Source

I got this problem from Rustan Leino, who says it was inspired by some basic probability questions mentioned in a lecture by Eric Hehner (and that can be formalized and solved by calculation using this Probability Perspective), and some subsequent discussions with him, Itay Neeman, Jim Woodcook, Ana Cavalcanti, and Leo Freitas.

I solved it and wrote up my solution.

Problem

The house next door has some new neighbors. They have two children, but you don't know what mix of boys and girls they are. One day, your wife tells you, "At least one of the children is a girl." What's the probability that both are girls?

Your wife then tells you, "The way I found out that at least one of the children is a girl is that I saw one of the children playing outside, and it was a girl." Now, what's the probability that both are girls?

Solution
Reveal

The answer to the first question is 1/3, and the answer to the second is 1/2, by the following reasoning.

For the first question, let $A$ be the event that both are girls, and $B$ be the event that at least one is a girl. We're interested in $P(A \mid B).$ This is $P(A \cap B)/P(B) = (1/4)/(3/4) = 1/3.$

For the second question, we know a little more information, so the answer is different. An arbitrarily chosen child is a girl, so they're both girls if the other one is a girl. The probability of this is $1/2.$