Randomness, Chaos, Quantum mechanical probability functions Can someone explain these 3 concepts into a unified framework.


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*Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision of the experiment) physics but, the complexity allows us to treat the system as random. (i.e we can always run an exact physical simulation to predict the coin's fall)

*Chaotic randomness: randomness as seen in weather or fluid flow, where the system is highly sensitive to initial conditions and as a result no matter what the precision of measurement/simulation is, we cannot predict the out come. 

*Quantum randomness: The inherent probability distributions that we study in QM, where the electron's position probablity distribution is set by QM. 
how do the concepts of randomness in coin toss, randomness in chaos and randomness in QM fit into a unified picture. Can someone please elaborate ? 
 A: Firstly, I love this question and I wish that others will provide more insights so I can learn more.
I would classify #1 and #2 as the same underlying substance, unless the system scaling is such that the flows can be affected by quantum behavior in which case #2 and #3 would be the same.  The idea behind a coin toss in the first place is that sensitivity to the Initial Conditions (ICs) is high enough (and could be said to be chaotic) so it is useful as a random number.  We don't know very well where the variation in the ICs come from, but we generally don't care because the coin flip experiment subjects those ICs to a transformation that produces a psuedorandom number in terms of heads and tails, just like the famous RANDU computer function.  RANDU notoriously does not produce a perfect random number (or even a good one), but this same argument applies for the coin flip, although it is chaotic randomness, meaning that the mathematics has a transition to chaos.
Contrast this to a radioactive decay experiment where the outcome is actually random.  Let me be clear that I do not understand Quantum Mechanics (QM), and I don't need to.  In fact, it seems like I can't make a statement about QM without being corrected by a physicist.  Radioactive decay appears to be the collapse of the wave function of a coherent state outside of the nucleus.  This is wrong and I don't know why.
I'm fairly sure we will all agree that in the pre-decay state there is a quantifiable probability for the emitted particle to exist outside the nucleus in $1/s$ units, which leads to the decay constant, and combined with the detection efficiency gets the average (or "expected") rate of the counting.  No one has ever provided me a sufficient explanation for how QM gives the described result.  In a literal interpretation of QM, I think that the particle would simultaneously be both inside the nucleus and flying away in all possible directions and times of decay.  Since the $\Delta m$ of the decay, and thus energy of the particle, is a very accurately known number, maybe the uncertainty of the position of the particle is great.  It wouldn't matter anyway because the entire point is that we're not predicting the position of individual particles and are, in fact, using the fact that we don't know as an engineering convenience.
A Defined QM Example
Since the question lacks a clear definition of a quantum experiment I will provide one.  Take a GM Tube radiation detector with a small dead time compared to the counting time.  Establish a counting time based on an expected value of 100 counts, then look at the count and record if it is odd or even.  This functionally does the same thing as a heads or tails coin toss.
Now, some purist is going to comment on this and tell me that the evens are more likely in said experiment.  Yes, that's right, I calculate one extra even for every $2.7 x 10^{44}$ experiments.  So while there is some imperfection in the experiment design, the experiment gives a random outcome in the true sense of randomness.
A: You pretty much know it already. "Random" is a broad word that we use to mean that we can't predict behavior. Each of the three cases of randomness that you cite is unpredictable for a different reason, though - that's the difference.
Dice are random because they are complicated, chaotic pendulums are random because we aren't good enough to measure their initial position perfectly, and quantum systems are random because they aren't deterministic.
Expanding on that:
The randomness of a coin toss or a dice roll is based on an imprecise model. In principle, if we knew everything about the coin (its initial position, the forces applied, the density of air that slows it down, etc) then you could predict whether it winds up heads or tails with certainty. In the real world, nobody bothers, since constructing this model is very difficult. It depends sensitively on the height you are flipping the coin from, its initial spot on your thumb, and so on.
Chaotic randomness is due to imprecision in initial measurements alone. Different initial conditions do not cause smooth changes in the final outcome. A good example is the chaotic motion of the planets - if we try to predict the position of Saturn in 500,000,000 years, we get a certain position based on where it is now and all of the forces acting on Saturn. But if we choose a slightly different initial condition - say, 10cm further along in its orbit to start - then we get a totally different answer potentially hundreds of thousands of km off. Then we look at an initial condition in between, 5cm further along - and the deviation is even worse - it's now millions of kilometers off! In other words, it chaotic randomness arises in systems where improving your accuracy of measurement does not help. The only way to get the "true" answer is to have the exact initial value.
Quantum randomness is due to fundamental laws of nature. Quantum particles behave randomly on their own because that is just what they do, axiomatically. There is no initial measurement which could even be exact. The outcome is fundamentally nondeterministic, not a limit based on our models or our measurements.
In some sense, quantum randomness guarantees that we can never "beat" chaos by getting a perfect initial measurement. But it arises from a fundamentally different origin. 
A: You should check out these two papers:


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*"The Nature of Randomness: Part 1 -
Knowable or Unknowable?" 2008,
Journal of Risk Finance, 9, 1.

*"The Nature of Randomness: Part 2 -
Cognitive Constraints," 2008, Journal
of Risk Finance, 9, 2.
They provide a framework for at least two of your randomness concepts (your second one, i.e. deterministic chaos, would be subsumed under the first one) and connect it with concepts from algorithmic complexity theory (Kolmogorov/Chaitin) + plus they are considerably easy to understand, too.
Esp. the drawn consequence is mind boggling - but you see for yourself...
Highly recommended!
A: There are not 3 categories but only 2.
Randomness without qualification doesn't exist in physical systems.
There is chaotic randomness which applies to classical systems and quantum randomness which applies to quantum systems.
Despite similar words, a chaotic system is not a classical limit of a quantum system - the source of randomness is different and that's why there is not a unique framework for "randomness" alone.


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*Chaotic randomness.
Here the word means that there exists an invariant probability distribution of future states. The chaotic variable itself is not computable but the distribution of future values exists and is invariant.
The system is ergodic and the framework to understanding the randomness in chaotic systems is the ergodic theory.
Billiard balls, rigid molecules, spinning coins, roulette ball, loto numbers, Lorenz system are all examples of ergodic chaotic systems.
For the simplest case, the perfect coin with zero thickness toss, there is an invariant probability distribution of future states which is independent of initial conditions and is 50% for head and 50 % for tails.
The word "randomness" applied to such systems just says that there exists an invariant probability distribution of future states.


Strictly speaking computable deterministic systems are a particular case of an ergodic chaotic system whose probability function is 100% for a state X0 and 0% for all the rest.
It is also necessary to add that not all chaotic systems are ergodic. Non ergodic chaos is both non computable and has no invariant probability distribution. Perhaps it could be called hyperrandomness.


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*Quantum randomness.
Here the randomness is fundamental to the theory. The square of the wave function gives the probability of presence in a volume dV. This is an axiom. From there you derive the formulas giving the probability of measure of any quantum variable. These formulas are experimentally proven with high accuracy. Here the framework to understanding the randomness is the whole of QM which allows to compute the wave function. The quantum randomness is the defining property of psi.

*Quantum chaos. This special category defined as the quantum counterpart of classical chaotic systems by applying the correspondence principle is still speculative. No real quantum chaotic systems have been sofar observed and it is even still discussed whether classical chaotic systems really have a quantum counterpart.
