It is commonly said that classical Newtonian mechanics is deterministic, while quantum mechanics is probabilistic. I used to believe this. I have changed my mind.
Classical mechanics is not deterministic in any practical sense. The inputs and outputs have uncertainties. Always. Even if the inputs were certain, the systems are frequently chaotic, making the output uncertain.
Quantum mechanics is often interpreted in terms of probabilities, but that is just one interpretation, and it is not required by any experiment. It is a convention. Physicists find the probabilistic interpretation to be convenient way of thinking about reality, but that is all. There are other conventions.
There are quantum experiments, like radioactive decay, that seem probabilistic. But that is just because the weak interaction is imprecisely modeled. It seems random in the same way that a coin toss seems random. There is nothing inherent in the math or the physics to make it truly random.
It is sometimes said that classical mechanics allows perfect prediction of the future, given perfect info about the past. This is more philosophy than science, because only a deity could have such perfect info. But the same perfect prediction could be made with quantum mechanics also.
Experimental papers in classical and quantum mechanics do not show any difference in their treatment of probabilities and statistics. They all report sigmas in the same way. Scientists all believe in doing repeatable experiments, so they rely on a belief that experiments determine results. This weak form of determinism seems essential. But strong determinism requires that all randomness be eliminated from the system. No one relies on strong determinism.
You can believe in strong determinism if you wish. I just don't think that physics gives any evidence for or against it, whether you use classical or quantum mechanics.
Here is an amusing story about how an American court recently had to rule on what is or is not "random". Of course it only had to do with some legal notion of fairness, and nothing to do with determinism.
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