Text":"well, we\u0027ve already calculated that."},{"Start":"01:54.965 ","End":"01:59.000","Text":"That\u0027s in the probability matrix but hold on."},{"Start":"01:59.000 ","End":"02:00.395","Text":"Before we get to that,"},{"Start":"02:00.395 ","End":"02:04.630","Text":"let\u0027s look at how we can best represent the data."},{"Start":"02:04.630 ","End":"02:07.775","Text":"This data is best represented as a tree."},{"Start":"02:07.775 ","End":"02:11.705","Text":"Why? Because we have conditional probabilities."},{"Start":"02:11.705 ","End":"02:16.824","Text":"Given that a person only saw the ad in the Internet,"},{"Start":"02:16.824 ","End":"02:20.145","Text":"what\u0027s the probability that he\u0027ll call Jack?"},{"Start":"02:20.145 ","End":"02:26.000","Text":"We first have the first level of these events."},{"Start":"02:26.000 ","End":"02:30.560","Text":"A person seeing the ad only in the Internet or only in the newspaper,"},{"Start":"02:30.560 ","End":"02:33.169","Text":"or both in the Internet and in the newspaper."},{"Start":"02:33.169 ","End":"02:38.195","Text":"The second level is whether the person will be calling Jack or not."},{"Start":"02:38.195 ","End":"02:40.175","Text":"Well, let\u0027s see."},{"Start":"02:40.175 ","End":"02:49.960","Text":"Let\u0027s take a look at the probability tree. That\u0027s right here."},{"Start":"02:49.960 ","End":"02:56.240","Text":"Again, we have the 4 events here."},{"Start":"02:56.240 ","End":"02:58.280","Text":"Here we have only 3."},{"Start":"02:58.280 ","End":"03:00.170","Text":"We have only the Internet,"},{"Start":"03:00.170 ","End":"03:02.195","Text":"only the newspaper,"},{"Start":"03:02.195 ","End":"03:04.339","Text":"the Internet and newspaper,"},{"Start":"03:04.339 ","End":"03:08.180","Text":"and the fourth one, it\u0027s not written down here but if a person didn\u0027t"},{"Start":"03:08.180 ","End":"03:12.430","Text":"see the ad in the Internet or in the newspaper."},{"Start":"03:12.430 ","End":"03:14.280","Text":"That means that we\u0027ve stopped here."},{"Start":"03:14.280 ","End":"03:18.630","Text":"Now, the next stage whether he calls or not."},{"Start":"03:18.630 ","End":"03:22.620","Text":"Well, if you only saw the Internet we"},{"Start":"03:22.620 ","End":"03:28.440","Text":"call Jack with a probability of 0.7, that\u0027s right here."},{"Start":"03:29.110 ","End":"03:33.770","Text":"If he calls Jack if you only saw the ad in the newspaper,"},{"Start":"03:33.770 ","End":"03:36.660","Text":"that\u0027s 0.6, that\u0027s right here."},{"Start":"03:36.830 ","End":"03:41.840","Text":"If you saw the ad both in the Internet and in the newspaper,"},{"Start":"03:41.840 ","End":"03:44.930","Text":"that\u0027s 0.9. That\u0027s right here."},{"Start":"03:44.930 ","End":"03:52.115","Text":"Again, in order to complete the probabilities for this level,"},{"Start":"03:52.115 ","End":"03:54.140","Text":"we have to go back to"},{"Start":"03:54.140 ","End":"04:00.690","Text":"our probability matrix and see what the probabilities are for these events."},{"Start":"04:02.110 ","End":"04:05.870","Text":"Now, here\u0027s our probability matrix."},{"Start":"04:05.870 ","End":"04:11.660","Text":"Again, we have to find out what the probabilities are for these events."},{"Start":"04:11.660 ","End":"04:19.070","Text":"Well, the probability of a and not b equals what?"