Say that a woman and a man (who are unrelated) each have two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy.
It follows that the chances that the man has two boys are about 1/2. What are the chances that the woman has two boys?
Note:
Each child is equally likely to be a boy or a girl, and the gender of one sibling does not influence the gender of the other.
1,000It depends on how we found out that the woman has one boy. If we shouted "hey, everybody with two children and at least one boy, come here" and then grabbed one of the women, the probability is 1/3. If we shouted "hey, everybody with two children regardless of their gender, come here" and then grabbed someone and said "tell me the gender of one of your children" and the woman said "a boy", than the probability's 1/2 (the same scenario as in the @LoganTurner's comment). But if we told her "tell me the gender of one of your children, but preferably answer 'a boy' if you can do so without lying", the probability is back to 1/3.
At the playground, you meet a woman who says she has two kids.
BB BG GB GG — 1/4
A boy yells "Mom!" and the woman waves back. That eliminates GG.
BB BG GB — 1/3
You consider asking if he is the oldest. Either he is…
BB BG — 1/2
…or he isn't.
BB GB — 1/2
Either way, the probability becomes 1/2, so you don't even need to ask. But then where did the extra information come from?
(Answer: When he yells "Mom", that rules out not only GG but also half of the BG and GB scenarios where the girl yells first.)