The Gaussian polytope is the convex hull of n independent standard normally distributed points in. We derive explicit expressions for the probability that contains a fixed point as a function of the Euclidean norm of x, and the probability that contains the point, where is constant and X is a standard normal vector independent of. As a by-product, we also compute the expected number of k-faces and the expected volume of, thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function and its complex version. The main tool used in the proofs is the conic version of the Crofton formula.