In contrast to the wide-spread opinion that any separable quantum state
satisfies every classical probabilistic constraint, we present a simple example
where a separable quantum state does not satisfy the original Bell inequality
although the latter inequality, in its perfect correlation form, is valid for
all joint classical measurements. In a very general setting, we discuss
inequalities for joint experiments upon a bipartite quantum system in a
separable state. We derive quantum analogues of the original Bell inequality
and specify the conditions sufficient for a separable state to satisfy the
original Bell inequality. We introduce the extended CHSH inequality and prove
that, for any separable quantum state, this inequality holds for a variety of
linear combinations.