A three-way duel (probability puzzle) This puzzle is taken from Mathematical Puzzles: A Connoisseur's Collection [Peter Winkler]. I don't understand the solution.
Alice, Bob, and Carol arrange a three-way duel. Alice is a poor shot, hitting her target only 1/3 of the time on average. Bob is better, hitting his target 2/3 of the time. Carol is a sure shot. They take turns shooting, first Alice, then Bob, the Carol, then back to Alice, and so on until one is left. What is Alice's best course of action?
The solution is that Alice is better of missing than hitting Carol or Bob, so she should shoot into the air. Indeed, then Bob will shot Carol, and it can be shown that it gives the greatest probability of survival for Alice. But I wonder if Bob should not voluntary shoot into the air too, so that Carol will do the same, and no one be shot. If this is the case, Alice survival probability is 1. What do you think of it? What is Alice survival probability?

Arno proved that if Alice, Bob and Carol want to survive, they will go to a slatemate, so as Daniel V points out, the goal should be to win the duel and not to survive. But assuming that the goal is to win the duel, if Alice and Bob shoot in the air, why Carol could not shoot in the air too? Because she will have to shoot Bob anyway. Indeed, if they keep shooting in the air, when Carol has two bullets left, she has to shoot someone because she prefers having a chance to win rather being sure not to win. As a matter of facts, if Bob does not shoot her dead in the begining, she will shoot him.
For Alice, the probability to win a duel against Bob when it's her turn to shoot is 1/2, and the probability to win a duel against Carol is 1/3, hence her probability to win is 5/12.
