The Lost Boarding Pass Problem

The puzzle:

One hundred people line up to board a plane with 100 seats. The first person in line is has lost his boarding pass, so he randomly chooses a seat. After that, each person entering the plane either sits in their assigned seat, if it is available, or, if not, chooses an unoccupied seat randomly.

When the 100th passenger finally enters the plane, what is the probability that she finds her assigned seat unoccupied?

Solution:

The probability, somewhat surprisingly, is 1/2!