Puzzles and Babies

I have seen this puzzle a few times recently, but don’t think I have posted it. Here you go, courtesy of John Allen Paulos:

Assume you know that a woman has two children, the older of whom is a boy. Given this knowledge, what is the probability that she has two boys? Let’s count the possibilities. The only two are an older boy and younger girl (B-G) or an older boy and a younger boy (B-B). Since these are equally likely, the probability that the woman has two boys is 1/2.

No problem there, but now consider this second scenario. Assume that you know that a woman has two children, at least one of whom is a boy. You know nothing about this boy except his sex. Given this knowledge, what is the probability that she has two boys?

Answers and related discussion here.

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