Boy or Girl?
Based on a conversation with somebody you meet for the first time you discover that this person has at least one son.
You subsequently discover that this person has two children.
What is the probability that the other child is a boy?
Assuming that any birth is equally likely to be a boy or girl (this is
not strictly correct but it does not really matter here), most people
answer 1/2.
In fact, the correct answer is 1/3.
The key thing to note is that, in any family of two children there are
four possible ways in which the children will have been born:
|
First Child
|
Second child
|
|
Boy
|
Boy
|
|
Boy
|
Girl
|
|
Girl
|
Boy
|
|
Girl
|
Girl
|
Since we are assuming any
birth is equally likely to be a boy or girl, it follows that each of
the four outcomes is, in the general case, equally likely. However, in
this particular case we know that the fourth outcome (Girl, Girl) is
impossible because at least one child is a boy. Hence the other
three outcome are equally likely and so each has probability 1/3.
Only one of these (Boy, Boy) leads to the 'other' child being a
boy. And so the probability the other child is a boy is 1/3, while the
probability the other child is a girl is 2/3 (since the other two
possibilities (Biy, Girl), (Girl, Boy) both lead to such an outcome).
What makes the example intuitively hard to understand is that the correct answer is 1/2 if either:
1) We had learnt from our initial conversation that the child was born first.
2) We had learnt from our initial conversation that the child was not born first.
Since one of these underlying two statements must be true, does this
mean we have a paradox? No. There is a diifference between knowing that
either 1 or 2 is true (which is the original situation we were in) and
knowing which one of 1 or 2 is true. Imagine somebody selecting
two balls (with replacement) from an urn containing an equal number
of white and black balls. If you see
the colour of the first ball drawn then, irrespective of what the
colour was, you know that the probability the second ball being a
black is 1/2. But if all you know is the colour of at least one of the
two balls drawn then, irrespective of whether that colour is white or
black, only one out of three equally likely possibilities for how they
were drawn leads to the other ball being the same colour. Specifically,
if you know one ball was black then the possibilities are: (B, B), (B,
W), (W, B); and if you know one ball is white then the possibilities
are: (W, W), (B, W), (W, B).
A detailed analysis of this problem can be found here.