  Mr. Smith has two children. Given that at least one of them is a boy, how
likely is it that Mr. Smith has two boys? It's a very standard puzzle in
elementary books on probability theory. Whoever asks you this question hopes
that you will answer "$\frac{1}{2}$", in which case they can say triumphantly
"Oh no, the answer is $\frac{1}{3}$". This is called the two-child puzzle. Some
authors have discussed a striking variation, which we'll call the Adam puzzle.
Again, Mr. Smith has two children. Given that one of them is a boy named Adam,
how likely is it that Mr. Smith has two boys? Astonishingly, now the answer is
$\frac{1}{2}$, at least approximately. (The exact answer depends a bit on
precise assumptions.) We give pictorial explanations of both puzzles. We then
point out that the answers usually given rely on a tacit assumption about how
the information that one of Mr. Smith's two children is a boy, or one of them
is a boy named Adam, is obtained. We give examples showing that the answers may
be different with different assumptions. We conclude with a discussion of why
the Adam puzzle is so confusing to most people.
