crazy man in the airplane A line of 100 airline passengers is waiting to
board a plane. they each hold a ticket to one of the 100 seats on that
flight. (for convenience, let's say that the nth passenger in line has a
ticket for the seat number n.)

Unfortunately, the first person in line is crazy, and will ignore the seat
number on their 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 (#100)?

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