Jason (jcreed) wrote,

Beautiful little probability/geometry problem here: what's the probability that four points chosen uniformly from the sphere lie in some hemisphere?.

A solution by J. G. Wendel does it by solving the much more general problem where you've got N random points in Rn that obey any joint probability distribution (not even necessarily IID!) that is reflection-invariant and guarantees that all N-point marginal distributions are linearly independent with probability 1. Out of the linear-independence assumption he notes that N hyperplanes in general position chop Rn up into a certain constant number of connected components (for which it's easy to get a nice recurrence), and from reflection invariance he gets that the number of components determines the probability the problem is asking for.

So cute!
Tags: math

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