The contextual approach to the Kolmogorov probability model gives the possibility to represent this conventional model as a quantum structure, i.e., by using complex amplitudes of probabilities (or in the abstract approach—in a Hilbert space). Classical (Kolmogorovian) random variables are represented by in general noncommutative operators in the Hilbert space. The existence of such a contextual representation of the Kolmogorovian model looks very surprising in the view of the orthodox quantum tradition. However, our model can peacefully coexist with various ‘no-go’ theorems (e.g., von Neumann, Kochen and Specker, Bell, …).