The specific nature of the randomness arising when quantum subsystems interact (collide) is analyzed. It is shown that the Birkhoff–Khinchin ergodic theorem—the key theorem for classical statistics—or its analog is absent, in principle, in the quantum theory. Thus quantum probabilities cannot be defined within the ergodic concept. A metric definition of probability, based on von Neumann's theory of measurement, is proposed as a measure of comparison of a posteriori physical situation with the a priori situation. The workability of the adopted approach is demonstrated for random walk problems and the theory of thermal equilibrium.