In this problem, we want SmallR to win. He is the first to play, so what's the probability that he wins right in the first attempt? a/b. Now, what's the probability that he shoots in the target in his second attempt? For this to happen, he and Zanoes need to miss their first try and then he needs to shoot in the target. Thats (1-(a/b))*(1-(c/d))*(a/b). You can easily perceive that, to win in his n-th try, they both need to miss n-1 times. That's (((1-(a/b))*(1-(c/d)))^(n-1))*(a/b) The answer is the sum of his probability of winning in any try. That's the sum of a geometric series with ratio (1-(a/b))*(1-(c/d)). If you do not know how to solve this, check this link.