Compute probability last passenger gets own seat

There are 100 passengers boarding a plane with 100 seats, numbered 1 to 100. Passenger `i` (for 1 ≤ i ≤ 100) has a ticket for seat `i`. The boarding process is as follows: 1. Passenger 1 is drunk and ignores the seat assignment. They choose **one of the 100 seats uniformly at random** and sit there. 2. For each subsequent passenger `k` from 2 to 100: - If seat `k` (their assigned seat) is still empty, they sit in seat `k`. - Otherwise (if their assigned seat is already occupied), they choose **uniformly at random** from all the remaining empty seats and sit in one of those. You are passenger 100. What is the probability that you end up sitting in your **own assigned seat** (seat 100)? Give the final probability as a simplified fraction or decimal.