A problem of distinguishing a finite number N of simple statistical hypotheses is considered. The decision is to be based on a sequence of experiments the length of which is random, but bounded by a finite number m. On each stage, we can choose one of L types of experiments. The cost of experiment depends on its type. \textit{Yu. B. Sindler} and the author [Optimal control criteria in truncated sequential analysis problems. Kibernetika 1985, No.4, 73-78 (1985); English translation in Cybernetics 21, 506-513 (1985)] have formulated this problem as a nonlinear programming problem and an equivalence of it to some dual problems has been shown. The present paper is devoted to methods of solving the dual problem. It is shown that the method of constructing the objective function of the dual problem agrees directly with the methods for determining the Bayes risk. An application of the results to the problem of detecting a signal in a noisy background is given.