  We compare the classical Kolmogorov and quantum probability models. We show
that the gap between these model is not so huge as it was commonly believed.
The main structures of quantum theory (interference of probabilities, Born's
rule, complex probabilistic amplitudes, Hilbert state space, representation of
observables by operators) are present in a latent form in the Kolmogorov model.
In particular, we obtain ``interference of probabilities'' without to appeal to
the Hilbert space formalism. We interpret ``interference of probabilities'' as
a perturbation (by a $\cos$-term) of the conventional formula of total
probability. Our classical derivation of quantum probabilistic formalism can
stimulate applications of quantum methods outside of microworld : in
psychology, biology, economy,...
