There is a 1/6 chance that he will lose on the first shot, and a 5/6 chance that he doesn't.
If he doesn't, the other man then also has a 1/6 chance (since the chamber is respun) of losing, and a 5/6 chance that he survives the second turn.
Therefore, there is a (5/6)(5/6)= 25/36 probability that the first man takes a second shot, at which point the game essentially starts over.

From this point, there is the same chance that he loses any time after the first shot as after the third shot, etc. Putting this into an equation:

P = 1/6 + (25/36)P
(11/36)P = 1/6
P = 6/11

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