The Absent-Minded First Patron

A sold-out recital hall has $80$ numbered seats and $80$ ticket holders who enter one at a time. The first patron has misplaced her ticket, so she sits in a uniformly random seat. Every patron after her sits in their own assigned seat if it is still empty; otherwise they pick a uniformly random empty seat. What is the probability that the very last patron to enter ends up in his own assigned seat?