  We in this Letter derive analytic formulas of Bell correlations in terms of
quantum probability statistics under the assumption of measuring
outcome-independence. For a spin-1/2 singlet state we find analytically that
the violations of Bell-type inequalities are really related to the quantum
non-local correlations. However, the Bell and Clauser-Horne-Shimony-Holt (CHSH)
inequalities are always satisfied for the spin-1 singlet states. More generally
the quantum non-locality does not lead to the violation of Bell and CHSH
inequalities for the integer-spin singlet since the non-local interference
effects cancel each other by the quantum statistical-average. Such a
cancellation no longer exists for the half-integer spin singlets due to the
nontrivial Berry phase, and thus the relevant Bell-type inequalities can be
violated. Specifically, our generic observations can be experimentally tested
with the entangled photon-pairs. Our arguments are based on the spin-singlet
states, but could be generalized to other bipartite quantum states.
