Is the probabilistic behaviour of quantum mechanics something "new" (not directly connected) to wave-particle duality? Is the probabilistic behaviour of quantum mechanics theory a direct consequence of the particle-wave duality ? Or is probabilistic behaviour an additive, independent feature (with respect to particle-wave duality) , of quantum mechanics.
To reformulate in another way : if we assume that a particle is actually both a particle or a wave, would the theory necessary have for consequence to be a probabilistic theory ?
If the answer is that probabilistic behaviour is a direct consequence of wave-particle duality, then I could not understand why Einstein was puzzled and against the idea of probabilistic behaviour ("god does not play with dice"). Indeed, Einstein did show that photon was both a particle and a wave, so he showed already the duality. If the consequence of duality is the probabilistic behaviour, how could he be shocked of probabilistic behaviour ?
 A: It's something new.
Generally, probability was seen as an outcome of not knowing everything about a system. This is how thermodynamiics explain the bulk properties of many particle systems.
In QM, on the other hand, probability was seen as ontologically basic. It's not due to the lack of information we have about the system.
Having said it is something new, Aristotle said that 'some philosophers claim that chance is a cause'. This means at the very beginnings of physics a claim was made that chance should be taken as ontologically basic. This was forgotten in the enthusiasm that greeted Newtonian physics which was wholly deterministic.
Quantum reality is value indefinite in some sense. And it's because of this that chance is ontologically basic. Though in this view, it is also derived, since by being indefinite and so indeterminiate we cannot say that something can be determined and so cause here must be by chance.
A: These are the postulates of quantum mechanics.
The wavefunction postulate is the one introducing probabilities in the theory of quantum mechanics:

The theory is being validated continuously. How the theory is used, whether for elementary particles, or composite ones, or excitations depends on the physical observables studied.
Wave particle duality emerges from the underlying probabilistic quantum theory.Let us take electons through a double slit:

It is very clear that individual electrons are dots, leaving particle traces. The accumulation of electrons displays an  interference pattern of a wave. Thus the  so called "duality" is not in space for a single electron,but for an accumulation of electrons with the same energy and boundary conditions, the probabilistic wave appears. The same is true with photons.

if we assume that a particle is actually both a particle or a wave,

as the animation  shows, a particle is not both, it has a probability to fall on a wave determined by its momentum and the boundary conditions of the problem, but a single particle is a point in the probability distribution.
Experiments with interference effects existed before quantum mechanics emerged as a strict theory. Once a theory is found, it supersedes previous phenomenological or hand waving models or hypothesis.
A: No, because of duality we do NOT have probability.  We have probability because the electron is so small and moving so fast and even trying to observe it with light is problem because light has many sizes and the wavelengths of these are much bigger than the electron.
Einstein was thinking that if we have enough information we could say that the electron has a starting position , it has a velocity, therefore we know where it will be ..... but the problem is that we can never measure the final position because the light we use has a significant error in it from the electrons point of view.
Einstein was of the opinion that as scientists get better equipment we can forget about probability ... but so far that is not the case.
