Re: Answer
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Bayes' Theorem is a way to adjust a prior probability given new information.
In this case, the prior probability of finding a bullet directly after a spin is 2/6 or 1/3.
The new information is that your friend pulled the trigger, and there was no bullet in the chamber. That tells you something about which chambers can or cannot be next. There are only four chambers that held no bullet in them. Thus, you know the gun must have been on one of those chambers when your friend fired.
Of those four chambers he could have fired, three of them have empty chambers next to them. Only one of those four has a bullet next. So while the prior probability of finding a bullet is 1/3, the probability GIVEN THAT YOUR FRIEND DIDN'T FIND A BULLET, is only 1/4. Thus, you should just shoot the next chamber.
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This answer assumes that the weight of the 2 bullets chambered side-by-side does not influence their position after a robust spin in an oiled, well-maintained, perfectly balanced revolver.
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