Is the quantum world really random?How can one be sure that there are no variables that can actually predict the outcome like they do in Newtonian physics?
I think J.G.'s answer is good, but doesn't fully address the question since you have the 'quantum-interpretations' tag set (maybe this was done by Qmechanic?), but this is really an unresolved question and highly metaphysical.
In brief, some interpretations of quantum mechanics rely on true randomness (such as most of the collapse interpretations), e.g. Copenhagen and the GRW models.
Other interpretations, such as Bohmian mechanics, only have apparent randomness, which is rather similar to classical effects such as Brownian motion with incorporated non-locality, where the way it works out it is impossible to ever know with certainty the result of measurements, but this isn't due to randomness but rather the observer being a part of the system they are measuring (thus affecting results of measurements---which aren't random---in unpredictable ways).
You then have a loose third set of interpretations that either make no assertion either way, or it's unclear whether randomness exists or not within then.
As I said though, this is all metaphysical; most people can just pick the interpretation they like the most (or none) and forget about it. In this case, J.G.'s answer is all you really need. If you'd like elaboration on anything feel free to ask.
A common misconception is that the difference between classical and quantum mechanics is that only the latter uses probability, and that we might therefore get away with "hidden variables" that explain the apparently stochastic behavior of particles. But actually the difference is that probability obeys different rules in the two theories (see here for a full explanation). In particular, Bell's inequalities apply to classical theories and are in general invalid in quantum mechanics. A simple explanation is given here.