This puzzle recently (10/02/04) appeared in the "Puzzlah" section of Car Talk, on NPR radio. I happen to think it's one of the most ingenious problems so far presented on the program (I am a long-time fan); my reason for talking about it here is that I want to discuss my own particular slant on the problem.

You are one of a hundred passengers waiting in line to board a commercial aircraft that has one hundred seats; you are the last in line. Each passenger has a boarding pass that indicates the seat assignment. However, as luck would have it (it wouldn't have it any other way), the first passenger loses his boarding pass after starting down the jetway, and decides (instead of trying to recover the boarding pass) to choose a seat at random, once he gets on the plane. Each succeeding passenger then either finds her/his own assigned seat; or, if it is erroneously occupied, chooses an empty seat at random and claims it for the duration. The problem is to determine the probability that, once you finally get on the airplane, you will be able to sit in your previously-assigned seat.