Red Dead Redemption

In a room stand $n$ armed and angry people. At each chime of a clock, everyone still standing simultaneously spins around and shoots one of the other people still standing, chosen uniformly at random and independently of everyone else. The persons shot fall dead and the survivors spin and shoot again at the next chime. Eventually, either everyone is dead or there is a single survivor. As $n\to\infty$, the probability that there is a survivor: