Okay, then let's start at the very beginning, the wave particle duality. This is nothing like a particle surrounded by a wave, the particle and the wave function live in entirely different spaces. The particle lives in spacetime while the wave function lives in an abstract state space.
Imagine a particle that can occupy any square of a chess board. The space the particle lives in is then a discrete two dimensional space with eight possible positions in each dimension. The wave function of this particle on the other hand lives in a 64 dimensional complex Hilbert space, one dimension for every possible state of the particle, one for every possible location of the particle. If the particle could additionally either have a charge of plus or minus one unit, then the state space would become 128 dimensional, 64 possible positions times 2 possible charges.
If there is only one particle, you can indeed visualize a probability wave on the chess board indicating the probability of finding the particle on one square or another, but that starts failing once you bring in a second particle. The state space for two particles on a chess board is 4096 dimensional, again one dimension for every possible state, one for every combination of the locations of the first and second particle.
And if the particles are entangled, you can no longer simply visualize the state [of one of the particles] with a probability wave on the chess board because the probability of finding one particle on a specific square may in general depend on the other particle. At best you can integrate over all possible states of the second particle but this will destroy a lot of information contained in the wave function.
Can I ask a dumb question? If I will create model of our Universe in a computer language and I will chose to use five-dimensional array for ease of computation, will our Universe have fifth dimension then?
Despite that randomness is used in calculations, it is very unlikely that our Universe is random, thus "probability wave" is just mathematical abstraction, disconnected from reality, not a real thing. It just a particle, with concentrated rings of magnetic field around it (like rings of magnetic field around magnets). It moves fast, and it motion is influenced a lot by fluctuations in quantum field, so it is impossible to measure position of a particle precisely, like motion of a single atom in plasma, so random distributions and averages are very helpful in calculation.
Well, if you want to model our universe, you have to model our universe. If you decide to use 5 dimensions - do you mean 5 or 4+1 dimensions? - the simulation will have 5 dimensions but that will obviously have no effect on our universe. I actually don't understand what you are aiming at.
Nobody knows whether the universe makes use of randomness or not. Quantum mechanics is essentially deterministic, the evolution of the wave function is described by a unitary operator, the Hamiltonian. Randomness only enters in the Copenhagen interpretation due to the Born rule and the collapse postulate. Note that the unitary evolution of the wave function and wave function collapse are not really compatible. What or who decides whether something constitutes a normal physical process with unitary evolution or whether it is a measurement that leads to the non-unitary collapse of the wave function? So this is definitely an open question whether the universe is deterministic or not.
I don't understand the rest of your comment, from the thing with the magnetic field around particles on, and that again doesn't sound right to me. Especially the uncertainty in the Heisenberg uncertainty relation on the one hand and the uncertainty in the description of a many particle system due to ignorance of the microscopic states on the other hand are of very different nature. The former one is fundamental, the later one is just due to ignorance.
I have seen those videos and they are kind of nice models for simple quantum mechanical systems, but one should really not try to stretch them to far, there is no way those system can accurately model quantum mechanics. They are classical systems and we know from the violation of Bell's inequality that quantum mechanics is not classical, so it is certain that those models will break down at some point.
Imagine a particle that can occupy any square of a chess board. The space the particle lives in is then a discrete two dimensional space with eight possible positions in each dimension. The wave function of this particle on the other hand lives in a 64 dimensional complex Hilbert space, one dimension for every possible state of the particle, one for every possible location of the particle. If the particle could additionally either have a charge of plus or minus one unit, then the state space would become 128 dimensional, 64 possible positions times 2 possible charges.
If there is only one particle, you can indeed visualize a probability wave on the chess board indicating the probability of finding the particle on one square or another, but that starts failing once you bring in a second particle. The state space for two particles on a chess board is 4096 dimensional, again one dimension for every possible state, one for every combination of the locations of the first and second particle.
And if the particles are entangled, you can no longer simply visualize the state [of one of the particles] with a probability wave on the chess board because the probability of finding one particle on a specific square may in general depend on the other particle. At best you can integrate over all possible states of the second particle but this will destroy a lot of information contained in the wave function.