  In this paper we propose a time-independent \textit{equality} and
time-dependent \textit{inequality}, suitable for an experimental test of the
hypothesis of realism. The derivation of these relations is based on the
concept of conditional probability and on Bayes' theorem in the framework of
Kolmogorov's axiomatics of probability theory. The equality obtained is
intrinsically different from the well known GHZ-equality and its variants,
because violation of the new equality might be tested in experiments with only
two microsystems in a maximally entangled Bell state $\big\vert\,
\Psi^-\,\big\rangle$, while a test of the GHZ-equality requires at least three
quantum systems in a special state $\big\vert\,
\Psi^{\textrm{GHZ}}\,\big\rangle$. The obtained inequality differs from Bell's,
Wigner's, and Leggett-Garg inequalities, because it deals with spin $s=1/2$
projections onto only two non-parallel directions at two different moments of
time, while a test of the Bell and Wigner inequalities requires at least three
non-parallel directions, and a test of the Leggett-Garg inequalities -- at
least three distinct moments of time. Hence, the proposed inequality seems to
allow one to avoid the "contextuality loophole". Violation of the proposed
equality and inequality is illustrated with the behaviour of a pair of
anti-correlated spins in an external magnetic field and also with the
oscillations of flavour-entangled pairs of neutral pseudoscalar mesons.
