Equal-covariance Gaussian classification error ($P_{\mathrm{error}}=\Phi (-D/2),D^2=({\mu }_1-{\mu }_2{)}^T{\Sigma }^{-1}({\mu }_1-{\mu }_2)$)

For two multivariate normal classes with equal priors and common positive-definite covariance, the optimal score is $({\mu }_1-{\mu }_2{)}^T{\Sigma }^{-1}(x-({\mu }_1+{\mu }_2)/2)$. Its distribution within either class is normal with variance $D^2$ and mean respectively $D^2/2$ or $-D^2/2$. Both class error probabilities are therefore the displayed normal tail.