I don't understand how our quantum Universe suddenly becomes classical. If quantum mechanics governs everything - and accordingly nothing is determined - then how do we have a past history that goes back in time, classically, to the big bang?
How do we know anything? We look at things and make observations.
How do we do that? You need a system with enough parts that reliably work together so it can react to the thing you want to observe without being confused by every single other thing happening in the universe. If you are distracted by shiny objects you won't observe what you are supposed to.
Without this ability you wouldn't see a classical world and you wouldn't be able to observe anything. And the emergence of the classical world is made possible by this. Becsuse if you can be insensitive to the rest of the universe then you can be insensitive to certain amounts of noise and error.
This means you can look at a large group of stuff and instead of observing it's exact state you observe some bulk average properties.
And when we talk about a classical world we don't mean that we see $10^25$ particles with particular locations and velocities. Our math might look like we think that bit it isn't actually what you see or experience.
You experience an interaction with your environment where you are sensitive to bulk properties of joint states of many systems.
And not everything happens just as often. And when you have many things it is harder for them to all act weird. It is like when you want to find the average height of people I'm some country. You could pick people at random, if you pick 1000 of them the odds that you picked the 1000 tallest people is unlikely. It could have happened. If you picked just one maybe the odds are poor you pocked on of the 1000 tallest but it becomes really harder if you picked 1000 at random that they were all so unusually tall.
So a lot of the possibilities just average out and you are set up to be sensitive only to the averages becsuse you have to be able to notice a bulk thing and not get distracted by every passing photon.
If you wanted to be sensitive to such things then an easy way is to shield something from all the distractions then have it interact with something that can be sensitive to that and then have something else interact with something sensitive to that and so on until you get something with bulk properties you can sensitive to.
But it is more than just that. There are two ways a configuration can evolve. One is based on be classical forces the system is subject to in that configuration. The other is a quantum effect that you can think of as a correct to that. And all the weirdest parts of that correction are due to interference effects which require that the two configuration evolve to be close two each other. Just one part of the configuration being different means no interference. When you interact with something you make the thing interacting become different. So you change the configuration of that part so the entire configuration changes.
This washes out the effects that we call the quantum effects. Or the other way around. The effects that aren't washed out by interactions are called the classical effects.
That said there are macroscopic quantum things. Bose-Einstein condensates, lasers, superconductors, magnets.
Unlike in classical physics, in quantum physics there exists such a thing as a correlated state, different degrees of freedom can get entangled. Perhaps paradoxically, this feature of quantum mechanics gives rise to classical mechanics. What happens is that the degrees of freedom of any system will get entangled with those of the environment. A statistical description of only the system by averaging out the effects of the environment is then possible and then yields the effective classical behavior of the system if the system is large enough.
Now, it can be shown that the very notion of a probability always arises from quantum mechanics. There is no known application of classical theory of probability such that the invoked probabilities don't ultimately arise from quantum effects.