[This problem 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.]
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 is 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 is the probability that both are girls?