Compute probability last passenger gets own seat

Q: Compute probability last passenger gets own seat

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Question

There are 100 passengers boarding a plane with 100 seats, numbered 1 to 100. Passenger i (for 1 ≤ i ≤ 100) has a ticket for seat i.

The boarding process is as follows:

Passenger 1 is drunk and ignores the seat assignment. They choose one of the 100 seats uniformly at random and sit there.
For each subsequent passenger k from 2 to 100:
- If seat k (their assigned seat) is still empty, they sit in seat k .
- Otherwise (if their assigned seat is already occupied), they choose uniformly at random from all the remaining empty seats and sit in one of those.

You are passenger 100. What is the probability that you end up sitting in your own assigned seat (seat 100)?

Give the final probability as a simplified fraction or decimal.

Compute probability last passenger gets own seat

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