Probability for certain outcomes with sets of independent events in sequence

I am developing a game that uses dice to resolve conflcts involving two players. The two players each toss a pair of standard d6 dice in matches until only one of them obtains his or her target number, thus winning the challenge.
Assume the two players are seeking numbers with different probabilities, like a 6 for 5/36 and a 7 for 6/36

I know the formulas for independent events to figure out the probability that player A and/or B will obtain their bumber. And from that value you subtract the probability that both will get their number.

But what I want to know is how to find the probability that only one player will obtain his or her number over muliple matches/trials, like doing binomial calculations but for trials involving independent events. In other words, I not only want to know the probability that only one player will obtain his or her number in one match, but also over two or three successive matches, such as if neither or both get their number in the 1st match, in the 2nd match, etc. The probability that only one or the other player gets their number of course increases with each additional match in the series.