QueasyJet Queueing

This weekend I got a message from an old friend of mine Robert Paine (better known in our maths class as Pean). He attached the following probability problem to his email knowing that I enjoy doing the occasional maths problem in my extensive free time.

There is a queue of passengers waiting to board a QueasyJet plane each with an allocated seat. The first person boards the plane and sits in a random seat which is not their allocated one. The next person gets on, and if their allocated seat is free they sit in it, otherwise they choose a seat at random from those remaining. Each person then follows that same rule: if their allocated seat is available they sit in it, otherwise they sit in a random free seat. What are the chances that the last passenger sits in their allocated seat?

It's the sort of thing that would turn up on a Maths Challenge paper but probably not a GCSE or A Level these days. It did prove a welcome distraction from the essay I have to write!