100 passengers are boarding a plane with 100 seats. The first passenger in line has lost his boarding pass and chooses a random seat to sit in. Each subsequent passenger takes their assigned seat if it is unoccupied. Otherwise, they take a random unoccupied seat. What is the probability that the last passenger will find their seat unoccupied?

Why this works

This riddle hinges on the behavior of the first passenger and the choices of the subsequent passengers. When the first passenger randomly chooses a seat, they could either choose their own assigned seat (which is seat 1), or one of the other 99 seats, including the last passenger's assigned seat (seat 100). As each passenger follows the rule of sitting in their assigned seat if it's available or choosing randomly from unoccupied seats if it's not, the outcome ultimately depends on whether the first passenger sat in seat 100 or not. If the first passenger doesn't take seat 100, then by the time the last passenger boards, their seat will still be available. Thus, there are only two critical scenarios: either the first passenger ends up in seat 1 (leaving seat 100 untouched), or they end up in seat 100 (making it occupied). This gives the last passenger a 50% chance of finding their seat unoccupied, as these are the only two relevant outcomes!