Ronald Meester: The shooting problem

The starting point (time t=0) is a crowd of n people in a room, all having a gun. At time t=1, all people in the room shoot a randomly chosen person in the room; it is possible that two people shoot at each other, but no one can shoot him or herself. We assume that every shot will be fatal and will kill the person shot at. After this first shooting round, some random number of people has survived, and at time t=2 we repeat the procedure with all survivors. We continue to do so, until we have reached the situation that either no one survived, or exactly one person survived. We denote by p(n) the probability that eventually there are no survivors. We are interested in the asymptotics of p(n), as n tends to infinity. The behavior of p(n), for n to infinity, turns out to be rather interesting.

(Joint work with Wouter Kager.)