The Three Men and A Duel

Intuition tells us that Bob, the perfect shot must have the best chance at survival. But what are his chances?

Alex and Chris will never fire at each other as long as Bob is alive- if they were to ever to kill the other, Bob would have the next shot and kill the survivor. Assume then that Alex and Chris fire all their shots at Bob. Assume as well that Bob, to maximize his chances of survival, finishes off Chris then Alex (Chris being the more dangerous opponent). That means Alex will have two shots at Bob, and Chris will have none (having died of lead poisoning by the time his turn rolls around).

Under these conditions, Bob has a 49% chance of survival. Pretty good.

So what are the odds for the other two?

If Alex makes his first shot, it becomes a simple showdown with Chris starting. In this case Alex's chance of survival is 3/13 (the sum of the infinite series of probabilities), or about 23%. But if Alex misses, it his probability of killing Bob on his next shot, or 30%.

This draws our attention to an interesting note- Alex's chances improve if he misses Bob on his first shot. Alex therefore should deliberately miss on his first shot. This raises Bob's chances to a whooping 70%, at the expense of any chance of Chris surviving. Not very nice to Chris, but if Alex really liked Chris in the first place, he wouldn't be trying to kill him.

So Bob really does have the best chance to survive, and his chances are even better than they first appeared!