Three Revolvers and One Bullet

Hi everyone,

I'm new here. My background is in both pure and applied mathematics with a strong emphasis on differential equations, algebra, group theory and number theory however I never studied too much probability theory. Me and some friends (mathematicians and engineers) discussed a problem out of a video game we had all played and for the life of us could not come to a satisfying conclusion. Imagine the following problem.

A man has three revolvers. In case you don't know what I'm on about let me explain. A revolver in the context of this question is a gun which holds six bullets. As one chamber is fired it moves along and there are now 5 more chambers to go. I suppose most of us already knew that but anyway, the important thing is that we've got 6 chambers and that there's an order to them.

Now, the man with the three revolvers picks one of them randomly and places one bullet into one of its chambers randomly. To ensure there's no bias he then spins the cylinder around before locking it in. It is now unknown exactly which chamber of the cylinder the bullet is in.

Now, he begins juggling the three guns. Whilst juggling them he randomly selects one from the air, points and pulls the trigger at an unfortunate person. In total, he does this six times.

What is the probability of the man being shot with the single bullet?

The problem seemed simple at first but we soon realised it wasn't so... at least we think. Sure there's a 1/3 chance of selecting the loaded gun and the loaded gun has a 1/6 chance of having the bullet in the chamber set to fire... but if it doesn't then the next time we fire it we'll have a 1/5 chance, then a 1/4 and so on. However, that increasing probability is affected by an external random choice.

Can this problem simply be solved by using expectation? That just doesn't seem very satisfying somehow.

Me and my friends would love to see any nice neat way to figure this out without resorting to a computer simulation (I don't want to have to remind myself how to write stuff like this in Mathematica).

Any takers?

The game was Metal Gear Solid 3: Snake Eater.