In
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 flo ...
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
-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
:
The statement is equivalent to
being the probability that the origin is not contained in the
convex hull 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
hyperplanes in general position in
divides it into
regions.
References
{{reflist
Probability theorems
Theorems in geometry