Players A and B roll a pair of dice in turn, with player A rolling first. A's objective is to obtain a sum of 6 (on one roll), while B's ojective is to obtain a sum of 7. The winner is the first player to reach his or her objective. What's the probability that player A wins?

Does my approach make any sense? I'm probably making this way too hard.

I let be the event that A wins, and be the event that player A rolls a sum of i (i=2,3,...,12) on his first roll.

so

when , , which is the probability that player B does not win on his first roll times the probability that A wins (assuming independence of course)

then

and then solve for

I get 30/61, or about 49%