Imagine I have a regular balanced dice and i roll it once. It is assumed that the probability of any number (1-6) is 1/6.
However, isn't this just an illusion we are feeding ourselves for our lack of knowledge regarding the environment and conditions under which the dice is thrown? If we could control the flick of the wrist throwing the dice, know all the characteristics of the surface on which the dice hits, control all the other variables involved, and we replicated the exact same circumstances, wouldn't we always get the same result?
My point is, can't we define Probabilities simply as the lack of knowledge/technology we have of a certain event, and that when (and if) we achieve the capability of controlling all variables in nature that this branch of Mathematics will be put aside as obsolete? That this field only exist as long as our knowledge is limited, and as such can't be considered a time-invariant "truth", as algebra or calculus is? That probabilities don't really exist in Nature, where there is cause-effect and not true randomness, and therefore is a simplification of reality that doesn't exist outside the human mind?
If you buy a high-quality die, like they use in Vegas, then saying "each side has a 1/6 probability of coming up" is not a statement of our lack of information about the die; on the contrary it is a statement of our knowledge of how symmetric and fair the die is. A 10-cent die you buy in the grocery store might not be fair, but the really good ones are fair to anyone's ability to measure.
"If we could control ... wouldn't we always get the same result?" The answer to that would seem to be yes, but the reality is that we simply cannot achieve that level of control. Consider a die roll at a craps table in Vegas: the die travels a couple meters bounces a few times on a felt surface, and then comes to rest. A tiny change in the angle of the die when it hits (eg. perfectly flat vs. slightly on the leading edge vs. slightly on the trailing edge) will yield a large change in the direction/spin of the die after it bounces. Compound that by several bounces, and you get to the point where deviation by the width of an atom will change the result.
Clearly, we're way beyond what can be achieved with flesh and bone. Even robots would fail to produce repeatable results due to variations in the airflow. Maybe in a vacuum? Maybe. For this example.
If we wanted something more complicated, like "throw a die down a 50-foot hallway such that it bounces at least ten times on the carpet", then the accuracy required will pass the point where Quantum-Mechanical fluctuations would take over. The result would be truly random.
Your comment "That probabilities don't really exist in Nature, where there is cause-effect and not true randomness," also fails to take into account QM. Every time a radioactive particle decays, the timing of the decay and the directions taken by the decay products are both entirely random. That information simply does not exist in this universe before the decay happens. All those random events add up: If you had access to all the information that exists right now, you could not tell us where it will be raining one year from today.