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