},{"Start":"04:19.070 ","End":"04:22.073","Text":"Well, here\u0027s a and here\u0027s not,"},{"Start":"04:22.073 ","End":"04:28.756","Text":"so the probability is 0.22 and the probability of not a and b,"},{"Start":"04:28.756 ","End":"04:32.345","Text":"well, that\u0027s not a and that\u0027s b."},{"Start":"04:32.345 ","End":"04:35.130","Text":"That\u0027s 0.42."},{"Start":"04:35.130 ","End":"04:38.890","Text":"The probability of a intersect b,"},{"Start":"04:38.890 ","End":"04:41.095","Text":"we\u0027ve calculated that before."},{"Start":"04:41.095 ","End":"04:46.480","Text":"That\u0027s 0.08, that\u0027s right here."},{"Start":"04:47.260 ","End":"04:51.830","Text":"Now, let\u0027s plug in these values back on to the tree."},{"Start":"04:51.830 ","End":"04:53.600","Text":"Well, that\u0027s a and not b,"},{"Start":"04:53.600 ","End":"04:55.250","Text":"that\u0027s only the Internet."},{"Start":"04:55.250 ","End":"04:58.620","Text":"Right here we\u0027ll write 0.22."},{"Start":"04:59.030 ","End":"05:02.060","Text":"Here not a and b,"},{"Start":"05:02.060 ","End":"05:03.784","Text":"that\u0027s only the newspaper."},{"Start":"05:03.784 ","End":"05:08.130","Text":"That means that the probability is 0.42."},{"Start":"05:08.170 ","End":"05:17.647","Text":"Here the probability of both the Internet and the newspaper, that\u0027s 0.08."},{"Start":"05:17.647 ","End":"05:24.410","Text":"Now, it doesn\u0027t matter what the probability is right here."},{"Start":"05:24.410 ","End":"05:31.160","Text":"Because if he doesn\u0027t see the ads either in the newspaper or in the Internet,"},{"Start":"05:31.160 ","End":"05:33.170","Text":"he is not going to call Jack."},{"Start":"05:33.170 ","End":"05:35.825","Text":"We\u0027re basically stopped here."},{"Start":"05:35.825 ","End":"05:41.330","Text":"Now, let\u0027s fill out the probabilities here."},{"Start":"05:41.330 ","End":"05:43.925","Text":"If you only saw the ad in the Internet,"},{"Start":"05:43.925 ","End":"05:47.570","Text":"the probability that he\u0027ll call Jack is 0.7 so"},{"Start":"05:47.570 ","End":"05:51.875","Text":"that the probability of not calling him is 0.3."},{"Start":"05:51.875 ","End":"05:57.600","Text":"Likewise here that will be 0.4 and that\u0027ll be 0.1."},{"Start":"06:00.200 ","End":"06:05.552","Text":"Now that we have all the probabilities filled out in the tree right here,"},{"Start":"06:05.552 ","End":"06:09.295","Text":"well, let\u0027s try to answer question number 1."},{"Start":"06:09.295 ","End":"06:14.590","Text":"It asks us, what\u0027s the probability of the person that\u0027s interested in buying a car,"},{"Start":"06:14.590 ","End":"06:17.545","Text":"what\u0027s the probability of him calling Jack?"},{"Start":"06:17.545 ","End":"06:20.650","Text":"Well, we have to figure out the probability of"},{"Start":"06:20.650 ","End":"06:25.242","Text":"this branch right here that he saw at only the Internet and he calls,"},{"Start":"06:25.242 ","End":"06:28.645","Text":"or if he saw the ad only in the newspaper and he calls,"},{"Start":"06:28.645 ","End":"06:34.015","Text":"or if he saw the ad both in the Internet and in the newspaper and he calls."},{"Start":"06:34.015 ","End":"06:43.980","Text":"Well, first the probability then of a person calling equals to 0.22"},{"Start":"06:43.980 ","End":"06:51.750","Text":"times 0.7 plus 0.42"},{"Start":"06:51.750 ","End":"07:00.615","Text":"times 0.6 plus 0.08 times 0.9,"},{"Start":"07:00.615 ","End":"07:09.230","Text":"and that equals to 0.478."