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)?

After a discussion with the wife, we came upon a slicker approach:

1) Each time a passenger does a random choice, the first and last seats are equally likely to be picked.
2) Once the first or last chair is picked, there are no more random choices.
3) The first or last chair must be picked before the last passenger comes aboard.

Thus, the last passenger has a 50% chance of being in the last chair and a 50% chance of being in the first chair.
Note the lack of dependency upon the number of passengers, and that there is a 50% chance of the last passenger not getting their chair if the first and last chair are different (n>1).

Posted by owl on 2005-08-18 02:33:43