The other answers provide an account of quantum theory as a source of the answer. But I would like to consider the question from a more general perspective: that of the meaning of "probability" especially in the classical context.
Specifically what is meant by "probability" of the dice value in this question and where has the value 1/6 come from? This may seem like a trivial point, but there are (at least) three different perspectives on the answer:
(1) Frequentist. This is the view that probability is a number determined empirically by a series of tests. So this view says that such tests have been conducted on the dice and in 1/6 of cases (subject to some statistical factor) a given number was returned.
(2) Absolutist. This is the view (although I am not sure anyone will promote it) that the probability (with its value here of 1/6) is a derived physical property of the dice (like temperature, say). As a physical property it has been determined by some detailed analysis. One possible such analysis might be on the phase space of all possible motions, which has been appropriately partitioned and it is found that 1/6 of the volume is associated with each resultant value. This analysis might be done for that dice, or for some idealised dice. So the absolutist would claim that physics determines (classical) probabilities, and would view the frequentist data as simply experimental (dis-)confirmation of a specific model.
(3) Relativist. This is really the Bayesian viewpoint as espoused by Jaynes. This is using conditional probability $P(A|C)$ to define a probability in a relative way, with Bayes Theorem - which also allows us to construct $P(C|A)$ - this allows discussion of Inference and Prior Probabilities. The Bayesian view (especially as promoted by Jaynes) would be that there is no absolute probability, but only levels of knowledge, "observer inferencing" and (changing) prior assumptions. So the 1/6 value would be determined subjectively, as a measure of your "expectation" for example. It would have nothing to do with any intrinsic physics of the dice.
So to return to Quantum Mechanics. It too deals with probabilities, which are calculated from $|\Psi(x)*\Psi(x)|$. So what do these probabilities mean now that we have done some analysis?
Jaynes liked to promote his Bayesian view (ie 3 above) that he claimed reconciled Bohr with Einstein, thus interpreting $\Psi$ as somewhat epistemological. Phrases I have seen in other Stack answers like "the $\Psi$ collapse is all in your head" when discussing $\Psi$ might suggest that some physicists share an aspect of this view.
However an alternative view might be that $\Psi$ really solves the Absolutists problem: namely it provides a well validated physical basis for calculating probabilities in an absolute sense.
If this were true then the only physical way to get to 1/6 in the dice would be to do some giant quantum calculation from its atoms!