You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?
If you list all possibilities for the sexes of two children, $BB, BG, GB, GG$, you see that $2$ of the $3$ pairs that have B (for boy) in them also have a girl, so the answer one could argue is $2/3$.
On the other hand, one could argue that the answer is $1/2$, since the probability that any one child is a girl is $1/2$, and intuitively (?) should be independent of the gender of its siblings.
Some background to possibly justify posting it here: the question was asked at an interview for an actuarial/insurance type position, and the interviewer said the answer was $2/3$, whereas my friend who was being interviewed (and has a masters in math) thought the answer was $1/2$, even after the interviewer explained his logic. My friend felt that the interviewer wasn't taking into account the fact that it is not equally likely that a boy will open the door in the $BB$ versus the $BG$ combination, and one has to take into account that fact. I have no idea which is the correct answer, both sound somewhat convincing to me (I have a Ph.D. in math, but I won't mention from where in an effort to avoid embarrassing my degree granting institution!). Anyways, any help would be appreciated and I apologize if this is too simple a question for this forum.