At least one girl, is the other a boy?

Question

A family has two children, each independently a boy or a girl with probability $\tfrac{1}{2}$ . You are told that at least one of them is a girl.

What is the probability that the family also has a boy, that is, that the two children are one of each?

Accepted Answer

Write the two children in birth order. The four equally likely families are BB, BG, GB, GG. "At least one is a girl" removes BB, leaving BG,\quad GB,\quad GG,\qquad 	ext{each with probability } 	frac13. The family "also has a boy" in exactly the mixed cases BG and GB, so P(	ext{has a boy} \mid 	ext{at least one girl}) = \frac{2}{3}. This is the flip side of the classic result. Given at least one girl, both-girls has probability 	frac{1}{3}, and the remaining 	frac{2}{3} is the chance of one of each. Many people guess 	frac{1}{2} here, again by treating "the other child" as a fresh coin…

At least one girl, is the other a boy?

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