1 00:00:02,910 --> 00:00:06,670 This is a simple example where we want to just apply the 2 00:00:06,670 --> 00:00:08,630 formula for conditional probabilities 3 00:00:08,630 --> 00:00:10,610 and see what we get. 4 00:00:10,610 --> 00:00:15,140 The example involves a four-sided die, if you can 5 00:00:15,140 --> 00:00:19,300 imagine such an object, which we roll twice, and we record 6 00:00:19,300 --> 00:00:21,830 the first roll, and the second roll. 7 00:00:21,830 --> 00:00:24,660 So there are 16 possible outcomes. 8 00:00:24,660 --> 00:00:28,350 We assume to keep things simple, that each one of those 9 00:00:28,350 --> 00:00:32,820 16 possible outcomes, each one of them has the same 10 00:00:32,820 --> 00:00:35,800 probability, so each outcome has the probability 1/16. 11 00:00:39,820 --> 00:00:43,250 Let us consider now a particular event B on which 12 00:00:43,250 --> 00:00:44,460 we're going to condition. 13 00:00:44,460 --> 00:00:49,286 This is the event under which the smaller of the two die 14 00:00:49,286 --> 00:00:53,990 rolls is equal to 2, which means that one of the dice 15 00:00:53,990 --> 00:00:57,970 must have resulted in two, and the other die has resulted in 16 00:00:57,970 --> 00:01:01,220 something which is 2 or larger. 17 00:01:01,220 --> 00:01:03,800 So this can happen in multiple ways. 18 00:01:03,800 --> 00:01:07,130 And here are the different ways that it can happen. 19 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. 20 00:01:14,100 --> 00:01:18,080 All of these are outcomes in which one of the dice has a 21 00:01:18,080 --> 00:01:20,200 value equal to 2, and the other die 22 00:01:20,200 --> 00:01:22,880 is at least as large. 23 00:01:22,880 --> 00:01:25,560 So we condition on this event. 24 00:01:25,560 --> 00:01:29,520 This results in a conditional model where each one of those 25 00:01:29,520 --> 00:01:33,820 five outcomes are equally likely since they used to be 26 00:01:33,820 --> 00:01:36,759 equally likely in the original model. 27 00:01:36,759 --> 00:01:38,539 Now let's look at this quantity. 28 00:01:38,539 --> 00:01:41,340 The maximum of the two die rolls-- 29 00:01:41,340 --> 00:01:43,960 that is, the largest of the results. 30 00:01:43,960 --> 00:01:47,710 And let us try to calculate the following quantity-- 31 00:01:47,710 --> 00:01:52,240 the conditional probability that the maximum is equal to 1 32 00:01:52,240 --> 00:01:55,420 given that the minimum is equal to 2. 33 00:01:55,420 --> 00:01:57,820 So this is the conditional probability of 34 00:01:57,820 --> 00:02:00,050 this particular outcome. 35 00:02:00,050 --> 00:02:02,480 Well, this particular outcome cannot happen. 36 00:02:02,480 --> 00:02:06,450 If I tell you that the smaller number is 2, then the larger 37 00:02:06,450 --> 00:02:10,669 number cannot be equal to 1, so this outcome is impossible, 38 00:02:10,669 --> 00:02:14,870 and therefore this conditional probability is equal to 0. 39 00:02:14,870 --> 00:02:17,550 Let's do something a little more interesting. 40 00:02:17,550 --> 00:02:20,910 Let us now look at the conditional probability that 41 00:02:20,910 --> 00:02:26,690 the maximum is equal to 3 given the information that 42 00:02:26,690 --> 00:02:29,190 event B has occurred. 43 00:02:29,190 --> 00:02:32,210 It's best to draw a picture and see what that event 44 00:02:32,210 --> 00:02:33,990 corresponds to. 45 00:02:33,990 --> 00:02:35,829 M is equal to 3-- 46 00:02:35,829 --> 00:02:38,110 the maximum is equal to 3-- 47 00:02:38,110 --> 00:02:44,500 if one of the dice resulted in a 3, and the other die 48 00:02:44,500 --> 00:02:48,190 resulted in something that's 3 or less. 49 00:02:48,190 --> 00:02:53,770 So this event here corresponds to the blue 50 00:02:53,770 --> 00:02:56,630 region in this diagram. 51 00:02:56,630 --> 00:03:00,610 Now let us try to calculate the conditional probability by 52 00:03:00,610 --> 00:03:03,110 just following the definition. 53 00:03:03,110 --> 00:03:07,910 The conditional probability of one event given another is the 54 00:03:07,910 --> 00:03:10,860 probability that both of them-- 55 00:03:10,860 --> 00:03:12,900 both of the two events-- 56 00:03:12,900 --> 00:03:18,810 occur, divided by the probability of the 57 00:03:18,810 --> 00:03:20,690 conditioning event. 58 00:03:20,690 --> 00:03:23,880 That is, out of the total probability in the 59 00:03:23,880 --> 00:03:27,000 conditioning event, we ask, what fraction of that 60 00:03:27,000 --> 00:03:31,060 probability is assigned to outcomes in which the event of 61 00:03:31,060 --> 00:03:33,730 interest is also happening? 62 00:03:33,730 --> 00:03:36,200 So what is this event? 63 00:03:36,200 --> 00:03:40,100 The maximum is equal to 3, which is the blue event. 64 00:03:40,100 --> 00:03:44,500 And simultaneously, the red event is happening. 65 00:03:44,500 --> 00:03:48,840 These two events intersect only in two places. 66 00:03:48,840 --> 00:03:52,020 This is the intersection of the two events. 67 00:03:52,020 --> 00:03:58,480 And the probability of that intersection is 2 out of 16, 68 00:03:58,480 --> 00:04:02,720 since there's 16 outcomes and that event happens only with 69 00:04:02,720 --> 00:04:04,910 two particular outcomes. 70 00:04:04,910 --> 00:04:09,360 So this gives us 2/16 in the numerator. 71 00:04:09,360 --> 00:04:11,210 How about the denominator? 72 00:04:11,210 --> 00:04:14,990 Event B consists of a total of five possible outcomes. 73 00:04:14,990 --> 00:04:20,360 Each one has probability 1/16, so this is 5/16, so the final 74 00:04:20,360 --> 00:04:23,860 answer is 2/5. 75 00:04:23,860 --> 00:04:28,470 We could have gotten that same answer in a simple and perhaps 76 00:04:28,470 --> 00:04:29,870 more intuitive way. 77 00:04:29,870 --> 00:04:33,409 In the original model, all outcomes were equally likely. 78 00:04:33,409 --> 00:04:37,470 Therefore, in the conditional model, the five outcomes that 79 00:04:37,470 --> 00:04:41,470 belong to B should also be equally likely. 80 00:04:41,470 --> 00:04:45,980 Out of those five, there's two of them that make the event of 81 00:04:45,980 --> 00:04:48,190 interest to occur. 82 00:04:48,190 --> 00:04:52,500 So given that we live in B, there's two ways out of five 83 00:04:52,500 --> 00:04:54,940 that the event of interest will materialize. 84 00:04:54,940 --> 00:04:57,159 So the event of interest has 85 00:04:57,159 --> 00:04:59,230 conditional probability [equal to] 86 00:04:59,230 --> 00:05:00,480 2/5.