Interference of probabilities in the classical probabilistic framework

The notion of context (complex of physical conditions) is basic in this paper. We show that 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 classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach, it is forbidden to consider abstract context-independent probabilities: “first context and then probability.” We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. Our contextual approach is important for demystification of quantum probabilistic formalism. This approach gives the possibility to apply quantum-like models in domains of science different from quantum theories, e.g., in economics, finance, social sciences, cognitive sciences, psychology.