Race between a sum of six and a sum of seven

Two players alternate throwing a pair of fair dice, with A throwing first. A wins the moment A's throw totals 6; B wins the moment B's throw totals 7. Turns continue until one of them wins. What is the probability that A is the winner?

$30/61$. A succeeds with probability $\tfrac{5}{36}$ and B with $\tfrac{6}{36}$. A wins in round $k$ after $k-1$ rounds where both fail, so $P(A)=\dfrac{5/36}{1-\tfrac{31}{36}\cdot\tfrac{30}{36}}=\tfrac{30}{61}$ — slightly under one half despite moving first.