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In this paper we present a method for approximating the risks and Bayes risk associated with a Bayes decision procedure. Additionally, our method leads to approximating the least favorable distribution and the risk associated with the minimax decision procedure. We assume two states of nature (or classes of patterns) and multivariate probability density functions. Taylor series expansions are used, and an nth-order polynomial equation derived from such expansions provides an approximation to one of the least favorable probabilities. An application to a normally distributed random vector of observables is presented with numerical comparisons. The method can be generalized to cases having more than two states of nature by using Taylor series expansions in several variables.