geometric probability Problems of the following type, and their solution techniques, were first studied in the 18th century, and the general topic became known as geometric probability. * (Buffon's needle) What is the chance that a needle dropped randomly onto a floo ...

theory, Wendel's theorem, named after James G. Wendel, gives the

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

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_=2^\sum_^\binom.

The statement is equivalent to

p_

being the probability that the origin is not contained in the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of the ''N'' points and holds for any probability distribution on 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_^\binom

regions.

References

{{reflist Probability theorems Theorems in geometry