Variations on the two-child problem

Abstract

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.