Who Took My Seat?!?!

For this problem, consider an airplane with one hundered seats. Every seat has a passenger that is assigned to sit in it and the passengers file in one at a time to take their seats. The first person gets on the plane and realizes he can't read what seat number is his. So, he picks a random seat out of the 100 seats on the plane. Now, each passenger after this boards the plane and, if their seat is already taken, will pick a random seat out of what ever is left. If their seat is empty, then of course they sit in their own seat. Here is the question: what is the probability that the last person to board the plane will sit in their own seat?

Hint: There is a bit of useless information in the problem....