Three children, at least one girl

Question

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

What is the probability that all three children are girls?

Accepted Answer

With three children in birth order there are 2^3 = 8 equally likely combinations, from BBB to GGG. The statement "at least one is a girl" removes exactly one combination, the all-boys family BBB. That leaves 7 equally likely families, and only one of them, GGG, is all girls. P(	ext{all girls} \mid 	ext{at least one girl}) = \frac{1}{7}. A quick sanity check by the complement: without any information, all girls has probability 	frac{1}{8} and at least one girl has probability 	frac{7}{8}, so the conditional probability is 	frac{1/8}{7/8}=	frac17. Takeaway. With three children the same "a…

Three children, at least one girl

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