Suppose that the first $n$ passengers have lost their boarding passes. Consider the $n+1$ seats consisting of the $n$ seats assigned to the first $n$ passengers, along with the last person's seat. The order in which these $n+1$ seats get filled is entirely random, as nobody will take any of these seats based on what their boarding pass says. The last passenger will get to sit in her correct seat if and only if that seat is the last of the $n+1$ seats to get filled, so the probability that the last passenger gets her correct seat is $$\frac{1}{n+1}.$$