Thanks, I’ll answer this later, but if possible, I’d like to know where in my post you think I’ve gone wrong.
-FrL-
Actually, the probability in this case is 2p/3 + (1-p)/2 where p is the probability that the informant actually means “They have two children, at least one of which is a girl.”
THe thing that amazes me is that nobody ever seems to bother to figure this out for themselves empirically. Here’s how you can do it, even without any computer skills:
- Take two coins and toss them. “Heads” = girl, “Tails”=boy.
- Write down your results.
- Repeat this, say, 30 times (that’ll be good enough to clearly see this effect, I think).
- Now, cross out all the TT. That represents the information that we know at least one child is a girl.
- Look at the remaining options. They’re about evenly divided between TH, HT, and HH. That is, there will be about 10 each of TH, HT, and HH.
- So in 10 cases, the girl’s sibling is another girl. In the other 20, it’s a boy. So, given the knowledge that one sibling is a girl, there’s a 2/3 chance that the other is a boy.
Please, do the experiment. It may help you understand why this works, and even if it doesn’t, it will show you that it works…possibly better than any argumentation would.
(The reason that it works, in short, is that TH and HT are both cases of “her sibling is a boy”, even though they’re, in some sense, distinct cases. It’s semantics as much as anything else.)
And JWK, I disagree.
If the problem were that people thought the problem was stating “exactly one child is a girl,” as opposed to “at least one,” they’d claim the probability that her sibling was a boy was 100%. I’ve never seen someone do that, so I’m quite sure that that’s not the misunderstanding.
“Exactly one child is a girl” hasn’t been brought into the discussion at all. (That is another possible reading, but, as you say, it makes the problem too absurdly trivial.) The question is the difference between, “This one is a girl,” and “At least one is a girl.”