The integration of some functions involving a 2D Gaussian probability integral is reduced to a single integral, thereby greatly facilitating its numerical evaluation. This is surprising because it is well known that a 2D Gaussian probability integral can only be computed using numerical quadrature. Consequently, one might expect that integrating such a function would necessitate double integrals. A numerical example demonstrates the clear advantages of our method. Applications include RF (radio frequency) design engineering coverage probability calculations.