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