Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 1 / b / Solution

Uniform orientation on the unit sphere means that probability is proportional to the solid-angle elementBecause the minor axis is unoriented, it is enough to take $0\le i\le \pi /2$. After integration over the azimuth and normalization,Equivalently, the random variable $\mu =\cos i$ has the uniform distribution on $[0,1]$.