The Lost Boarding Pass Problem

A hundred passengers board a plane one at a time, each with an assigned seat. The first passenger has lost their boarding pass and sits in a random seat instead. Every passenger after that sits in their own assigned seat if it's free, and otherwise picks a random empty seat. What's the probability the last passenger ends up in their own assigned seat?

The instinct is to imagine tracking all hundred passengers, but the seat assignments only ever "matter" in two ways at any point: either the first passenger's own seat is still free, or the very last seat is still free. Once someone (the first passenger, or a later bumped passenger) is forced to choose randomly, they either take their own seat, the last passenger's seat, or someone else's seat — and only the last two possibilities actually change anything. If they take a random passenger's seat, that just pushes the same problem one step down the line with one fewer seat in play. So the only two outcomes that ever get "resolved" are: the first passenger's real seat gets taken, or the last seat gets taken. By symmetry, of the two special seats floating around, either one is equally likely to be the one finally chosen at each decision point.

That symmetry argument gives the answer directly: at the moment either "special" seat is finally selected, it's equally likely to be seat 1's or seat 100's, so the last passenger's own seat is free with probability exactly one half — regardless of how many passengers there are.