A method is given which can be used to construct procedures for discriminating among distributions on a Euclidean space with continuous distribution functions. The decision space used includes "partial" decisions and the probabilities of errors are random variables with beta distributions. Emphasis is upon control of the distribution of the conditional overall probabilities of errors. These procedures can be used in a wide class of discrimination problems, such as, for example, discriminating among multivariate normal distributions with unknown, unequal dispersion matrices. A number of other writers have suggested nonparameteric discrimination procedures. The first work in this area, to the knowledge of these writers, was by Fix and Hodges [2]. Since the work by those authors has remained unpublished, a brief statement of its approach and results is given in Section 4 for comparison. Procedures have been suggested also by Stoller [3], Anderson [1] and Kendall [8].