},{"Start":"07:09.230 ","End":"07:10.835","Text":"Now, question 2 asks,"},{"Start":"07:10.835 ","End":"07:15.370","Text":"a person that\u0027s interested in buying a used car calls Jack."},{"Start":"07:15.370 ","End":"07:20.675","Text":"The question is, what\u0027s the probability that he saw both ads?"},{"Start":"07:20.675 ","End":"07:23.300","Text":"Well, let\u0027s take a look."},{"Start":"07:23.300 ","End":"07:26.870","Text":"We know that this is a conditional probability because we\u0027re"},{"Start":"07:26.870 ","End":"07:31.370","Text":"given that a person that was interested in buying a car, he call Jack."},{"Start":"07:31.370 ","End":"07:35.180","Text":"We\u0027re asked, what\u0027s the probability that the saw both ads?"},{"Start":"07:35.180 ","End":"07:38.765","Text":"That means that we can write this down."},{"Start":"07:38.765 ","End":"07:44.834","Text":"Given that a person saw both ads a and b,"},{"Start":"07:44.834 ","End":"07:51.840","Text":"what\u0027s the probability that he saw both ads given that he calls?"},{"Start":"07:52.990 ","End":"08:01.830","Text":"Now, the probability of that is that he saw both a and b,"},{"Start":"08:01.830 ","End":"08:08.000","Text":"that he saw the ad both in the Internet and in the newspaper and that he"},{"Start":"08:08.000 ","End":"08:16.530","Text":"calls divided by the probability of a person calling."},{"Start":"08:17.410 ","End":"08:22.535","Text":"We\u0027ve just calculated that right up here."},{"Start":"08:22.535 ","End":"08:25.870","Text":"The probability of a person calling is what?"},{"Start":"08:25.870 ","End":"08:28.880","Text":"That\u0027s 0.478."},{"Start":"08:28.880 ","End":"08:35.105","Text":"What\u0027s the probability of a person seeing both ad and that he calls?"},{"Start":"08:35.105 ","End":"08:38.420","Text":"Well, if we take a look at our tree,"},{"Start":"08:38.420 ","End":"08:40.475","Text":"that\u0027s this guy right here,"},{"Start":"08:40.475 ","End":"08:46.775","Text":"he saw the ad both in the Internet and in the newspaper and that he calls."},{"Start":"08:46.775 ","End":"08:55.145","Text":"That means that that\u0027s 0.08 times 0.9."},{"Start":"08:55.145 ","End":"08:59.430","Text":"That equals to 0.15."}],"ID":12956},{"Watched":false,"Name":"Exercise 8","Duration":"11m 21s","ChapterTopicVideoID":12478,"CourseChapterTopicPlaylistID":245035,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"This question is taken from the field of electrical engineering."},{"Start":"00:03.945 ","End":"00:08.580","Text":"Now, consider the following electrical system. Here we go."},{"Start":"00:08.580 ","End":"00:11.820","Text":"We have point A, we have point B,"},{"Start":"00:11.820 ","End":"00:17.355","Text":"and we have 5 units that are configured in this manner."},{"Start":"00:17.355 ","End":"00:24.420","Text":"Each unit works independently with the probability of p. Now,"},{"Start":"00:24.420 ","End":"00:26.175","Text":"in order for the system to work,"},{"Start":"00:26.175 ","End":"00:31.170","Text":"current must flow from point A to point B."},{"Start":"00:31.170 ","End":"00:35.300","Text":"We\u0027re asked to prove that"},{"Start":"00:35.300 ","End":"00:40.415","Text":"the probability of the system working is this expression right here."