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Drunk passenger boarding

A plane has 100 seats labeled 1 through 100. One hundred passengers board in order.

Passenger 1 (the only drunk) sits in a uniformly random seat instead of seat 1.

Passengers 2 through 100 are sober. Each one:

  • Sits in their assigned seat if it is empty;
  • Otherwise chooses uniformly at random among the remaining empty seats.

What is the probability that passenger 100 sits in seat 100?

1/2.

Reframe the process: whenever someone finds their seat taken, “wake” the drunk and move him to another empty seat. Then only the drunk is ever in the wrong seat.

When passenger 100 boards, the drunk must occupy one of the two remaining empty seats, and seat 100 vs. the drunk’s seat are symmetric—so the last passenger gets seat 100 with probability 1/2.