Crazy Passenger Puzzle

Here's an interesting puzzle which I just encountered:

The Crazy Passenger

There are 100 airline passengers lining up to board the plane. They each hold a ticket to one of the 100 seats on that flight.

Unfortunately, the first person in line is crazy, and will ignore the seat number on his ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random.

What is the probability that the last (100th) person to board the plane will sit in his proper seat?"

I found the puzzle (along with others) here. Unfortunately, I saw the numerical answer before giving the question much thought, but in my defense I did work out the solution almost immediately after.