A group of people are standing around in a circle.

Every ten seconds, every person simultaneously shoots a random other person (with a laser tag system, of course. Nobody is actually hurt). If you're hit, you're out. Everyone is a great shot: they don't miss. Whoever they picked at random to take out is out.

Ten seconds later, once again, everyone still standing shoots someone else at random. And so it goes on until one of two things happen: either there's a last lucky person standing, or there's nobody left.

Question: what's the probability that nobody's left at the end of the game? Your response should contain 550+ words.
user