Dice Trouble

Two players $A$ and $B$ alternately throw two fair six-sided dice. Player $A$ wins if he scores 6 points before $B$ scores 7 points. Otherwise, player $B$ wins. Calculate the probability that $A$ wins, knowing that this player starts the game.

If the value of this probability can be written as $\displaystyle{\frac{m}{n}}$ , where $m$ and $n$ are relatively prime integers, find $m+n$ .