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In many multi-stage decision processes we face the problem of dealing with random variables whoe distributions are initially imperfectly known, but which become known with increasing accuracy as the process continues. In this paper we shall show how Dynamic Programming may be used to treat a class of such problems, which are currently called adaptive processes. After discussing the general theory, we shall illustrate the techniques by a specific example. For this example we derive simple computational algorithms, which are typical of those obtained for the whole class of problems under consideration.