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Probability: Fallacies, Myths and Puzzles: Main Page

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.



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Norman Fenton


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