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?

Alice should miss because:


*

*Should Alice actually kill Carol OR Bob, then Alice is immediately the only remaining target. 

*If she misses, Bob will surely target Carol. If Carol dies, it’s Alice’s turn again.
If Bob misses, Carol will shoot at Bob, and it’s Alice’s turn again.
No math required. She should miss so the others can shoot at each other. 
