WEBVTT 00:00:02.910 --> 00:00:06.670 This is a simple example where we want to just apply the 00:00:06.670 --> 00:00:08.630 formula for conditional probabilities 00:00:08.630 --> 00:00:10.610 and see what we get. 00:00:10.610 --> 00:00:15.140 The example involves a four-sided die, if you can 00:00:15.140 --> 00:00:19.300 imagine such an object, which we roll twice, and we record 00:00:19.300 --> 00:00:21.830 the first roll, and the second roll. 00:00:21.830 --> 00:00:24.660 So there are 16 possible outcomes. 00:00:24.660 --> 00:00:28.350 We assume to keep things simple, that each one of those 00:00:28.350 --> 00:00:32.820 16 possible outcomes, each one of them has the same 00:00:32.820 --> 00:00:35.800 probability, so each outcome has the probability 1/16. 00:00:39.820 --> 00:00:43.250 Let us consider now a particular event B on which 00:00:43.250 --> 00:00:44.460 we're going to condition. 00:00:44.460 --> 00:00:49.286 This is the event under which the smaller of the two die 00:00:49.286 --> 00:00:53.990 rolls is equal to 2, which means that one of the dice 00:00:53.990 --> 00:00:57.970 must have resulted in two, and the other die has resulted in 00:00:57.970 --> 00:01:01.220 something which is 2 or larger. 00:01:01.220 --> 00:01:03.800 So this can happen in multiple ways. 00:01:03.800 --> 00:01:07.130 And here are the different ways that it can happen. 00:01:07.130 --> 00:01:14.100 So at 2, 2, or 2, 3, or 2, 4; then a 3, 2 and a 4, 2. 00:01:14.100 --> 00:01:18.080 All of these are outcomes in which one of the dice has a 00:01:18.080 --> 00:01:20.200 value equal to 2, and the other die 00:01:20.200 --> 00:01:22.880 is at least as large. 00:01:22.880 --> 00:01:25.560 So we condition on this event. 00:01:25.560 --> 00:01:29.520 This results in a conditional model where each one of those 00:01:29.520 --> 00:01:33.820 five outcomes are equally likely since they used to be 00:01:33.820 --> 00:01:36.759 equally likely in the original model. 00:01:36.759 --> 00:01:38.539 Now let's look at this quantity. 00:01:38.539 --> 00:01:41.340 The maximum of the two die rolls-- 00:01:41.340 --> 00:01:43.960 that is, the largest of the results. 00:01:43.960 --> 00:01:47.710 And let us try to calculate the following quantity-- 00:01:47.710 --> 00:01:52.240 the conditional probability that the maximum is equal to 1 00:01:52.240 --> 00:01:55.420 given that the minimum is equal to 2. 00:01:55.420 --> 00:01:57.820 So this is the conditional probability of 00:01:57.820 --> 00:02:00.050 this particular outcome. 00:02:00.050 --> 00:02:02.480 Well, this particular outcome cannot happen. 00:02:02.480 --> 00:02:06.450 If I tell you that the smaller number is 2, then the larger 00:02:06.450 --> 00:02:10.669 number cannot be equal to 1, so this outcome is impossible, 00:02:10.669 --> 00:02:14.870 and therefore this conditional probability is equal to 0. 00:02:14.870 --> 00:02:17.550 Let's do something a little more interesting. 00:02:17.550 --> 00:02:20.910 Let us now look at the conditional probability that 00:02:20.910 --> 00:02:26.690 the maximum is equal to 3 given the information that 00:02:26.690 --> 00:02:29.190 event B has occurred. 00:02:29.190 --> 00:02:32.210 It's best to draw a picture and see what that event 00:02:32.210 --> 00:02:33.990 corresponds to. 00:02:33.990 --> 00:02:35.829 M is equal to 3-- 00:02:35.829 --> 00:02:38.110 the maximum is equal to 3-- 00:02:38.110 --> 00:02:44.500 if one of the dice resulted in a 3, and the other die 00:02:44.500 --> 00:02:48.190 resulted in something that's 3 or less. 00:02:48.190 --> 00:02:53.770 So this event here corresponds to the blue 00:02:53.770 --> 00:02:56.630 region in this diagram. 00:02:56.630 --> 00:03:00.610 Now let us try to calculate the conditional probability by 00:03:00.610 --> 00:03:03.110 just following the definition. 00:03:03.110 --> 00:03:07.910 The conditional probability of one event given another is the 00:03:07.910 --> 00:03:10.860 probability that both of them-- 00:03:10.860 --> 00:03:12.900 both of the two events-- 00:03:12.900 --> 00:03:18.810 occur, divided by the probability of the 00:03:18.810 --> 00:03:20.690 conditioning event. 00:03:20.690 --> 00:03:23.880 That is, out of the total probability in the 00:03:23.880 --> 00:03:27.000 conditioning event, we ask, what fraction of that 00:03:27.000 --> 00:03:31.060 probability is assigned to outcomes in which the event of 00:03:31.060 --> 00:03:33.730 interest is also happening? 00:03:33.730 --> 00:03:36.200 So what is this event? 00:03:36.200 --> 00:03:40.100 The maximum is equal to 3, which is the blue event. 00:03:40.100 --> 00:03:44.500 And simultaneously, the red event is happening. 00:03:44.500 --> 00:03:48.840 These two events intersect only in two places. 00:03:48.840 --> 00:03:52.020 This is the intersection of the two events. 00:03:52.020 --> 00:03:58.480 And the probability of that intersection is 2 out of 16, 00:03:58.480 --> 00:04:02.720 since there's 16 outcomes and that event happens only with 00:04:02.720 --> 00:04:04.910 two particular outcomes. 00:04:04.910 --> 00:04:09.360 So this gives us 2/16 in the numerator. 00:04:09.360 --> 00:04:11.210 How about the denominator? 00:04:11.210 --> 00:04:14.990 Event B consists of a total of five possible outcomes. 00:04:14.990 --> 00:04:20.360 Each one has probability 1/16, so this is 5/16, so the final 00:04:20.360 --> 00:04:23.860 answer is 2/5. 00:04:23.860 --> 00:04:28.470 We could have gotten that same answer in a simple and perhaps 00:04:28.470 --> 00:04:29.870 more intuitive way. 00:04:29.870 --> 00:04:33.409 In the original model, all outcomes were equally likely. 00:04:33.409 --> 00:04:37.470 Therefore, in the conditional model, the five outcomes that 00:04:37.470 --> 00:04:41.470 belong to B should also be equally likely. 00:04:41.470 --> 00:04:45.980 Out of those five, there's two of them that make the event of 00:04:45.980 --> 00:04:48.190 interest to occur. 00:04:48.190 --> 00:04:52.500 So given that we live in B, there's two ways out of five 00:04:52.500 --> 00:04:54.940 that the event of interest will materialize. 00:04:54.940 --> 00:04:57.159 So the event of interest has 00:04:57.159 --> 00:04:59.230 conditional probability [equal to] 00:04:59.230 --> 00:05:00.480 2/5.