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the last person to shoot will never have fired more shots

If the odds of losing are 1/6 for every shot he fired at himself, then the total odds of losing are 1/6 raised to the S (for shots fired at himself)
The last person to shoot will at times have shot one fewer than his oponent and at times shot the same. He will never have fired more shots. The number of shots fired will dictate the odds of losing. Shoot last and you won't shoot most - guaranteed.

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