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What is the probability that n random points on a sphere, sliced by m planes passing through the center, all fall on the same hemisphere defined by one of the m planes? Assume that each plane is perpendicular to a vector from the center of the sphere to one corner of an embedded cube (m=8/2 or 4 unique planes), or in general, a 3D lattice with m=k^3 vertices where k is the number of levels in each dimension.

For example, 5 points are placed on a sphere divided into 14 regions by 4 planes.

This is a generalization of Probability that $n$ random points on a circle, divided into $m$ fixed and equal sized slices are contained in less than $m/2$ adjacent slices.

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    The fact that you haven't had answers is maybe that your problem (very connected to the problem of "all points in "a" same hemisphere", without predefined planes) has not a unique answer because it depends on the relative positions of the planes... – 2017-02-01
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    Good point. I added a constraint that the planes are arranged according to the corners of an embedded cube. – 2017-02-03

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