Here is the game:
$A$ plays against $B$. Each round, $A$ has probability $a$ of winning, $B$ has a probability $b$ of winning, and the probability of a draw is $d$. So $a+b+d=1$. Each round is independent. Winning criteria is the one who first reaches a streak of $N$. Let's say $N=10$. If a draw occurs, the streak is reset. So the game continues until there is a winner.
How to calculate the probability of $A$ or $B$ winning the game?
What are some approaches I can take? Is there a mathematical way to calculate this rather than running simulations? (I tried running simulations, but I'm not satisfied with the result. If $N$ is fairly big, it takes a long time to run.)