In a game two players A and B take turns in throwing a pair of fair...

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

$P ( A\text{wins} ) = P ( A ) + P ( \overset{¯}{A} ) \cdot P ( \overset{¯}{B} ) \cdot P ( A ) + P ( \overset{¯}{A} ) P ( \overset{¯}{B} ) \cdot P ( \overset{¯}{A} ) \cdot P ( \overset{¯}{B} ) \cdot P ( A ) + \ldots \ldots$

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

Thus, probability that $A$ wins $= \dfrac{P ( A )}{1 - P ( \overset{¯}{A} ) P ( \overset{¯}{B} )}$
$= \dfrac{\dfrac{5}{36}}{1 - \dfrac{31}{36} \cdot \dfrac{30}{36}} = \dfrac{30}{61}$