Variations on the two-child problem

TL;DR

This paper explores variations of the two-child probability puzzle, analyzing how different assumptions about information acquisition affect the likelihood of Mr. Smith having two boys, and discusses why the Adam puzzle confuses many.

Contribution

It provides pictorial explanations and highlights how assumptions influence the solutions to the two-child and Adam puzzles, clarifying common misunderstandings.

Findings

Answers depend on assumptions about information acquisition.

Different assumptions lead to different probability outcomes.

The Adam puzzle's confusion stems from subtle informational nuances.

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.