Opening probability question in almost any quant screen.
Two children, one is a boy
The question
A family has two children. You are told at least one of them is a boy. What is the probability both are boys?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
1/3.
Write the equally likely orderings of two births: BB, BG, GB, GG.
The information rules out GG, leaving three cases, each still equally likely.
Only one of those three is BB, so the probability is 1/3.
The classic error is answering 1/2 by silently conditioning on a specific child rather than on the statement 'at least one'. If instead you were told 'the elder is a boy', the answer really would be 1/2, because that names a child.
What it is testing
Whether you condition on the information given rather than the information you expected.
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