The two-children problem

Question

A family has two children. Assume each child is independently a boy or girl with probability 	frac12. (a) You are told at least one is a boy. What is the probability that both are boys? (b) You are told the elder child is a boy. Now what is the probability that both are boys? Explain why the two answers differ.

Accepted Answer

List the four equally likely outcomes by birth order (elder, younger): BB,\ BG,\ GB,\ GG. (a) "At least one is a boy." This rules out only GG. The remaining equally likely cases are \{BB,\ BG,\ GB\}. Of these three, only BB has two boys. So P(	ext{both boys} \mid 	ext{at least one boy}) = \frac{1}{3}. (b) "The elder is a boy." This rules out GB and GG (any outcome whose elder child is a girl). The remaining equally likely cases are \{BB,\ BG\}. Of these two, only BB has two boys. So P(	ext{both boys} \mid 	ext{elder is a boy}) = \frac{1}{2}. Why they differ. "At least one is a boy" is a s…

The two-children problem

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