The prompt
One hundred people line up to board an airplane. Each has a boarding pass with assigned seat. However, the first person to board has lost his boarding pass and takes a random seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied seats at random otherwise. What is the probability that the last person to board gets to sit in his assigned seat? [Winkler, Peter. Mathematical puzzles: a connoisseur's collection. CRC Press, 2003]
The attempt
- After passenger #2 is seated, seat #2 is taken (either by passenger 2, or by 1). But the key point is that, the seat is taken.
- After passenger #3 is seated, seat #3 is taken (either by passenger 1,2 or 3).
Key point again being, seat 3 is taken after passenger 3 is seated. - similarly
- After passenger #99 is seated, seat #99 is taken (either by the passengers before them, or by themself).
- When passenger 100 sits, seats #2 - #99 are taken.
This means, either seat #1 or seat #100 are available for passenger 100. Therefore, they can choose their own seat (#100) with the probability 1/2.