A line of 100 airline passengers is waiting to board an aeroplane. They each have a ticket to one of the 100 seats of that flight. (For convenience, let's suppose that the kth passenger in the line has a ticket for the seat number k.)
Unfortunately, the first person in line of these 100 passengers is crazy, and will ignore the seat number on the ticket, picking up a random seat to occupy. All the other 99 passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, then they will search for a free seat to sit in, at random.
What is the probability that the last (100th) person in the line to board the plane will sit in his/her proper seat (#100)?