Uniform orientation on the <unit sphere> means that probability is proportional to the solid-angle element
$$
d\Omega=\sin i\,di\,d\varphi.
$$
Because the minor axis is unoriented, it is enough to take $0\leq i\leq\pi/2$. After integration over the azimuth and normalization,
$$
\boxed{p_i(i)=\sin i,
\qquad 0\leq i\leq\frac\pi2}.
$$
Equivalently, the <random variable> $\mu=\cos i$ has the <uniform distribution> on $[0,1]$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-320/1/b/solution