today's fun with probability

[eub]

1) You meet a man at the bowling alley [this is how it was posed, honest]. He has two children, bowling on lane 17. You see one girl there; the other child doesn't happen to be visible.

2) Man, bowling, two children. You ask if he has any daughters. He says yes.

In each case (any difference?), what's the probability that he has two daughters?

Assume male and female are mutually exclusive and 50-50 and independent between children. Make any further assumptions you need.

[edit: spoilers in comments, natch]

Comments

[eub.livejournal.com]

Hey hey hey, slummin' on LJ. How's tricks?

That reply is correct. Once you look at it in terms of four possibilities (which is where I was leading with the (2)-(3)-(4) sequence), you're going to get the right answer.

It's tempting, though, to think that one is a girl, the other is 50-50, so it's 50-50 that both are. The question is how to reeducate this intuition.

The reason that these two situation are different is that in the first, we can fix the gender of a child that is chosen at random from the two, whereas the second situation does nothing to specify the child.

Does this do it? Not for me; my tempting statement was in terms of "one" and "the other" anyway.

The key has to be that "the one is a girl" and "the other is a girl" are not independent, given that at least one is. If one is not a girl, then the other must be. This dependence means you can't break down the probability in the tempting manner. But why, the devil says, can't we talk "given that either the elder is, or the younger is", in either of which cases...? The math doesn't work, but I can't think of a nice intuitive appeal.