The Two-Children Problem

A family has two children. You're told at least one is a girl. What's the probability both are girls?

Most people answer $1/2$, reasoning that the other child is a coin flip. The textbook answer is $1/3$. Both can be defended — the puzzle turns on a detail interviewers often leave out on purpose: how did you learn "at least one is a girl"?

The standard reading

List the four equally likely ways two children can be born, older child first: BB, BG, GB, GG. "At least one is a girl" rules out BB, leaving three equally likely outcomes: BG, GB, GG. Only one of those three has two girls, so

$$P(\text{both girls} \mid \text{at least one girl}) = \frac{1}{3}.$$

The trap is treating "the other child" as an independent coin flip, which silently assumes you know which child is the girl. You don't — BG and GB are both still live, and that third outcome is what drags the answer below $1/2$.

Why "1/2" is also defensible

Change how the fact arrived: you meet one specific child and she happens to be a girl. Now you've pinned down one child's sex directly, and the other really is an independent coin flip — $P(\text{both girls}) = 1/2$. The sample space didn't change; the selection procedure did. Ruling out BB is different from observing one particular child and finding her female, because the second procedure never gives GB and BG a chance to both survive as distinct cases — meeting "the older child" as a girl kills BB and GB, leaving BG and GG, a fair coin on the younger one.

This is the generalizable lesson: conditioning on "does the family satisfy property X" and conditioning on "I sampled one member and it had property X" are different experiments, even when the English sentence describing them looks identical. The interview move is to always ask how the information was generated before touching a formula — the notorious "Tuesday boy" variant (one child is a boy born on a Tuesday) pushes the answer to $13/27$ purely by changing how finely the conditioning event is sliced, which only matters because it's answering the "which family qualifies" version, not the "I met a child" version.