To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates θ and ϕ from uniform distributions θ element [0, 2π) and ϕ element [0, π], since the area element d Ω = sin ϕ d θ d ϕ is a function of ϕ, and hence points picked in this way will be "bunched" near the poles (left figure above). n random points can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n]. To obtain points such that any small area on the sphere is expected to contain the same number of points (right figure above), choose U and V to be random variates on (0, 1).
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