Answer to three-week-old Friday Puzzle

A few weeks back I posted a Friday puzzle, and forgot to answer it. This was the question:

You meet a woman and ask how many children she has; she replies “two.” You ask if she has any boys, and she replies “yes”. What is the probability of the other child also being a boy?

As was figured out in the comments, the answer is 1/3. This is because there are four combinations of children: BB, BG, GB, GG. You know there is at least one boy, so you can remove the GG option, leaving BB, BG and GB. Only in BB is the other child also male, hence 1/3.

If you rephrase the question ever so slightly, the answer changes:

You meet a woman and her son, and ask how many children she has; she replies “two.” What is the probability of the other child also being a boy?

In this case, the answer is 1/2. In the first question ‘the other child’ referred to either of two children, resulting in three possibilities. This time ‘the other child’ refers to only one child – the one that isn’t the boy – which can obviously be only a boy or a girl.

I figured out the first question – it’s similar to Monty Hall – but the second confused me for ages, even after I’d seen the correct answer 🙂 This from Simon in the comments makes it clearer, I think:

The first question is similar to this. I put before you two boxes, each containing a child. I tell you one is a boy. In working out the probability the trap people will fall into is this: They will assign one of the boxes the boy and then consider the second box. They will fail to consider that the first box may have contained a girl and thus failed to acknowledge one of the three cases (if the first box contains a girl the second had to contain the boy).

The second question is more like me placing a single box containing a child in front of you and placing a boy next to it. Here there are only two cases and the boy is totally irrelevent.

Further discussion on two Skeptics’ Guide forum pages, plus this, which discusses the ambiguity that arises if the question isn’t formulated just so.