Reaching a two-point lead with a 30 percent edge

Alice and Bob play a series of basketball games starting from a score of $0$ each. Alice wins any game with probability $0.3$, independently. The winner of a game gains a point and the loser drops a point, and the set goes to the first player to reach $2$ points. What is the probability Alice wins the set?

$\tfrac{9}{58}$. Points move in lockstep, so the set ends when one player wins two games in a row from a tied position; a split pair resets it. Alice wins a pair with probability $0.09$ and Bob with $0.49$, so Alice wins with probability $\tfrac{0.09}{0.58}$.