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Taking Seats on a Plane

The 120 seats on a Northeast Airlines flight were completely booked, with each of the 120 passengers having different assigned seats. The passengers entered the plane one-by-one. Unfortunately, the first passenger couldn’t read their boarding pass and sat in a (uniformly) random seat. Each subsequent passenger sat in their assigned seat if it was available when they entered and sat in a (uniformly) random empty seat otherwise. What is the probability that the last passenger sat in their assigned seat?

My intuition says this is 1/2, but I don't know how to go about this formally.

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marked as duplicate by Aryabhata, Did, Henry, Rudy the Reindeer, Martin Sleziak Jan 25 '12 at 8:14

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
The usual intuition about this problem is to think that the answer must depend on the number $n$ of passengers/seats (and oddly, people often suggest that the probability is $\frac1n$...). Curious to know how your intuition leads you to the answer $\frac12$. –  Did Jan 25 '12 at 8:00

1 Answer 1

There are 2 important seats: Seat 1 and Seat 120. If Seat 1 gets occupied, then everyone else get their right place (including last person). If Seat 120 gets occupied, then the last person won't get his place.

It is possible to show, that persons 1, .., 119 choose Seat 1 with the equal probability than Seat 120 (possibly with 0 probability).

The last person gets either Seat 1 or Seat 120, 50% chance.

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