Do I want to start first?

The probability of getting 6 points when we throw two dice is $\frac{5}{36}$ . Getting 7 points is slightly more probable: $\frac{6}{36}$ .

A throws first. For $A$ to win, he must get either 6 points in the first turn, or in the third after $B$ loses the second, or in the fifth after $B$ loses the fourth... But notice that every odd turn is like starting the game over. This allows us to skip the infinite sum and write a simple equation for $p$ :

$p=\frac{5}{36}+(1-\frac{5}{36})\cdot(1-\frac{6}{36})\cdot p \implies p=\frac{5}{36}+\frac{31}{36}\cdot\frac{30}{36}\cdot p$

Solving for $p$ we get $\boxed{p=\dfrac{30}{61}}$

The odds are slightly against player $A$ .