On 6/12/06, Jason Holt <jason@lunkwill.org> wrote:
On Mon, 12 Jun 2006, James Propp wrote:
"A family has two children, at least one of whom is a boy; what is the probability that both are boys?"
and
"A family has two children, at least one of whom is a boy named Bartholomew; what is the probability that both are boys?"
have different answers.
That doesn't sound right to me.
It didn't sound right to me either. I get easily confused, though. One thing that helps me with these puzzles is to think of it as someone asking the question, like this: A: I have two children. B: Do you have at least one boy? A: Yes, I do. Then P(two boys) is 1/3 (under the usual assumptions). In the alternative, A: I have two children. B: Do you have a boy named Bartholomew? A: Yes, I do. So what's the difference? I'll suppose here that 1/100 of boys are named Bartholomew. I'll use b for a boy not named Bartholomew and B for a boy named Bartholomew. Then, when A says "two children", it could be GG Gb GB bG BG bb bB Bb and maybe BB, too, I'm not sure if we'll allow parents to do that or not. Then, when A says "yes", we have (under the assumption for convenience that 1% of boys are named Bartholomew, and so 1/200 kids are boys named Bartholomew): GB: 1/400 BG: 1/400 bB: 99% of 1/400 Bb: 99% of 1/400 and maybe BB, too, 1% of 1/400 So you can see that as long as Bartholomew is a reasonably uncommon name, the chance of two boys and the chance of two girls is almost the same. Not quite 1/2, but certainly a lot closer to 1/2 than 1/3. Thanks, Jim, for this comment! I'd never seen this variation before, and indeed the answer surprised me. --Joshua Zucker