Difficulty: Easy · Puzzle 3 of 6 · Topic: Probability Puzzles
A family has two children. Given that at least one child is a boy, what is P(both are boys)? Assume boy/girl equally likely and independent.
Write conditioned sample space explicitly.
1) Unconditioned equiprobable outcomes: BB, BG, GB, GG.
2) Condition 'at least one boy' removes GG.
3) Remaining equally likely outcomes: BB, BG, GB.
4) Only BB satisfies 'both boys'.
5) Therefore probability is 1/3.Answer: P(both boys | at least one boy) = 1/3.
Checked by running a model of the rule in a Python model. A computational check: enumerating the four equally likely families gives 1/3 for 'at least one boy' and 1/2 for 'older child is a boy'.
Answering 1/2 by saying 'the other child is a coin flip'. The condition 'at least one boy' keeps BB, BG and GB, so BB is 1 case out of 3, not 1 of 2.
How does answer change if condition is 'older child is a boy'? Model answer: it becomes 1/2. The condition keeps only BB and BG (older is B), so BB is 1 of 2. Naming which child you know about breaks the symmetry between BG and GB.
Let me list the equally likely families: boy-boy, boy-girl, girl-boy and girl-girl. Knowing that at least one is a boy removes girl-girl, which leaves three equally likely cases. Only one of those three is boy-boy, so the answer is one third. The trap is saying one half because the other child looks like a coin flip, but that ignores that boy-girl can happen in two orders.