Jack and Jill throw die, where the first person to throw a double wins. Jack throws first. What is the probability of Jill winning on her second turn.

Question

Jack and Jill are taking turns throwing a pair of dice. The first person to throw a double wins the game. Jack rolls first. Find the probability that Jill wins the game on her second turn.

I believe that the answer is $\frac{125}{1296}$, and I got this by multiplying each expected event by each other ($\frac56\cdot\frac56\cdot\frac56\cdot\frac16$). Could anybody please confirm this, or otherwise tell me how to do the question if I am wrong.

Answer 1

This is infact correct, the probability of Jill winning on her second turn = p(Jack doesn't throw a double)*p(Jill doesn't throw a double)*p(Jack doesn't throw a double)*p(Jill throws a double)= $\frac56\cdot\frac56\cdot\frac56\cdot\frac16=\frac{125}{1296}$