Assigned flight seating


Suppose there are 100 passengers waiting to board a flight. The flight has exactly 100 seats, and each seat is reserved for a passenger. However, when the first passenger boards they decide to sit in a randomly chosen seat instead of their assigned seat (with all seats equally likely). Each passenger thereafter takes their assigned seat if it's available, otherwise they too sit in a randomly chosen seat.

Given this information, what is the probability that the final passenger to board the flight ends up sitting in their assigned seat?


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