In a room stand $N$ armed and angry people. At each chime of a clock, everyone simultaneously spins around and shoots a random other person. The persons who get 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 $N$ grows, what is the limiting probability that there will be a survivor?

Warning: I could not solve it. The solution only shows what I tried.