What does it mean that an electron's position is described by probability?

I just read an answer to a Phys.SE question about why electrons don't collide with protons where an answering user said:

In quantum mechanics, an electron doesn't have a definite position or momentum. It has a wave function from which the probability of finding it at a particular position or momentum can be calculated. An electron bound to a proton will probably be very near the proton.

Is an electron's path inherently non-deterministic -- i.e., is it unlike billiards in that if you know the starting velocity and mass of one ball you can calculate the rest of its path entirely. Or is it that, it is just impossible for observers to figure out this data, and thus we have to rely on probability calculations?

If it is not simply a case of being unable for us to observe, then I will be very confused. Does that mean "randomness" exists? That would seem extremely... illogical? But obviously I'm missing some pieces of understanding.

• Uncertainty is not randomness. The evolution of an isolated quantum process is perfectly deterministic. Unlike in classical mechanics the final state of that evolution of that system can simply have multiple possible classical outcomes. Quantum mechanics leaves the future open to us, more importantly, still, it allows us to chose which aspect of the future we prefer. The price we have to pay for those choices is that they are not fully repeatable when we set up multiple classically identical experiments. That's why there will only be statistical distributions at the end. – CuriousOne Feb 22 '16 at 21:09
• I like pathintegral's assertion that "QM is not as deterministic as you would wish." Remember, QM is not reality: QM is a model that, as far as it goes, is consistent with reality. The problem is, it does not go as far as you might wish. Could some other model go farther? QM will not tell you the answer to that one, and right now, QM is the best model that we have. – Solomon Slow Feb 22 '16 at 22:15

However, quantum mechanics is not "as deterministic as you would wish", in the sense that there is Heisenberg uncertainty principle. The uncertainty principle arises from the fundamental property that quantum observables do not always commute with each other, i.e. generally $\hat A\hat B\neq \hat B\hat A$, and as a result, if a quantum state has a definite value for $A$, it cannot at the same time have a definite value for $B$, and vise versa. This is in sharp contrast with the way we perceive the world works -- for example we say we know the state of a billiard ball only if we know both its position $x$ and its velocity $v$. Well quantum mechanics says this is simply impossible for quantum states (which means for all states in real life).
But things are not that bad. The uncertainty principle says that the uncertainties $\Delta A$ and $\Delta B$ satisfy $\Delta A\Delta B\sim \hbar$. You see that while it is impossible to be absolutely certain about both ($\Delta A$ and $\Delta B$ cannot both be zero), but you are still pretty certain, in the sense that the "average uncertainty" $\hbar$ is really really a small number.