For each possible point on the sphere, the probability that all the particles are in the hemisphere centered on that point is 2^{-n}. Pull the constant out and integrate over the sphere to get 2^{-n} * 1. Or am I missing something? On Tue, Sep 17, 2013 at 1:02 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
A spherical vessel contains n molecules of gas. What is the probability that all the molecules can be found in one hemisphere? For a given hemisphere, it is 1/2^n. For 6 hemispheres arranged like the faces of a cube, inclusion-exclusion gives 6/2^n - 12/4^n + 8/8^n. But what is the probability if any hemisphere is allowed? I'm stuck on this problem.
In two dimensions, the probability that all n molecules lie in some semicircle is 2n/2^n.
A related question is: Given n vectors x[1], ... , x[n], how do we test if they all lie in some half-space, i.e. does there exist a vector u such that u.x[i] > 0 for all i? I'm stuck on this as well.
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