A line of 100 airline passengers is waiting to board the plain. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let's say that the Nth passenger in line has a ticket for the seat number N.)
Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random.
What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?
The last passenger has a 50% chance of finding their seat empty. This can be seen by
imagining those who find someone in their seat actually taking that seat and
sending the person occupying their seat to a random seat. This is completely equivalent to what
happens in the problem as given, since it replicates the order in which
(interchangeable) bodies are seated in seats. Viewed this way, everyone gets seated
correctly except the person in the wrong seat, who must move, and this is always the "crazy man". This passenger will be moved often until he lands in, and stays in either
his correct seat or the last passenger's seat (with equal likelihood). He is in one of these two when the last
passenger boards.
simpler solution
