The title is slightly misleading. I really want to know if the randomness and probabilities observed in quantum mechanics is really just the result of a chaotic (yet deterministic) system.
If it is not simply a chaotic system, how do we know?
I'm guessing that when you talk about randomness you're thinking about the collapse of the wavefunction and that the the result of the collapse is apparently random. If so, most us currently believe that the randomness is only apparent and is the result of decoherence.
Decoherence describes the interaction of a quantum system with the environment around it. It's impossible to completely isolate any system from it's surroundings and that means the system must interact with the surroundings and be affected by them. For example if you measure the spin of an electron to find out if it's up or down, you and your equipment are interacting with the electron. Because the environment (i.e. you) has so many degrees of freedom the interaction looks random but it's actually deterministic and the randomness is just a reflection of our limited knowledge.
Whether quantum mechanics is an emergent theory that arises from some deeper theory is an open question. See for example Deterministic quantum mechanics on this site. There are differing views on this and both sides include well respected physicists. I don't think there is any way for bystanders like us to judge who is correct.
I want to examine whether the elucidated question:
can be answered in the positive or the negative.
From the wikipedia entry one has an adequate definition of deterministic chaos :
Do quantum mechanical solutions and expectation values fall within this definition?
does not describe quantum mechanical solutions, which do give long term predictions in a probabilistic format.
Quantum mechanical solutions give probabilities of realization, and thus the future behavior is not fully determined.
So no, Quantum Mechanics does not fall within the province of mathematics known as deterministic chaos theory, with its strange attractors etc.
Yes, you can interpret it as a chaotic, yet deterministic system. But with principally unknown initial conditions (of both observed system and the observer).
This interpretation is called Bohm mechanics.