Is quantum physics truly random or does it just appear that way because of Heisenberg uncertainty principle

The behavior of an electron (and other tiny things) is said to be probabilistic because we can't say where an election will be when we measure it, but only where it will probably be. As I understand it, the Heisenberg uncertainty principle says the more we know about momentum the less we know about location and vice versa.

Is there some property of nature that makes the behavior of an electron random, or does it simply appear random to us because our ability to predict its location in the future is limited by our inability to determine both momentum and location in the present? Or, as seems likely to me, is it simply impossible for us to know whether the behavior is random because we are limited in our ability to observe the details of what is happening (again because of Heisenberg)?

UPDATE: What I really asking is whether we have an answer to the question, "Does God play dice with the universe."
a. Yes - Even if we knew everything about the state of the universe and could violate HUB by knowing both momentum and position precisely we still wouldn't be able to predict the future. b. No - It's a clockwork but we'll never be able to make predictions because there is a limit to know we can know about the current state of the universe. c. The question is unanswerable because the dice/clockwork are so small that we can't ever see them and the behaviors we are capable of observing are the same regardless.

Answer

The Heisenberg Uncertainty Principle (HUP) holds for special observables, as energy and time, space and momentum, ..

To every observable there corresponds a quantum mechanical operator. Quantum mechanical operators either commute or not commute, and are seen in the commutation relationships. Observables that do not commute are what the HUP is about.

It is the HUP that characterizes the probabilistic behavior of elementary particles and the framework of physics when sizes become small enough that h_bar is significant enough to be seen in the behavior of observables, like momentum and location.

Is there some property of nature that makes the behavior of an electron random,

There are effective random distributions in classical statistical mechanics, and these are defined by the gaussian distribution and the standard deviation that describes the randomness.

or does it simply appear random to us because our ability to predict its location in the future is limited by our inability to determine both momentum and location in the present?

It appears random but the distribution is not a gaussian with its standard deviation giving the error. The distribution is strictly defined by the quantum mechanical equations that give the solutions for the specific boundary conditions.

Or, as seems likely to me, is it simply impossible for us to know whether the behavior is random because we are limited in our ability to observe the details of what is happening (again because of Heisenberg)?

With our measurements we measure the probability distributions and see that they are not gaussian, so we know that there is no randomness. The distributions fit the calculations from the quantum mechanical solutions.

a. Yes - Even if we knew everything about the state of the universe and could violate HUP by knowing both momentum and position precisely we still wouldn't be able to predict the future.

This statement is true for classical statistical mechanics, as h_bar there is effectively 0, because of the immense complexity of the ~10^23 molecules per mole. We would still have to work with the gaussian probabilities.

If there exists a deterministic underlying layer below quantum mechanics, the same would hold true, the complexity would be such that the probabilistic form calculated and validated by quantum mechanics would have to hold . A number of physicists are working on this, not popular, direction as 't Hooft who has also contributed on discussions of his proposals on this site.

The arguments against quantum mechanics being an emergent level from a deterministic one come from space time considerations.

b. No - It's a clockwork but we'll never be able to make predictions because there is a limit to know we can know about the current state of the universe.

Physics never says never for new discoveries.

c. The question is unanswerable because the dice/clockwork are so small that we can't ever see them and the behaviors we are capable of observing are the same regardless.

same as in b.