The Airplane Seating Problem

The problem
P1 🍷P100?100 seats · passenger 1 picks at randomdoes P100 get seat 100?

100 passengers board a plane with 100 numbered seats. Each has a ticket for a specific seat. Passenger 1 is drunk, forgets his ticket, and picks a seat at random. Every subsequent passenger:

  • Sits in their assigned seat if it's free.
  • Otherwise, picks a random empty seat.

What's the probability that passenger 100 ends up in seat 100?

Tempting (but wrong)
99 cascading displacements · feels chaoticgut says either ≈ 1/n or ≈ 1 — both wrong

The two most common guesses:

  • "Very small" — maybe 1/1001/100 or 1/991/99. With 99 passengers worth of cascading randomness, intuition says it's nearly impossible passenger 100 gets the right seat.
  • "Very high" — maybe 99/10099/100. Each passenger after the drunk usually sits correctly; the chaos must mostly dissipate.

Both miss the structure. The answer is exact, beautiful, and the same for any n≥2n \geq 2.