100 people line up to board a plane with 100 seats. The first person has lost his boarding pass and picks a seat randomly. Each subsequent person takes their assigned seat if available, otherwise picks a random empty seat. What is probability last person gets their assigned seat?

100 people line up to board a plane with 100 seats. The first person has lost his boarding pass and picks a seat randomly. Each subsequent person takes their assigned seat if available, otherwise picks a random empty seat. What is probability last person gets their assigned seat?

Approach: Observe that when someone picks a random seat, the problem restarts.

1/2. Let P_n be probability for n people. P_2=1/2. For n>2, P_n = 1/n * 1 + 1/n * 0 + (n-2)/n * P_{n-1} = 1/2. Induction shows all equal 1/2.

Follow-up: What if first k people lose passes?

Key concepts: recursive probability, symmetry, seat switching.