Drunk Passenger

A line of 100 airline passengers are 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 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; in that case, they will randomly choose a free seat. You're person number 100. What is the probability that you end up in your seat (i.e., seat #100)?