Coverage of antipodal points by a randomly oriented sector ($\mathbb{P}(\text{both covered}\mid \Phi =\varphi )=\max (\varphi -\pi ,0)/\pi$)

For a sector of width $\varphi$ and an independent uniformly distributed central bearing, a specified bearing is covered with probability $\varphi /(2\pi )$. Two antipodal bearings cannot both be covered when $\varphi <\pi$. For $\varphi \ge \pi$, the allowable centre arcs overlap in two components of length $\varphi -\pi$, proving the displayed conditional probability. Averaging over the width's probability density function gives unconditional coverage probabilities.