A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if he gets a total of 7. If A starts, the probability of winning the game by A in third throw of the pair of dice is :

$\frac{775}{7776}$

The correct answer is Option (3) → $\frac{775}{7776}$

A wins for → $\{(2,4),(1,5),(4,2),(5,1),(3,3)\}$

B wins for → $\{(1,6),(2,5),(3,4),(4,3),(5,2)\}$

$⇒P(A)=\frac{5}{36},P(B)=\frac{1}{6}$

for A to win in 3rd thrown

$P=P(\overline A)P(\overline B)P(A)$

$=\frac{31×5×5}{36^3}=\frac{775}{7776}$