The new tomographic formulation of quantum mechanics is used to develop a method which can reconstruct the entire density matrix of a two-particle spin state in terms of positive classical distributions of probabilities of the values of certain observables. It is shown that to obtain a complete description of the mixed spin state it is necessary to know not only the probabilities of the spin projections as functions of the coordinates of the points on a unit sphere but also the probabilities defining the contributions of the pure states to the mixed ones. With the help of this method the Einstein-Podolsky-Rosen paradox is analysed. It is shown that to remove the paradox the observer must strictly fix at every moment the set of observables and describe the transformation of one set into another. Such a description is performed with the help of the technique of selective and nonselective measurements as defined by Schwinger.