A Probability Puzzle. Radford Neal, 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.