Quantum mechanics is spectacularly successful on the technical level but the meaning of its rules remains shrouded in mystery even more than seventy years after its inception. Quantum-mechanical probabilities are often considered as fundamentally different from classical probabilities, in disre-gard of the work of Cox (1946) -and of Schrödinger (1947) -on the foundations of probability theory. One central question concerns the superposition principle, i. e. the need to work with inter-fering wave functions, the absolute squares of which are probabilities. Other questions concern the relationship between spin and statistics or the collapse of the wave function when new data become available. These questions are reconsidered from the Bayesian point of view. The superposition principle is found to be a consequence of an apparently little-loiown mathematical theorem for non-negative Fourier polynomials published by Fejer in 1915 that implies wave-mechanical inter-ference already for classical probabilities. Combined with the classical Hamiltonian equations for free and accelerated motion, gauge invariance and particle indistinguishability, it yields all basic quantum features -wave-particle duality, operator calculus, uncertainty relations, Schrödinger equation, CPT invariance and even the spin-statistics relationship -which demystifies quantum mechanics to quite some extent.