Last morning, I was having a rather humorous email exchange with some of my friends from my previous job when one of them, Gasan, sent a very interesting puzzle. This is not that simple. What is surprising is his kids got this problem as a homework problem. Also Raj let me know that the answers are all over the internet. So, if you want to exercise your brain, do not Google it up. After you have given it whatever time you want to give, feel free to look up the answer.

Meanwhile, just to show how much fun we had as a team – or rather the team had at my expense, I am going to copy Gasan’s articulation of the problem as is…

A line of 100 airline passengers are waiting to board a plane. They each hold a ticket to one of the 100 seats on the flight. (for some reference, let’s say that the nth passenger in line has a ticket for the seat number n.)
Unfortunately, the first person in line is a crazy bald headed guy named Rajib, and will ignore the seat number on their ticket, picking a random seat to occupy. All the other passengers including Gasan, Chris, Sunjay, even Raj and Karthik 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)?