In this short paper, an older Efron's result is extended to obtain a cutting plane integral formula for the mean volume of a random simplex in any d dimensions.
Comment: The assumption that the convex hull of d+2 points in R^d is either a d-simplex or a bi d-simplex is true only in d<4. In higher dimensions, there are more simplical polytopes, among which the cyclic polytope maximalizes the number of facets. As a consequence, there is no simple linear relation between the number of vertices and facets in d>3, from which one could connect the expected values