Russian Roulette 2

You and your Russian friend Anatoly are playing a suspenseful game of Russian Roulette. The game involves a six-chambered revolver loaded with a single bullet. Initially, the cylinder of the revolver is spun to ensure the chambers' order is randomized. You and Anatoly take turns pulling the trigger, and the cylinder is re-spun after each turn. Anatoly decides to go first. What is the probability that you will win the game?

Answer

Let $$p$$ be the probability that you eventually win the game. Consider the events in the first round:

p = \underbrace{\frac{1}{6}}_{\text{Anatoly dies on first try}} \;+\; \underbrace{\frac{25}{36} \cdot p}_{\substack{\text{both survive first round (each survives with }5/6)\\\text{and the game resets}}}.

Solving for $$p$$ :

p = \frac{1}{6} + \frac{25}{36}p \implies p - \frac{25}{36}p = \frac{1}{6} \implies \frac{11}{36}p = \frac{1}{6},

p = \frac{1}{6} \cdot \frac{36}{11} = \frac{6}{11}.

Thus, the probability that you will win the game is $\displaystyle \frac{6}{11}$ .