Expected Value, Linearity & Advanced Probability · Superchallenge problem

On a full flight of $100$ passengers, the first passenger sits in a uniformly random seat. Each later passenger takes their own assigned seat if free, otherwise a uniformly random free seat. Find the probability that the $100$th passenger sits in their own assigned seat.

The answer is a fraction $\tfrac{m}{n}$ in lowest terms; find $m + n$.