},{"Start":"00:40.415 ","End":"00:44.750","Text":"This isn\u0027t as difficult as it seems."},{"Start":"00:44.750 ","End":"00:50.075","Text":"Let\u0027s get started and let\u0027s analyze the system a little bit more closely."},{"Start":"00:50.075 ","End":"00:56.220","Text":"In order for current to flow from A to B,"},{"Start":"00:56.220 ","End":"00:59.205","Text":"it has to go in 1 of 2 directions."},{"Start":"00:59.205 ","End":"01:01.325","Text":"It has to go in this direction,"},{"Start":"01:01.325 ","End":"01:04.260","Text":"or it has to go in this direction."},{"Start":"01:04.450 ","End":"01:10.360","Text":"Let\u0027s define this as C."},{"Start":"01:10.360 ","End":"01:14.360","Text":"C will be the event that the current"},{"Start":"01:14.360 ","End":"01:18.350","Text":"is able to pass from this point right here to this point right here,"},{"Start":"01:18.350 ","End":"01:20.670","Text":"in order to get to B."},{"Start":"01:20.680 ","End":"01:25.040","Text":"Likewise, we\u0027ll define this as D,"},{"Start":"01:25.040 ","End":"01:28.130","Text":"and that\u0027s the event of current being able to"},{"Start":"01:28.130 ","End":"01:32.960","Text":"pass from here to here in order to get to B."},{"Start":"01:32.960 ","End":"01:41.190","Text":"Now, the whole system will work if C will work and D doesn\u0027t work,"},{"Start":"01:41.190 ","End":"01:47.160","Text":"because then a current can go through C or if D works and C doesn\u0027t work,"},{"Start":"01:47.160 ","End":"01:50.865","Text":"or if both C and D work,"},{"Start":"01:50.865 ","End":"01:53.360","Text":"and 1 with the system not work,"},{"Start":"01:53.360 ","End":"01:58.050","Text":"when both C and D do not work."},{"Start":"01:58.910 ","End":"02:05.855","Text":"Let\u0027s see how the system can work and what\u0027s the probability of the system working."},{"Start":"02:05.855 ","End":"02:08.240","Text":"Here, we have the system again."},{"Start":"02:08.240 ","End":"02:10.940","Text":"Here, we\u0027ve defined event C,"},{"Start":"02:10.940 ","End":"02:15.490","Text":"and here we\u0027ve defined event D. Now,"},{"Start":"02:15.490 ","End":"02:22.544","Text":"the probability of the system working is equal to what?"},{"Start":"02:22.544 ","End":"02:27.995","Text":"As we said, that the current will go either from C to"},{"Start":"02:27.995 ","End":"02:35.585","Text":"B or from D to B or it can go through both C and D to get to B."},{"Start":"02:35.585 ","End":"02:40.145","Text":"That\u0027s quite complicated to calculate all the probabilities,"},{"Start":"02:40.145 ","End":"02:46.025","Text":"but we know that the system will not work when both C and D do not work."},{"Start":"02:46.025 ","End":"02:48.935","Text":"Now, what\u0027s the probability of a system working?"},{"Start":"02:48.935 ","End":"02:55.110","Text":"That\u0027s a probability of 1 minus the system not working."},{"Start":"02:55.960 ","End":"02:59.165","Text":"When does the system not work?"},{"Start":"02:59.165 ","End":"03:07.995","Text":"Well, that\u0027s the probability of C not working and D not working."},{"Start":"03:07.995 ","End":"03:14.015","Text":"Because each unit here is independent,"},{"Start":"03:14.015 ","End":"03:18.515","Text":"that means that both C and D are independent."},{"Start":"03:18.515 ","End":"03:21.485","Text":"That means I can write because of independent."