Consider a simple game: you and I alternate flipping coins, and (arbitrarily) I flip first. I win if I flip a total of $M$ heads before you flip a total of $N$ heads. For integers $M$ and $N$, what is the probability that I will win?
I can easily write a program to determine the probability through random simulations, so I'm not interested in answers about that. What I'd like to know is whether any well-known probability distribution can model this situation, or whether there is any closed-form solution that can give an answer, even if only approximately.
I'm aware of the negative binomial distribution which seems related, but doesn't seem to apply directly. Of course generalizations (e.g. biased coins, simultaneous flips) are interesting too.