[I got this problem from Rajeev Joshi, who got it from Alex Groce.]
An airplane has 50 seats, and its 50 passengers have their own assigned seats. The first person to enter the plane ignores his seat assignment and instead picks a seat on random. Each subsequent person to enter the plane takes her assigned seat, if available, and otherwise chooses a seat on random. What is the probability that the last passenger gets to sit in her assigned seat?
Follow-up question: What is the probability that the penultimate passenger gets to sit in her assigned seat?