This is a story problem that is based on a true story (oh boy!). I went with some friends to dinner on a rainy night, and one of my friends left his coat on the coat rack in the front of the restaurant. Since I'm by nature extremely distrusting and paranoid, I took my coat to my chair and dripped all over the floor. Bah.
In any event, when we got up to leave the restaurant, it turns out that said friend's coat was no longer on the rack. Sure, there were plenty of other coats on the rack, but not the one we were looking for. Grrr!
Anyway, here's the problem:
There are N people in a restaurant, each of which has placed his coat on the coat rack out front.
The first person to leave is either drunk or unscrupulous (or both), and grabs a coat completely at random from the coat rack, puts it on, and walks out.
One by one, the other N-1 people pick up their coats and leave the restaurant. If, however, a patron's coat turns out to be missing (gasp!), said patron gets angry and steals someone else's coat at random.
What is the probability that the last person to exit the restaurant gets his/her own coat?
I haven't written the solution yet. I'll do that later.