A and B take turns rolling a die. A starts, the winner is the first one that rolls a 6. What is the probability that A wins?

Probability that A wins on first throw = 1/6

Probability that A and B lose their first throws and A wins on his 2nd throw = (5/6)²1/6

Probability that A and B lose their first two throws and A wins on his 3rd throw = (5/6)⁴1/6

etc.

So probability that A wins = (1/6)(1 + (5/6)² + (5/6)⁴ + ...)

This is a geometric progression with sum = (1/6)/(1 - (5/6)²) = 6/11

So the probability that A wins = 6/11

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wow, thank you so much Alan, you just saved me.

I like this solution too Alan, it is really neat