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.