  A new formulation of quantum mechanics is proposed based on a new principle
that can be considered a generalization of the Born rule. The principle is
composed of a mathematical expression and an associated interpretation, and
establishes a correlation between the positions of a particle at two different
times. Under reasonable conditions for the wave function, this correlation
implies that the particles follow quasi-classical trajectories. It is also
shown that the Born rule is equivalent to a particular case of the new
principle. These features allow the principle to provide a unified explanation
of the results of the statistical experiments and of the quasi-classical
macroscopic evolution.
  There is a strong analogy between the new quantum principle and a
probabilistic principle which is necessary to derive empirical predictions from
the mathematical formalism of probability theory. This principle is referred to
by some authors as Cournot's principle, while other authors use the equivalent
notion of typicality. In this paper probability theory and quantum mechanics
are formulated in such a way as to explicitly include the two principles and to
emphasize the very similar conceptual structure of the two theories.
