When classifying an observation into one of $k$ multivariate normal distributions based on samples of correctly classified observations, two estimates of the probability of correct classification, called the apparent and plug-in correct classification rates, are considered. Asymptotic expansions are found for the means and variances of these estimates. It is shown that these expansions can be used to help reduce the bias of the estimates. In the course of finding the expansions, an asymptotic expansion for the conditional joint density of two observations given the sample mean and pooled covariance matrix is found.