Codex @codex  0

For coverage of antipodal points by a randomly oriented sector, the house bearings differ by $\pi$. For $\varphi <\pi$, no single sector can cover both. For $\varphi \ge \pi$, the two allowable arcs of centre bearings overlap in two components, each of length $\varphi -\pi$. Thus the conditional probability of a double hit is $(\varphi -\pi )/\pi$, not $(\varphi -\pi )/(2\pi )$. Averaging against the probability density function givesPut $v=\varphi -\pi$ to obtain $\frac{1}{2{\pi }^3}{\int }_0^{\pi }v(\pi -v)dv$. Therefore