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 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?
I have been trying to solve this problem for a while but couldn't find a good approach. Any insight is appreciated! Here are the probabilites for the small values of $n$:
$n = 2$ -> $0$
$n = 3$ -> $3 / 4$
$n = 4$ -> $48/81$
For much larger values of n, I ran some computer simulation and found that the answer gets close to $1 / 2$.