Geometric Die

Can you solve this issue on the probability of dice?

Can you solve this issue on the probability of dice? I play a dice game with my friend George. We take turns rolling die impartial standard, and I have the leading role. The game ends when a 1 or 2 is rolled, and to the person who wins the roll is 1 or 2 of the game. However, if two 6s are rolled in a row, then the game ends in a tie and nobody wins / loses. Find the probability that I win the game. I'm really stuck on the way out! Your help is greatly appreciated! (Nb: the sum of a infinite geometric sequence should be used in this issue) or partitions, I'm not sure ...

let W denote gain (1 or 2), 1 / 3 odds on a single roll OPPORTUNITY means a draw is open on the next turn (6), 1 / 6 probability X means the possibility draw closed the next round (3, 4 or 5), 1 / 2 probability only focus on the "I" winner, which means draws or losses do not occur. Sequences in which I can make to follow that model W: 1 / 3 XXW: [1 / 2] [1 / 2] = W [3 / 12] W DXW: [1 / 6] [1 / 2] W = [1 / 12] W XDW: [1 / 2] [1 / 6] W = [1 / 12] W This follows a geometric series with a 1 = / 3, r = 5 / 12 P [I won] = A / (1-r) = [1 / 3] / [7 / 12] = 4 / 7 or 57.14% edit: ------ not change the above requirements. roundwise the number of victories seems Fibonacci: 1,2,3,5 ... with "I" will win in the playoffs strange but probabiliies are committed to making complex earns see if you can take advantage

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