There is a fully booked flight, which has just begun boarding passengers. Each passenger has already been assigned a seat. The first passenger to board is distracted and does not look at their assigned seat, but sits down in a random seat. For the other passengers, when they board, if their assigned seat is free they sit there. Otherwise they sit at a random seat that is free.
What is the probability that the last passenger to board sits in their assigned seat?
The last passenger has a 50% chance of finding their seat free.
Proof
From the point of view of the last passenger, it is an equivalent problem if the previous passengers all kick the distracted passenger out of the seat if it belongs to them. In this situation the distracted passenger chooses a different free seat.
They continue being dislodged until they ends up in either their own seat, or the seat of the last passenger. Both of these situations are equally likely, and hence the last passenger has a 50% chance of finding their seat free.