},{"Start":"03:21.485 ","End":"03:23.435","Text":"I can write this down,"},{"Start":"03:23.435 ","End":"03:34.115","Text":"that that equals to 1 minus the probability of not C times the probability of not D. Now,"},{"Start":"03:34.115 ","End":"03:39.800","Text":"what\u0027s the probability of a system not working or a unit not working?"},{"Start":"03:39.800 ","End":"03:44.090","Text":"We know that a unit works with a probability of p that\u0027s given to us."},{"Start":"03:44.090 ","End":"03:48.125","Text":"What\u0027s the probability of a unit not working?"},{"Start":"03:48.125 ","End":"03:52.730","Text":"Well, that\u0027s 1 minus p. In the case of event D,"},{"Start":"03:52.730 ","End":"03:54.380","Text":"we only have 1 unit."},{"Start":"03:54.380 ","End":"04:02.610","Text":"The probability of D not working is 1 minus p. Let\u0027s write this out."},{"Start":"04:02.710 ","End":"04:07.730","Text":"The probability of the system working is 1 minus"},{"Start":"04:07.730 ","End":"04:13.430","Text":"the probability of either C not working times the probability of D not working."},{"Start":"04:13.430 ","End":"04:18.935","Text":"We said that equals to 1 minus p. Excellent."},{"Start":"04:18.935 ","End":"04:26.645","Text":"Now, all we have to do is to calculate the probability of C not working."},{"Start":"04:26.645 ","End":"04:31.655","Text":"Let\u0027s take a look at C. Now,"},{"Start":"04:31.655 ","End":"04:39.195","Text":"as we can see, C is comprised of 2 modules that are built in a series."},{"Start":"04:39.195 ","End":"04:40.770","Text":"It\u0027s called a C_1,"},{"Start":"04:40.770 ","End":"04:42.105","Text":"it\u0027s called a C_2."},{"Start":"04:42.105 ","End":"04:43.725","Text":"These are the modules."},{"Start":"04:43.725 ","End":"04:46.530","Text":"In order for C_2 to work,"},{"Start":"04:46.530 ","End":"04:52.010","Text":"C_1 has to work, and C_2 has to work because they\u0027re built in a series."},{"Start":"04:52.010 ","End":"04:55.820","Text":"The probability of C working is equal to"},{"Start":"04:55.820 ","End":"05:01.550","Text":"the probability of C_1 working and the probability of C_2 working."},{"Start":"05:01.550 ","End":"05:06.815","Text":"Because C_1 and C_2 are independent, why is that again?"},{"Start":"05:06.815 ","End":"05:09.720","Text":"That\u0027s because each unit here is independent,"},{"Start":"05:09.720 ","End":"05:17.045","Text":"then we can write this as the probability of C_1 times the probability of C_2."},{"Start":"05:17.045 ","End":"05:19.655","Text":"Now, we\u0027re looking for,"},{"Start":"05:19.655 ","End":"05:21.475","Text":"not the probability of C,"},{"Start":"05:21.475 ","End":"05:26.480","Text":"but the probability of not C. That equals to"},{"Start":"05:26.480 ","End":"05:33.363","Text":"1 minus the probability of C. That equals to 1 minus,"},{"Start":"05:33.363 ","End":"05:36.500","Text":"and here\u0027s the probability of C,"},{"Start":"05:36.500 ","End":"05:41.070","Text":"probability of C_1 times the probability of C_2."},{"Start":"05:41.350 ","End":"05:44.345","Text":"Now, if that\u0027s the case,"},{"Start":"05:44.345 ","End":"05:47.395","Text":"what\u0027s the probability of C_1 and C_2?"},{"Start":"05:47.395 ","End":"05:50.270","Text":"Well, since they\u0027re identical,"},{"Start":"05:50.270 ","End":"05:52.070","Text":"and we can see that they\u0027re identical,"},{"Start":"05:52.070 ","End":"05:55.490","Text":"they have 2 units working in parallel."