On a sold out flight, 100 people line up to board the plane. The first passenger in the line has lost his boarding pass, but was allowed in, regardless. He takes a random seat. Each subsequent passenger takes his or her assigned seat if available, or a random unoccupied seat, otherwise. What is the probability that the last passenger to board the plane finds his seat unoccupied?
Amazingly enough it is 50%. The last passenger sees one empty seat and it can only be his seat or the seat that was originally allocated to the first passenger. If it belonged to one of the other 98 passengers it would have been taken.