It depends on if he's good enough to hit the target.

When Charlie shoots first, there are four possible outcomes:

Target Adam, HIT Adam
Target Adam, MISS Adam
Target Brian, HIT Brian
Target Brian, MISS Brian

Charlie has a 50% chance that he will score a successive hit. This applies to each person in the group when it is their turn. However, looking at it a different way, the person being aimed at has a 25% chance that they will be hit. So, ultimately, the best way for Charlie to survive is to miss his shot.

It's more likely then, that Brian will attempt to shoot Adam. If he is sucessful, then the gun is passed back to Charlie who still has that 50% chance of sucess.

If Brian were to be shot by Charlie, then the gun is passed to Adam and he is unlikely to miss his target of Charlie.