},{"Start":"05:55.490 ","End":"05:58.610","Text":"Let\u0027s just focus on 1 of them."},{"Start":"05:58.610 ","End":"06:04.530","Text":"Let\u0027s say C_1, and see what\u0027s the probability of C_1."},{"Start":"06:04.530 ","End":"06:10.430","Text":"Well, again, we have the current can flow through this unit right here,"},{"Start":"06:10.430 ","End":"06:12.170","Text":"or through this unit right here,"},{"Start":"06:12.170 ","End":"06:13.220","Text":"or through both of them."},{"Start":"06:13.220 ","End":"06:15.500","Text":"Because they\u0027re in parallel,"},{"Start":"06:15.500 ","End":"06:18.890","Text":"it\u0027s easier to take the 1 minus approach."},{"Start":"06:18.890 ","End":"06:22.760","Text":"That means that the probability of C_1 working equals"},{"Start":"06:22.760 ","End":"06:27.720","Text":"1 minus the probability of C_1 not working."},{"Start":"06:27.720 ","End":"06:30.647","Text":"Now, when does C_1 not work?"},{"Start":"06:30.647 ","End":"06:34.110","Text":"when both of these units won\u0027t work."},{"Start":"06:34.210 ","End":"06:37.970","Text":"What\u0027s the probability of each 1 working?"},{"Start":"06:37.970 ","End":"06:42.560","Text":"That\u0027s p. The probability of each 1 not"},{"Start":"06:42.560 ","End":"06:48.525","Text":"working is 1 minus p, now squared."},{"Start":"06:48.525 ","End":"06:50.840","Text":"Why squared? Because we have both of them."},{"Start":"06:50.840 ","End":"06:53.345","Text":"That\u0027s 1 minus, that\u0027s from here."},{"Start":"06:53.345 ","End":"06:58.940","Text":"The probability of C_1 not working is when both units do not work."},{"Start":"06:58.940 ","End":"07:04.215","Text":"That\u0027s that 1 minus p times 1 minus p. That\u0027s here."},{"Start":"07:04.215 ","End":"07:10.355","Text":"Excellent. Now that we have the probability of C_1,"},{"Start":"07:10.355 ","End":"07:12.785","Text":"we can input that here."},{"Start":"07:12.785 ","End":"07:18.509","Text":"The probability of not C equals 1 minus,"},{"Start":"07:18.509 ","End":"07:20.775","Text":"now, what\u0027s the probability of C_1?"},{"Start":"07:20.775 ","End":"07:22.710","Text":"That\u0027s this guy right here."},{"Start":"07:22.710 ","End":"07:25.245","Text":"So that\u0027s 1 minus,"},{"Start":"07:25.245 ","End":"07:32.570","Text":"1 minus p squared times the probability of C_2."},{"Start":"07:32.570 ","End":"07:36.590","Text":"That equals the probability of C_1 because they\u0027re identical."},{"Start":"07:36.590 ","End":"07:41.435","Text":"We just said that. Let\u0027s just square that."},{"Start":"07:41.435 ","End":"07:47.450","Text":"Excellent. Now, we have the probability of not C,"},{"Start":"07:47.450 ","End":"07:49.625","Text":"and this is what we\u0027re looking for."},{"Start":"07:49.625 ","End":"07:54.125","Text":"Let\u0027s input that into this equation right here."},{"Start":"07:54.125 ","End":"07:59.210","Text":"Now, the probability of the whole system"},{"Start":"07:59.210 ","End":"08:08.675","Text":"working equals 1 minus the probability of not C. Well,"},{"Start":"08:08.675 ","End":"08:12.525","Text":"the probability of not C is this guy right here."},{"Start":"08:12.525 ","End":"08:15.655","Text":"That\u0027s 1 minus square brackets,"},{"Start":"08:15.655 ","End":"08:17.460","Text":"1 minus round brackets,"},{"Start":"08:17.460 ","End":"08:19.409","Text":"1 minus p squared,"},{"Start":"08:19.409 ","End":"08:25.960","Text":"squared times 1 minus p."