Radford shared with us this probability puzzle of his from 1999:

A couple you’ve just met invite you over to dinner, saying “come by around 5pm, and we can talk for a while before our three kids come home from school at 6pm”.

You arrive at the appointed time, and are invited into the house. Walking down the hall, your host points to three closed doors and says, “those are the kids’ bedrooms”. You stumble a bit when passing one of these doors, and accidently push the door open. There you see a dresser with a jewelry box, and a bed on which a dress has been laid out. “Ah”, you think to yourself, “I see that at least one of their three kids is a girl”.

Your hosts sit you down in the kitchen, and leave you there while they go off to get goodies from the stores in the basement. While they’re away, you notice a letter from the principal of the local school tacked up on the refrigerator. “Dear Parent”, it begins, “Each year at this time, I write to all parents, such as yourself, who have a boy or boys in the school, asking you to volunteer your time to help the boys’ hockey team…” “Umm”, you think, “I see that they have at least one boy as well”.

That, of course, leaves only two possibilities: Either they have two boys and one girl, or two girls and one boy. What are the probabilities of these two possibilities?

NOTE: This isn’t a trick puzzle. You should assume all things that it seems you’re meant to assume, and not assume things that you aren’t told to assume. If things can easily be imagined in either of two ways, you should assume that they are equally likely. For example, you may be able to imagine a reason that a family with two boys and a girl would be more likely to have invited you to dinner than one with two girls and a boy. If so, this would affect the probabilities of the two possibilities. But if your imagination is that good, you can probably imagine the opposite as well. You should assume that any such extra information not mentioned in the story is not available.

As a commenter pointed out, there’s something weird about how the puzzle is written, not just the charmingly retro sex roles but also various irrelevant details such as the time of the dinner. (Although I can see why Radford wrote it that way, as it was a way to reveal the number of kids in a natural context.)

The solution at first seems pretty obvious: As Radford says, the two possibilities are:

(a) 2 boys and 1 girl, or

(b) 1 boy and 2 girls.

If it’s possibility (a), the probability of the random bedroom being a girl’s is 1/3, and the probability of getting that note (“I write to all parents . . . who have a boy or boys at the school”) is 1, so the probability of the data is 1/3.

If it’s possibility (b), the probability of the random bedroom being a girl’s is 2/3, and the probability of getting the school note is still 1, so the probability of the data is 2/3.

The likelihood ratio is thus 2:1 in favor of possibility (b).

Case closed . . . but is it?

Two complications arise. First, as commenter J. Cross pointed out, if the kids go to multiple schools, it’s not clear what is the probability of getting that note, but a first guess would be that the probability of you seeing such a note on the fridge is proportional to the number of boys in the family. Actually, even if there’s only one school the kids go to, it might be more likely to see the note prominently on the fridge if there are 2 boys: presumably, the probability that at least one boy is interested in hockey is an higher if there are two boys than if there’s only one.

The other complication is the prior odds. Pr(boy birth) is about .512, so the prior odds, which are .512/.488 in favor of the 2 boys and 1 girl, rather than 2 girls and 1 boy.

This is just to demonstrate that, as Feynman could’ve said in one of his mellower moments, God is in every leaf of every tree: Just about every problem is worth looking at carefully. It’s the fractal nature of reality.