  We show that the so-called quantum probabilistic rule, usually presented in
the physical literature as an argument of the essential distinction between the
probability relations under quantum and classical measurements, is not, as it
is usually considered, in contrast to the rule for the addition of
probabilities of incompatible events. The latter is, of course, valid under all
measurement situations upon classical and quantum objects. We discuss also
quantum measurement situation, which is similar to the classical one,
corresponding to classical Bayes' formula for conditional probabilities. We
show the compatibility of description of this quantum measurement situation in
the frame of purely classical and experimentally justified straight ward
frequency arguments [Khrennikov (1999, 2000, 2001)] and in the frame of the
quantum stochastic approach to the description of generalized quantum
measurements (Loubenets (2000); Barndorff-Nielsen and Loubenets (2001)]. In
view of derived results, we argue that even in the classical probability the
classical Bayes' formula describes a particular case of the considered
measurement situation, which is specific for context-independent measurements.
The similarity of the forms of the relation between the transformation of
probabilities, which we derive in the frame of quantum stochastic approach and
in the frame of straight ward frequency arguments, underlines once more that
the projective (von Neumann) measurements correspond only to a very special
kind of measurement situations in quantum theory.
