A marksman fires at a target which consists of s zones. At each stage, he shoots independently of his previous efforts and has a probability p 0 of missing the target which is a function of the probabilities p 1 …, ps of hitting zones 1, …, s, respectively. The distribution of the number of hits Xn on the target after n shots is obtained and used to study the marksman's optimal strategy for the case and α 0 ≦ 1.