},{"Start":"08:28.010 ","End":"08:32.240","Text":"All we\u0027re left with is an algebraic problem."},{"Start":"08:32.240 ","End":"08:40.895","Text":"Let\u0027s solve this. Let\u0027s take this expression and let\u0027s expand this part right here."},{"Start":"08:40.895 ","End":"08:46.190","Text":"That will equal to 1 minus squiggly brackets,"},{"Start":"08:46.190 ","End":"08:49.265","Text":"1 minus square brackets."},{"Start":"08:49.265 ","End":"08:52.445","Text":"Now, 1 minus."},{"Start":"08:52.445 ","End":"08:54.410","Text":"Now let\u0027s expand this guy."},{"Start":"08:54.410 ","End":"09:00.090","Text":"That\u0027s 1 minus 2p plus p squared."},{"Start":"09:03.310 ","End":"09:12.810","Text":"This needs to be squared times 1 minus p. Let\u0027s continue."},{"Start":"09:14.870 ","End":"09:20.065","Text":"That will be 1 minus squiggly brackets,"},{"Start":"09:20.065 ","End":"09:24.295","Text":"1 minus square brackets."},{"Start":"09:24.295 ","End":"09:30.785","Text":"Let\u0027s just simplify what\u0027s in the square brackets here now,"},{"Start":"09:30.785 ","End":"09:33.950","Text":"1 minus 1, that\u0027s 0."},{"Start":"09:33.950 ","End":"09:35.210","Text":"Cancels each other out,"},{"Start":"09:35.210 ","End":"09:38.075","Text":"but the minus switches the signs here."},{"Start":"09:38.075 ","End":"09:47.095","Text":"What we have here is 2p minus p squared, squared."},{"Start":"09:47.095 ","End":"09:56.285","Text":"Let\u0027s return the squiggly brackets times 1 minus p. Now,"},{"Start":"09:56.285 ","End":"10:04.280","Text":"let\u0027s take the 1 minus p right here and multiply it in the squiggly brackets."},{"Start":"10:04.280 ","End":"10:06.295","Text":"We get 1 minus."},{"Start":"10:06.295 ","End":"10:08.810","Text":"What do we have inside the squiggly brackets?"},{"Start":"10:08.810 ","End":"10:14.660","Text":"That\u0027s 1 times 1 minus p. That\u0027s 1 times this guy."},{"Start":"10:14.660 ","End":"10:18.680","Text":"Minus 1 minus p times what?"},{"Start":"10:18.680 ","End":"10:20.585","Text":"Times what we have in the square brackets."},{"Start":"10:20.585 ","End":"10:24.045","Text":"That\u0027s 2p minus p squared,"},{"Start":"10:24.045 ","End":"10:30.870","Text":"squared. That\u0027s squiggly brackets."},{"Start":"10:30.870 ","End":"10:33.610","Text":"Now, let\u0027s simplify."},{"Start":"10:34.460 ","End":"10:37.545","Text":"We have 1 minus,"},{"Start":"10:37.545 ","End":"10:39.780","Text":"now 1 times 1 minus p,"},{"Start":"10:39.780 ","End":"10:42.530","Text":"1 minus 1 cancels each other out."},{"Start":"10:42.530 ","End":"10:48.975","Text":"Minus minus p equals p, minus minus,"},{"Start":"10:48.975 ","End":"10:52.140","Text":"this expression is plus this expression,"},{"Start":"10:52.140 ","End":"11:02.000","Text":"which is 1 minus p times 2p minus p squared squared."},{"Start":"11:02.000 ","End":"11:07.550","Text":"Now, if we look at this expression and this expression right here,"},{"Start":"11:07.550 ","End":"11:09.745","Text":"we see that they\u0027re the same."},{"Start":"11:09.745 ","End":"11:14.510","Text":"That means that we\u0027ve proved that the probability of"},{"Start":"11:14.510 ","End":"11:21.390","Text":"the whole system working is this guy right here. Congratulate us."}],"ID":12957}],"Thumbnail":null,"ID":245035}]

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