Bohm-Bell type experiments: Classical probability approach to (no-)signaling and applications to quantum physics and psychology

We consider the problem of representation of quantum states and observables in the framework of classical probability theory (Kolmogorov's measure-theoretic axiomatics, 1933). Our aim is to show that, in spite of the common opinion, correlations of observables $A_1, A_2$ and $B_1,B_2$ involved in the experiments of the Bohm-Bell type can be expressed as correlations of classical random variables $a_1, a_2$ and $b_1, b_2.$ The crucial point is that correlations $\langle A_i, B_j \rangle$ should be treated as conditional on the selection of the pairs $(i, j).$ The setting selection procedure is based on two random generators $R_A$ and $R_B.$ They are also considered as observables, supplementary to the "basic observables" $A_1, A_2$ and $B_1, B_2.$ These observables are absent in the standard description, e.g., in the scheme for derivation of the CHSH-inequality. We represent them by classical random variables $r_a$ and $r_b.$ Following the recent works of Dzhafarov and collaborators, we apply our conditional correlation approach to characterize (no-)signaling in the classical probabilistic framework. Consideration the Bohm-Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science.