Probability of getting two consecutive 7s without getting a 6 when two dice are rolled Two dice are rolled at a time, for many time until either A or B wins. A wins if we get two consecutive 7s and B wins if we get one 6 at any time.
what is the probability of A winning the game??

Let $p_A$ and $p_B$ the winning probabilities for $A$ and $B$, and $p_6$ and $p_7$ the probabilities to roll 6 and 7.
Now, by regarding the different possibilities for the first roll (and if one starts with 7, also the second roll), we find:
$$p_A=p_6 \cdot 0 + p_7 (p_6 \cdot 0 +p_7+(1-p_6-p_7)p_A) + (1-p_6-p_7)p_A.$$
This is a linear equation for $p_A$.
We have $p_6=\frac5{36}$, $p_7=\frac{6}{36}$, and therefore:
$$p_A=\frac{p_7^2}{p_7p_6+p_7^2+p_6} =\frac{6}{41}.$$
