Given m points on the unit sphere in n-space, the hyperplanes tangent to the sphere at the given points bound a convex polyhedron with m facets. If the points are chosen independently at random from the uniform distribution on the sphere, the number $V_{mn} $ of the vertices of the polyhedron is a random variable. We obtain an integral expression for $EV_{mn}$ and asymptotic bounds of the form \[ \alpha ^n n^{(n - 6)/2} m\leqq EV_{mn} \leqq \beta ^n n^{(n - 5)/2} m. \]