1 00:00:00,900 --> 00:00:04,100 Let us now conclude with a fun problem, which is also a 2 00:00:04,100 --> 00:00:05,970 little bit of a puzzle. 3 00:00:05,970 --> 00:00:08,210 We are told that the king comes from a 4 00:00:08,210 --> 00:00:10,460 family of two children. 5 00:00:10,460 --> 00:00:13,980 What is the probability that his sibling is female? 6 00:00:13,980 --> 00:00:17,220 Well, the problem is too loosely stated, so we need to 7 00:00:17,220 --> 00:00:19,370 start by making some assumptions. 8 00:00:19,370 --> 00:00:22,800 First, let's assume that we're dealing with an anachronistic 9 00:00:22,800 --> 00:00:25,840 kingdom where boys have precedence. 10 00:00:29,930 --> 00:00:34,640 In other words, if the royal family has two children, one 11 00:00:34,640 --> 00:00:38,960 of which is a boy and one is a girl, it is always the boy who 12 00:00:38,960 --> 00:00:43,830 becomes king, even if the girl was born first. 13 00:00:43,830 --> 00:00:48,380 Let us also assume that when a child is born, it has 50% 14 00:00:48,380 --> 00:00:51,820 probability of being a boy and 50% 15 00:00:51,820 --> 00:00:53,320 probability of being a girl. 16 00:00:58,890 --> 00:01:03,790 And in addition, let's assume that different children are 17 00:01:03,790 --> 00:01:07,195 independent as far as their gender is concerned. 18 00:01:10,070 --> 00:01:12,850 Given these assumptions, perhaps we 19 00:01:12,850 --> 00:01:15,240 can argue as follows. 20 00:01:15,240 --> 00:01:18,620 The king's sibling is a child which is 21 00:01:18,620 --> 00:01:20,600 independent from the king. 22 00:01:20,600 --> 00:01:24,710 Its gender is independent from the king's gender, so it's 23 00:01:24,710 --> 00:01:29,160 going to be a girl with probability 1/2. 24 00:01:29,160 --> 00:01:34,750 And so this is one possible answer to this problem. 25 00:01:34,750 --> 00:01:36,770 Is this a correct answer? 26 00:01:36,770 --> 00:01:37,640 Well, let's see. 27 00:01:37,640 --> 00:01:40,710 We have to make a more precise model, so let's 28 00:01:40,710 --> 00:01:42,180 go ahead with it. 29 00:01:42,180 --> 00:01:47,460 We have two children, so there are four possible outcomes-- 30 00:01:47,460 --> 00:01:53,240 boy, boy; boy, girl; girl, boy; and girl, girl. 31 00:01:55,780 --> 00:02:01,290 Each one of these outcomes has probability 1/4 according to 32 00:02:01,290 --> 00:02:02,920 our assumptions. 33 00:02:02,920 --> 00:02:05,750 For example, the probability of a boy followed by a boy is 34 00:02:05,750 --> 00:02:10,960 1/2 times 1/2, where we're also using independence. 35 00:02:10,960 --> 00:02:15,150 So each one of these four outcomes has the same 36 00:02:15,150 --> 00:02:16,460 probability, 1/4. 37 00:02:19,660 --> 00:02:23,740 Now, we know that there is a king, so there must be at 38 00:02:23,740 --> 00:02:25,720 least one boy. 39 00:02:25,720 --> 00:02:30,560 Given this information, one of the outcomes becomes 40 00:02:30,560 --> 00:02:35,050 impossible, namely the outcome girl, girl. 41 00:02:35,050 --> 00:02:39,150 And we're restricted to a smaller universe with only 42 00:02:39,150 --> 00:02:41,620 three possible outcomes. 43 00:02:41,620 --> 00:02:45,470 Our new universe is this green universe, which includes all 44 00:02:45,470 --> 00:02:48,100 outcomes that have at least one boy, so that 45 00:02:48,100 --> 00:02:50,930 we can get a king. 46 00:02:50,930 --> 00:02:54,530 We should, therefore, use the conditional probabilities that 47 00:02:54,530 --> 00:02:57,590 are appropriate to this new universe. 48 00:02:57,590 --> 00:03:01,920 Since these three outcomes inside the green set have 49 00:03:01,920 --> 00:03:05,170 equal unconditional probabilities, they should 50 00:03:05,170 --> 00:03:08,460 also have equal conditional probabilities. 51 00:03:08,460 --> 00:03:12,230 So each one of these three outcomes should have a 52 00:03:12,230 --> 00:03:14,460 conditional probability equal to 1/3. 53 00:03:17,079 --> 00:03:21,340 In two of these outcomes the sibling is a girl and 54 00:03:21,340 --> 00:03:25,275 therefore, the conditional probability given that there 55 00:03:25,275 --> 00:03:28,930 is a king and therefore given that there is a boy, the 56 00:03:28,930 --> 00:03:34,570 conditional probability is going to be 2/3. 