One hundred passengers are waiting to board an aircraft. The first passenger forgets her boarding pass and therefore takes a random seat. From here on, the passengers who follow take their own, if available, or take a seat randomly. When the 100th passenger arrives, what is the probability that she gets the right spot?
The answer is a surprising 1/2, and the only seats that remain for the 100th person are the seat of the first person or her correct one. Let’s run this exercise and prove the answer by induction.
Let the seat number of the first person who lost the pass be #37. When she comes inside, she has three possibilities to select (at random).
1) Select her seat, #37: In that case, everyone else, including 100th one, will board correctly.
2) Select the seat of the 100th person, say #13. Here, everyone else will sit correctly, and the last person has #37.
3) Select a random seat other than #37 or #13. She chooses #79.
All other passengers board properly until #79 arrives. She has three choices:
1) Take #37: This is the actual seat of the first passenger. If this happens, then onwards, everyone gets their respective.
2) Take #13. It is the last one’s seat. All others, except the last, get their assigned seat and #37 is available empty for the last one.
3) Take a random one from the remaining seats other than #37 or #13, only for the next unlucky one to repeat the game!
