As mentioned in the comments, the correspondence principle enters, and how it does mathematically is shown using the density matrix formalism on the quantum mechanical solutions.
Physics is about mathematically modeling data and observations so that new phenomena can be predicted.
Probability functions do predict new phenomena, with a probability attached to them. There is determinism in probability except "what" is determined is the probability distribution its self. If you throw a true dice 1.000.000 times and plot the frequency of the six numbers appearing, the plot is predicted (by probability theory) to be flat. To the accuracy of the eye looking at it, it is flat.
It is the same with quantum mechanical probabilities,except due to the complicated mathematical modeling, a complicated mathematical "summation " explains it rigorously. The classical mathematics describing the macroscopic world , ie Newtonian mechanics, classical electrodynamics etc, emerge by the use of the density matrix formalism. These lecture notes may help in the mathematical path, from simple systems to many body systems.
In a hand waving manner: the density matrix has the quantum mechanical states in rows and columns, so that the off diagonal elements show the quantum mechanical phases which determine the quantum mechanical probabilistic behavior due to a coherence of the quantum mechanical phases. When the number of entries becomes large, and by avogadro's number macroscopic states are composed of order 10^23 molecules, because of the large distances involved between atoms, most of the off diagonal elements tend to zero, except for the ones in the proximity of individual atoms, which are not seen macroscopically. Macroscopically the quantum mechanical density matrix ends up with only diagonal elements reflecting the classical system of no extra probabilistic interference between particles ( except in special cases, like superconductivity). It is part of what is called "decoherence"