Probability representation for quantum states on examples of oscillator and qubit

In conventional formulation of quantum mechanics system states are described by wave functions, density operators and their representations as the Wigner function, the Husimi function, the Glauber-Sudarshan function. Many decades (practically century) the idea to construct usual probability distributions to describe the states of quantum systems was not realized. Only recently such probability distribution representation for quantum states was constructed for all systems. The spin (qudit) states are described by conditional probability distribution w(X|j) and oscillator states are described by quantum tomograms (symplectic tomographic probability distribution function). In the talk explicit expressions for these probabilities are given. The tomograms describing oscillator states are given by Gaussian probability distributions which contain all information about usual wave functions of the states and density matrices of the state. Two qubit states are described by conditional probabilities and usual density matrices of quantum states are expressed in terms of these probabilities. The Schrodinger equation for wave function is mapped onto kinetic equation for the introduced probability distributions. The general approach for quantization based on the formalism of star-product of functions-symbols of operators is reviewed.