Last passenger's seat after a random first pick

A plane has $100$ seats and $100$ passengers board in order, passenger $i$ holding a ticket for seat $i$. The first passenger ignores the ticket and sits in a uniformly random seat. Every later passenger takes their own seat if free and otherwise a uniformly random free seat. You board last. What is the probability you get your own seat?

$1/2$. The process ends when someone takes either seat $1$ or seat $100$. Every random choice treats those two seats symmetrically, so seat $1$ is equally likely to be claimed first, in which case you get your seat.