There are n seats on an airplane and n passengers have bought tickets. Unfortunately, the first passenger to enter the plane has lost his ticket and, so he just chooses a seat at random and sits in it. Thereafter, each of the remaining passengers enters one at a time and either sits in their assigned seat if it is empty, or, if someone is sitting in their seat, chooses a seat at random from those that are empty.
a) If n=2, what is the probability that the last passenger to enter will end up sitting in her assigned seat?
b) If n=3, what is the probability that the last passenger to enter will end up sitting in her assigned seat?
c) For general n > 2, what is the probability that the last passenger to enter will end up sitting in her assigned seat? (We will need a proof here).