= Coverage of antipodal points by a randomly oriented sector
{title2=$\mathbb P(\text{both covered}\mid\Phi=\phi)=\max(\phi-\pi,0)/\pi$}
For a sector of width $\phi$ and an independent uniformly distributed central bearing, a specified bearing is covered with probability $\phi/(2\pi)$. Two antipodal bearings cannot both be covered when $\phi<\pi$. For $\phi\ge\pi$, the allowable centre arcs overlap in two components of length $\phi-\pi$, proving the displayed <conditional probability>. Averaging over the width's <probability density function> gives unconditional coverage probabilities.
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