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Let us now conclude with a fun
problem, which is also a
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little bit of a puzzle.
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We are told that the
king comes from a
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family of two children.
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What is the probability that
his sibling is female?
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Well, the problem is too loosely
stated, so we need to
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start by making some
assumptions.
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First, let's assume that we're
dealing with an anachronistic
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kingdom where boys
have precedence.
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In other words, if the royal
family has two children, one
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of which is a boy and one is a
girl, it is always the boy who
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becomes king, even if the
girl was born first.
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Let us also assume that when
a child is born, it has 50%
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probability of being
a boy and 50%
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probability of being a girl.
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And in addition, let's assume
that different children are
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independent as far as their
gender is concerned.
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Given these assumptions,
perhaps we
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can argue as follows.
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The king's sibling is
a child which is
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independent from the king.
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Its gender is independent from
the king's gender, so it's
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going to be a girl with
probability 1/2.
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And so this is one possible
answer to this problem.
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Is this a correct answer?
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Well, let's see.
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We have to make a more precise
model, so let's
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go ahead with it.
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We have two children, so there
are four possible outcomes--
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boy, boy; boy, girl; girl,
boy; and girl, girl.
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Each one of these outcomes has
probability 1/4 according to
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our assumptions.
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For example, the probability of
a boy followed by a boy is
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1/2 times 1/2, where we're
also using independence.
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So each one of these four
outcomes has the same
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probability, 1/4.
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Now, we know that there is a
king, so there must be at
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least one boy.
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Given this information, one
of the outcomes becomes
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impossible, namely the
outcome girl, girl.
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And we're restricted to a
smaller universe with only
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three possible outcomes.
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Our new universe is this green
universe, which includes all
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outcomes that have at least
one boy, so that
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we can get a king.
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We should, therefore, use the
conditional probabilities that
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are appropriate to this
new universe.
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Since these three outcomes
inside the green set have
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00:03:01,920 --> 00:03:05,170
equal unconditional
probabilities, they should
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also have equal conditional
probabilities.
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So each one of these three
outcomes should have a
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conditional probability
equal to 1/3.
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In two of these outcomes the
sibling is a girl and
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therefore, the conditional
probability given that there
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is a king and therefore given
that there is a boy, the
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conditional probability
is going to be 2/3.
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So this is actually the official
answer to this
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problem, and this answer
is incorrect.
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Are we satisfied with
this answer?
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Maybe yes, maybe no.
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Actually, some more assumptions
are needed in
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order to say that 2/3 is
the correct answer.
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Let me state what these
assumptions are.
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We assume that the royal family
decided to have exactly
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two children.
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So the number two that we
have here is not random.
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It was something that
was predetermined.
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Once they decided to have the
two children, they had them.
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At least one turned out
to be a boy and that
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boy became a king.
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Under this situation, indeed,
the probability that the
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sibling of the king
is female is 2/3.
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But these assumptions that I
just stated are not the only
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possible ones.
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Let's consider some alternative
assumptions.
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For example, suppose
that the royal
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family operated as follows.
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They decided to have children
until they get one boy.
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What does this tell us?
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Well, since they had two
children, this tells us
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something--
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that the first child
was a girl.
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So in this case, the probability
that the king's
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sibling is a girl
is equal to 1.
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The only reason why they had two
children was because the
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first was a girl and then
the second was a boy.
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Suppose that the royal family
made some different choices.
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They decided to have children
until they would get two boys,
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just to be sure that the line
of succession was secured.
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In this case, if we are told
that there are only two
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children, this means that there
were exactly two boys,
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because if one of the two
children was a girl, the royal
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family would have continued.
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So in this particular case,
the probability that the
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sibling is a girl is
equal to zero.
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And you can think of other
scenarios, as well, that might
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give you different answers.
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So 2/3 is the official answer,
as long as we make the precise
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assumptions that the number of
children, the number two, was
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predetermined before anything
else happened.
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The general moral from this
story is that when we deal
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with situations that are
described in words somewhat
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vaguely, we must be very careful
to state whatever
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assumptions are being made.
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And that needs to be done before
we are able to fix a
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particular probabilistic
model.
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This process of modeling will
always be something of an art
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in which judgment calls
will have to be made.