Mixed-up Airplane Seats

Source

I got this problem from Rustan Leino, who got it from Rajeev Joshi, who Rustan thinks said he heard it from Jay Misra.

I solved it and wrote up my solution.

Problem    

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?

Solution     Reveal