On the mystery of quantum probabilistic rule: trigonometric and hyperbolic probabilistic behaviours

We demonstrate that the origin of so called quantum probabilistic rule (which differs from the classical Bayes’ formula by the presence of cos θ-factor) might be explained in the framework of ensemble fluctuations which are induced by preparation procedures. In particular, quantum rule for probabilities (with nontrivial cos θ-factor) could be simulated for macroscopic physical systems via preparation procedures producing ensemble fluctuations of a special form. We discuss preparation and measurement procedures which may produce probabilistic rules which are neither classical nor quantum; in particular, hyperbolic