57 00:03:34,570 --> 00:03:37,680 So this is actually the official answer to this 58 00:03:37,680 --> 00:03:40,710 problem, and this answer is incorrect. 59 00:03:43,610 --> 00:03:45,960 Are we satisfied with this answer? 60 00:03:45,960 --> 00:03:48,760 Maybe yes, maybe no. 61 00:03:48,760 --> 00:03:52,579 Actually, some more assumptions are needed in 62 00:03:52,579 --> 00:03:56,290 order to say that 2/3 is the correct answer. 63 00:03:56,290 --> 00:04:00,160 Let me state what these assumptions are. 64 00:04:00,160 --> 00:04:04,960 We assume that the royal family decided to have exactly 65 00:04:04,960 --> 00:04:06,830 two children. 66 00:04:06,830 --> 00:04:11,900 So the number two that we have here is not random. 67 00:04:11,900 --> 00:04:14,760 It was something that was predetermined. 68 00:04:14,760 --> 00:04:20,140 Once they decided to have the two children, they had them. 69 00:04:20,140 --> 00:04:23,720 At least one turned out to be a boy and that 70 00:04:23,720 --> 00:04:26,460 boy became a king. 71 00:04:26,460 --> 00:04:30,780 Under this situation, indeed, the probability that the 72 00:04:30,780 --> 00:04:36,540 sibling of the king is female is 2/3. 73 00:04:36,540 --> 00:04:39,760 But these assumptions that I just stated are not the only 74 00:04:39,760 --> 00:04:41,840 possible ones. 75 00:04:41,840 --> 00:04:46,040 Let's consider some alternative assumptions. 76 00:04:46,040 --> 00:04:48,190 For example, suppose that the royal 77 00:04:48,190 --> 00:04:51,110 family operated as follows. 78 00:04:51,110 --> 00:04:56,930 They decided to have children until they get one boy. 79 00:05:00,820 --> 00:05:02,610 What does this tell us? 80 00:05:02,610 --> 00:05:06,080 Well, since they had two children, this tells us 81 00:05:06,080 --> 00:05:06,660 something-- 82 00:05:06,660 --> 00:05:10,300 that the first child was a girl. 83 00:05:13,170 --> 00:05:16,110 So in this case, the probability that the king's 84 00:05:16,110 --> 00:05:20,340 sibling is a girl is equal to 1. 85 00:05:20,340 --> 00:05:23,350 The only reason why they had two children was because the 86 00:05:23,350 --> 00:05:27,570 first was a girl and then the second was a boy. 87 00:05:27,570 --> 00:05:30,890 Suppose that the royal family made some different choices. 88 00:05:30,890 --> 00:05:36,670 They decided to have children until they would get two boys, 89 00:05:36,670 --> 00:05:41,190 just to be sure that the line of succession was secured. 90 00:05:41,190 --> 00:05:45,520 In this case, if we are told that there are only two 91 00:05:45,520 --> 00:05:51,460 children, this means that there were exactly two boys, 92 00:05:51,460 --> 00:05:54,220 because if one of the two children was a girl, the royal 93 00:05:54,220 --> 00:05:56,170 family would have continued. 94 00:05:56,170 --> 00:05:59,720 So in this particular case, the probability that the 95 00:05:59,720 --> 00:06:02,970 sibling is a girl is equal to zero. 96 00:06:02,970 --> 00:06:05,970 And you can think of other scenarios, as well, that might 97 00:06:05,970 --> 00:06:09,610 give you different answers. 98 00:06:09,610 --> 00:06:15,330 So 2/3 is the official answer, as long as we make the precise 99 00:06:15,330 --> 00:06:19,120 assumptions that the number of children, the number two, was 100 00:06:19,120 --> 00:06:24,280 predetermined before anything else happened. 101 00:06:24,280 --> 00:06:27,070 The general moral from this story is that when we deal 102 00:06:27,070 --> 00:06:30,440 with situations that are described in words somewhat 103 00:06:30,440 --> 00:06:34,010 vaguely, we must be very careful to state whatever 104 00:06:34,010 --> 00:06:36,090 assumptions are being made. 105 00:06:36,090 --> 00:06:39,909 And that needs to be done before we are able to fix a 106 00:06:39,909 --> 00:06:43,050 particular probabilistic model. 107 00:06:43,050 --> 00:06:47,710 This process of modeling will always be something of an art 108 00:06:47,710 --> 00:06:50,590 in which judgment calls will have to be made.