Probability question: names in hats

[robvalue]

This is a question someone posed to me many years ago. It may be well known, and the answer might be on the internet. I would ask that if anyone does go look it up, that they please don't spoil it for everyone else (and me) by posting the solution here.

I have as yet been unable to solve this, not that I've been trying constantly! I put a few hours into it here and there, and I felt I was coming close but came up empty. Here is the question:

There are ten people, who each write their name on a piece of paper. These are then all put into a hat.

Each of the ten people, in turn, select a name from the hat using the following rule: (the order of the people is not important)

1) They select a piece of paper at random from those remaining in the hat.
2) If the name is not their own name, they keep the piece of paper.
3) If the name is their own name, they pick again randomly, and then return their name to the hat.

The question is: what is the probability that the tenth person is left with their own name in the hat?

Drawing a tree diagram will drive you insane! It's the "putting back your own name" that really makes this a tough puzzle. Regular probability and combination tricks don't apply as neatly.