The Plane Seat Problem
I came across this riddle about aeroplanes and probability (two of my favourite topics!): There are 100 passengers lined up (in a random order) to board a plane. The plane is fully booked, meaning there are exactly 100 seats available, assigned to the 100 passengers. The first passenger chooses a seat at random, with all seats equally likely. Then, each of the other passengers proceeds as follows: if their assigned seat is free, they will sit in it; otherwise, they will take a random available seat. What is the probability that the last passenger will sit in their assigned seat? Here's my analysis (using the terminology P1-P100 for the passengers, S1-S100 for the assigned seats, and DP1-DPn for the “displaced” passengers): For passenger P1: P1 (randomly) makes one of three choices: Sits in his own seat (S1) , in which case everyone sits in their assigned seat. Sits in the last passenger’s seat (S100) , in which case everyone else sits in their assigned seat, except P1...