The probabilities of misclassification involved in the use of estimated discriminant functions are subject to chance variations. The author's purpose in this paper is to derive the distribution laws that the probabilities of misclassification follow and to obtain their expected values. The parent populations are assumed to be normal. The first part of the paper considers the univariate case and the second part the multivariate case. The discussion of the multivariate case proceeds in three stages of increasing complexity. When the exact results are complicated, asymptotic results or approximations are given. Finally, the problem of estimating the expected probabilities of misclassification is considered. Interval estimates as well as point estimates are given.