The Three Men and A Duel

I will never forget the immemorable tale I heard in Puzzlania of the three man duel. There were once three men who were in competition for the hand of a young woman. So they decided to have a three person duel to see which of them would have to marry her. Wait, that's not right- it was two men and the woman's father. Or was it three brothers competing for their father's fortune?

No, I remember now, they were dueling to see which of them was the greatest Puzzler of all time. Yes that was it. I forgot their names, so we'll call them Alex, Bob, and Chris. Alex was an expert swordsman, which was really too bad, since by tradition this was going to be a pistol duel. As a pistol shot, he was third-rate. Alex hits his target only 30% of the time. Chris was quite a bit better, hitting his target 50% of the time. And Bob was the best of all, never missing a shot.

To compensate for the obvious inequities in their marksmanship skills, they would fire in turns, beginning with Alex, followed by Bob and then Chris. The cycle would repeat until there was but one man standing (with breaks to reload if necessary). And that man would be remembered for all time as the Greatest Puzzler of All Time.

I don't recall the outcome of the duel, but assuming each knew the others' ability and used the optimum strategy, who had the best chance of surving, and what were his odds?