At least one girl among three, are all three girls?

Question

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

Given that, what is the probability that all three are girls?

Accepted Answer

With three children there are 2^3 = 8 equally likely birth orders, from BBB to GGG. The clue "at least one girl" removes exactly one case, the all-boys outcome BBB, leaving: 8 - 1 = 7 	ext{ equally likely outcomes}. Only one of those seven is all girls, namely GGG: P(	ext{all girls} \mid 	ext{at least one girl}) = \frac{1}{7}. You can see the pattern: the clue eliminates just the single "none" outcome, so the denominator is 2^n - 1. For two children it was 1/3; for three it is 1/7. Takeaway. "At least one" conditioning simply deletes the all-opposite case, so the answer is 1 divided by (to…

At least one girl among three, are all three girls?

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