= Pairwise acute central angles of three uniform sphere points
{title2=$P=(\pi-1)/(4\pi)$}
For three independent uniform points on a <sphere>, condition on the central angle $\theta$ between the first two. The third point makes acute angles with both exactly when it lies in the intersection of their positive <hemispheres>. That intersection is a <spherical lune> of opening $\pi-\theta$, hence has probability $(\pi-\theta)/(2\pi)$. The first two points also need $\theta<\pi/2$. Integrating against the <central angle between independent uniform sphere points> density gives $(4\pi)^{-1}\int_0^{\pi/2}(\pi-\theta)\sin\theta\,d\theta=(\pi-1)/(4\pi)$.
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