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Correlations of spins in a system of entangled particles are inconsistentwith Kolmogorov's probability theory (KPT), provided the system is assumed tobe non-contextual. In the Alice-Bob EPR paradigm, non-contextuality means thatthe identity of Alice's spin (i.e., the probability space on which it isdefined as a random variable) is determined only by the axis \alphai chosen byAlice, irrespective of Bob's axis \betaj (and vice versa). Here, we studycontextual KPT models, with two properties: (1) Alice's and Bob's spins areidentified as Aij and Bij, even though their distributions are determined by,respectively, \alphai alone and \betaj alone, in accordance with theno-signaling requirement; and (2) the joint distributions of the spins Aij,Bijacross all values of \alphai,\betaj are constrained by fixing distributions ofsome subsets thereof. Of special interest among these subsets is the set ofprobabilistic connections, defined as the pairs \left(Aij,Aij'\right) and\left(Bij,Bi'j\right) with \alphai\not=\alphai' and \betaj\not=\betaj' (thenon-contextuality assumption is obtained as a special case of connections, withzero probabilities of Aij\not=Aij' and Bij\not=Bi'j). Thus, one can achieve acomplete KPT characterization bof the Bell-type inequalities, or Tsirelson'sinequalities, by specifying the distributions of probabilistic connectionscompatible with those and only those spin pairs \left(Aij,Bij\right) that aresubject to these inequalities. We show, however, that quantum-mechanical (QM)constraints are special. No-forcing theorem says that if a set of probabilisticconnections is not compatible with correlations violating QM, then it iscompatible only with the classical-mechanical correlations. No-matching theoremsays that there are no subsets of the spin variables Aij,Bij whosedistributions can be fixed to be compatible with and only with QM-compliantcorrelations.

No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics