Connectivity of three acute-angle sphere points ($P(\text{connected})=(\pi +2)/(4\pi )$)

Join three independent uniform points on a sphere when their central angle is acute. Given angle $\gamma$ between the first pair, the third point's two favorable hemispheres overlap in a spherical lune of area fraction $(\pi -\gamma )/(2\pi )$. If $\gamma <\pi /2$, connection requires membership in their union, of probability $1/2+\gamma /(2\pi )$. If $\gamma >\pi /2$, it requires membership in their intersection, of probability $1/2-\gamma /(2\pi )$. Integrating these against the central-angle density $\sin \gamma /2$ proves the displayed probability. Connectivity is weaker than all three pairwise angles being acute.