In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of…

Sum $6\to (1,5),(5,1),(3,3),(2,4),(4,2)$
Sum $7\to (1,6),(6,1),(5,2),(2,5),(3,4),(4,3)$

$P(A \text{wins})=P\left(A\right)+P\left(\overline{A}\right)·P\left(\overline{B}\right)·P\left(A\right)+P\left(\overline{A}\right)P\left(\overline{B}\right)·P\left(\overline{A}\right)·P\left(\overline{B}\right)·P\left(A\right)+\dots \dots$

We see that this is an infinite G.P. with common ratio $P\left(\overline{A}\right)\times P\left(\overline{B}\right)$

Thus, probability that $\mathrm{A}$ wins $=\frac{\mathrm{P}\left(\mathrm{A}\right)}{1-\mathrm{P}\left(\overline{\mathrm{A}}\right)\mathrm{P}\left(\overline{\mathrm{B}}\right)}$
$=\frac{\frac{5}{36}}{1-\frac{31}{36}·\frac{30}{36}}=\frac{30}{61}$