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)?

For 2 people, the probability is obviously 1/2. Suppose the probability is 1/2 for all numbers from 2 to n. Now, suppose there are n+1 people. Then, the first person picks some number x from 1 to n+1. When it is the xth person's turn, it is a similar situation with n+2-x people. Whatever x is, the probability is 1/2. Therefore, if there are n+1 people, then the probability is 1/2. By induction, the probability is 1/2 for all n. Then, if n=100, then the probability is 1/2.

Posted by Math Man on 2018-10-20 14:36:32