Wendel's theorem

In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that N points distributed uniformly at random on an $(n-1)$-dimensional hypersphere all lie on the same "half" of the hypersphere. In other words, one seeks the probability that there is some half-space with the origin on its boundary that contains all N points. Wendel's theorem says that the probability is
 * $$ p_{n,N}=2^{-N+1}\sum_{k=0}^{n-1}\binom{N-1}{k}. $$

The statement is equivalent to $$ p_{n,N}$$ being the probability that the origin is not contained in the convex hull of the N points and holds for any probability distribution on $R^{n}$ that is symmetric around the origin. In particular this includes all distribution which are rotationally invariant around the origin.

This is essentially a probabilistic restatement of Schläfli's theorem that $$N$$ hyperplanes in general position in $$\R^n$$ divides it into $$ 2\sum_{k=0}^{n-1}\binom{N-1}{k} $$ regions.