As noted in the comments this is a much studied question. Einstein, Podolsky and Rosen wrote a paper on it, "Can Quantum-Mechanical Description of Reality Be Considered Complete?", published in Physical Review in 1935, and universally known today as the EPR paper.
They considered a particular situation, and their paper raised the question of "hidden variables", perhaps similar to the microstates which undergird thermodynamics. Several "hidden variable" theories have been proposed, including one by David Bohm which resurrected de Broglie's "Pilot Wave" model. These are attempts to create a quantum theory which gets rid of the random numbers at the foundations of quantum mechanics.
In 1964 Bell analyzed the specific type of situation which appears in the EPR paper, assuming that it met the conditions Einstein et al had stipulated for "physical reality". Using this analysis he then showed some specific measurements that are in agreement with any such hidden-variable, classical theory would satisfy a set of inequalities; these are today known as the Bell inequalities. They are classical results.
He then showed that for ordinary quantum mechanics that the Bell inequalities are violated for certain settings of the apparatus. This means that no hidden variable theory can replace quantum mechanics if it also meets Einstein's conditions for "physical reality".
The EPR abstract reads: "In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete."
In fact, one can run quantum mechanical experiments that routinely violate Bell's inequalities; I'm currently involved in setting one up which will be validated by violating Bell's inequalities. People have been doing this for over 40 years. The main argument against closing this chapter are the various "loopholes" in the experiments. Recently it has been claimed that a single experiment has simultaneously closed all of the loopholes. If that is true, then there are no classical hidden variable theories which can replace regular quantum mechanics unless they are grossly non-local. Einstein certainly would not think that these were an improvement!