These notes present a probabilistic framework that enables a formulation of classical probability theory, thermodynamics, and quantum probability with a common set of four principles or axioms. It explains everything that usual quantum mechanics and classical probability theory does. We emphasize that this framework is not an interpretation of quantum mechanics, such as ''many worlds``, ''Kopenhagen interpretation``, or others. It is a probability algorithm that computes probabilities of future events and additionally enables a reconstruction of quantum theory, thermodynamics, diffusion, and Wiener processes. We distinguish strictly between possibilities and outcomes. Moreover, we use a time concept based on the classification of future, present, and past. Well-known paradoxes are resolved. The superposition principle obtains a new meaning. The inclusion-exclusion principle, well-known in probability theory and number theory, is generalized to complex numbers. Our probabilistic framework is not based on the Hilbert space formalism. It requires only simple set theory and complex numbers. Thus, this theory can be taught in schools. Our framework may be viewed as an axiomatic approach to probability in the sense of Hilbert, who asked for an axiomatic probability theory in his sixth of the twenty-three open problems presented to the International Congress of Mathematicians in Paris in 1900. We have applied our probabilistic algorithm to several problems, including classical problems, statistical mechanics and thermodynamics, diffraction at multiple slits, light reflection, interferometer, delayed-choice experiments, and Hardy's Paradox.