A Strange New Probability Puzzle

John Allen Paulos in his Who's Counting column at ABC News:

Nm_Its_A_Boy_100929_mn 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?

You might jump to the conclusion that the answer is […] 1/2, reasoning that the sex of one child has no bearing on the sex of the other. This conclusion is incorrect, however, since you don't know whether the boy you know about is the older or the younger child.

So let's look at the possibilities. Listing two children in the order in which they might be born, we note four possibilities: B-B, B-G, G-B, G-G. Since you know that at least one of the two children is a boy, the G-G possibility is eliminated. Of the three remaining equally likely possibilities (B-B, B-G, and G-B) only one results in two boys. Therefore the correct conclusion in this case is that the probability the woman has two boys is 1/3, not 1/2…

Now for the odd result. Suppose that when children are born in a certain large city, the season of their birth, whether spring, summer, fall, or winter, is noted prominently on their birth certificate. The question is: Assume you know that a lifetime resident of the city has two children, at least one of whom is a boy born in summer. Given this knowledge, what is the probability she has two boys?

More here.