  Pick $d+1$ points uniformly at random on the unit sphere in $\mathbb R^d$.
What is the expected value of the angle sum of the simplex spanned by these
points? Choose $n$ points uniformly at random in the $d$-dimensional ball. What
is the expected number of faces of their convex hull? We answer these and some
related questions of stochastic geometry. To this end, we compute expected
internal angles of random simplices whose vertices are independent random
points sampled from one of the following $d$-dimensional distributions: (i) the
beta distribution with the density proportional to $(1-\|x\|^2)^{\beta}$, where
$x$ is belongs to the unit ball in $\mathbb R^d$; (ii) the beta' distribution
with the density proportional to $(1+\|x\|^2)^{-\beta}$, where
$x\in\mathbb{R}^{d}$. These results imply explicit formulae for the expected
face numbers of the following random polytopes: (a) the typical Poisson-Voronoi
cell; (b) the zero cell of the Poisson hyperplane tessellation; (c) beta and
beta' polytopes defined as convex hulls of i.i.d. samples from the
corresponding distributions.
