Drunk Passenger?

A line of $100$ airline passengers is waiting to board a plane. They each hold a ticket to one of the $100$ seats on that flight. For convenience, let's say that the $n^{th}$ passenger in line has a ticket for the seat number '$n$'. Being drunk, the first person in line picks a random seat (equally likely for each seat). All of the other passengers are sober, and will go to their proper seats 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 $100^{th}$ person to board the plane will sit in their proper seat $\#100$?