Analytic quantification of the singlet nonlocality for the first Bell's inequality

Recently an alternative way to quantify Bell nonlocality has been proposed \[Phys. Rev. A {\bf 92}, 030101(R) (2015)\]. In this work we further develop this concept, the volume of violation, and analytically calculate its value for the spin-singlet state with respect to the settings of the first Bell's inequality. These settings correspond to three directions in space, or three arbitrary points on the unit sphere. It is shown that the triples of directions that lead to violations in local causality correspond to 1/3 of all possible configurations. From the perspective of quantum communications, this means that two distant parties that were capable of align their measurements in one direction only (the remaining direction in each site being random), have a probability of about 33.3\% to be able to certify their entanglement.