First to roll a six in an alternating duel

Question

Two players, Ana and Ben, take turns rolling a fair die, with Ana rolling first. The first player to roll a 6 wins immediately. What is the probability Ana wins?

Accepted Answer

Let P be the probability the player about to roll eventually wins. On this roll they win with probability 	frac16. Otherwise (probability 	frac56) the turn passes to the opponent, who is now the player-about-to-roll and wins with probability P; so the original roller wins with probability 1 - P in that branch: P = 	frac16 + 	frac56(1 - P). Expand: P = 	frac16 + 	frac56 - 	frac56 P = 1 - 	frac56 P, so 	frac{11}{6}P = 1 and P = 	frac{6}{11}. Ana rolls first, so P(	ext{Ana wins}) = \frac{6}{11} \approx 0.545. Takeaway. In an alternating race to a rare event, the mover's edge follows P…

First to roll a six in an alternating duel

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