Passengers and Random Seats in Airplane

Problem :

100 passengers are boarding an airplane with 100 seats.
Everyone has a ticket with his seat number. These 100 passengers boards the airplane in order.
However, the first passenger lost his ticket so he just took a random seat.
For any subsequent passenger, he either sits on his own seat or, if the seat is taken, he takes a random empty seat.
What's the probability that the last passenger would sit on his own seat ?

Solution:

Let us say that everyone is sitting except for the 100th passenger. Consider where can he sit - he can sit in either seat numbered 100 or 1. There cannot be any other seat vacant for him.
We can prove this using contradiction. Lets say that a seat numbered n=55 is vacant for 100th passenger. Since passenger 55 came earlier he should be sitting in his seat if its vacant. So seat 55 cant be vacant. This logic can be applied to all seats except for seat 1 and 100.
So all permutations of the seating arrangement would result in last person sitting in either seat 1 or seat 100. Both the options are equally likely.
So answer = 1/2.