Puzzle

daniele posed this problem last night:

A plane is booked full. Each passenger has a boarding pass with a specific assigned seat. Boarding is about to begin.

Normally each passenger would board the plane, go to his or her assigned seat, and sit down. But today, the first passenger to board is a little crazy, and takes a seat completely at random.

Here is how the remaining passengers will behave: they'll board one at a time. Each one will go first to his or her assigned seat and sit down if possible. But if someone's already sitting there, that passenger will pick an unoccupied seat at random (somebody else's seat) and sit there.

As it happens, the last passenger to board the plane is a VIP who must have his assigned seat, seat 1-A, or heads will roll. What is the probability that when the VIP boards, seat 1-A will be unoccupied?

Neat.