The Airplane Seat problem

I thought I’d drop a little problem to think about for a while. If you’ve seen it before, please don’t ruin it for those who haven’t by posting a solution in the comments.

Let’s imagine an airplane with 100 seats. Instead of the cattle-call boarding procedures we know, passengers board the plane one at a time. On this flight, the first passenger has lost his ticket, so he just picks a seat at random to sit in. When the second passenger boards, if her own seat is available she sits in it. If not, she picks a random empty seat. This repeats until the plane fills. What is the probability that the 100th passenger gets to sit in his own seat?

Assume that all the random choices are made with equal probability of picking any currently-empty seat.