Suppose we put two people in a room and ask them to toss coins. We expect them to get half heads and half tails. We also expect that when we look at both their coins together they will get opposites, head-and-tail, as often as they get the same, head-and-head or tail-and-tail. If the two coins come up randomly, but they're always either both heads or both tails, we never see head-and-tail, that's a correlation, and we might look for an explanation of that, even though we wouldn't have to look for any explanation of the way each coin came up. The Reichenbach common cause principle expresses precisely this requirement.
At this point I'll point out (coarsely) the difference between an empiricist and a realist (and noting that there are instrumentalists, naive or structural realists, etc.). An empiricist looks at the correlations and, finding that those correlations are always there, figures out a way to use them, without much caring what explanation or causes there might be for the correlations. Some people are temperamentally quite happy not to care how or why, they're happy to use it. A realist cares about how and why it happens. The proportions of Physicists who think in one way or the other, and who think in both ways depending on the situation they face in their research, goes in cycles. Of course an empiricist is still curious, it's just that their curiosity runs in different pathways. It's important that there are both ways of thinking and that there's an interplay between them.
Suppose now that we do a different experiment. A psychologist gives two people two buttons each, and asks them to press either one or the other, and then looks to see both what the patterns of the individual button presses are and what correlations there are between them. The psychologist carries out lots of tests, for a whole day at a time, and pays the subjects more if they maintain an average of 100 button presses per minute. That's 50000 events per day, say, in contrast to that many events per second, or far more still, in a Physics experiment. Nonetheless, lots of data. In each test, different feedback is given about what the other person has pressed. In one test, we give audible feedback immediately after the button is pressed, in another half a second after the button is pressed, in another we give visual feedback, perhaps of different colors and perhaps in different positions. We try lots of different alternatives, and discover that there are different correlations in each case. People are not very good at being random, but after an hour of something as boring as this someone who is taking the project seriously would get into a vibe with the other person. Finally, we could also introduce a third person whose job it is to try to get the two other people to be synchronized. They can control the feedback lights however they like. Imagine any configuration and different kinds of feedback you like and speculate on what the results would be.
The physics does just this. There are two detectors and a central source. If there's no source, the detectors "trigger" every now and then, which is called the "dark rate". When we introduce different central sources, we see different statistics for both the rate at which we see events and for the correlations between the events. We also see different rates and correlations if we introduce different detectors. Without knowing why there are different rates and correlations, we can see that there are different rates and correlations. We can use those different rates and correlations.
Where quantum theory differs from classical probability is that we describe the results we get for a single type of source for all different types of detectors, and, separately, the results for a single type of detector for all different types of sources, whereas for classical probability we describe the results we get for each single type of source and detector.
There are so many ramifications that I'm uncertain where to stop. For example, a "detector" can have a compound structure. It can be two detectors, your A and B, either close together or far apart, perhaps set to trigger as a pair only when they trigger at the same time. The electronics in real experiments can be byzantine, although it's the job of experimentalists to be able to tell other experimentalists what their electronics does so they can do the same thing, ideally in a different way.
Notoriously, there are certain kinds of explanations for the statistics we observe in Bell-inequality violating experiments that are not possible. They can't be explained by the idea that the central source sends what might be called classical particles to the two detectors. People have proposed other kinds of causal models, but Physicists have for the most part not found those models useful.
Again, this is embarrassingly incomplete, but the nature of your Question and of your response to John Schanck's more-or-less passable account in terms of the nitty gritty of quantum mechanics suggested to me that this type of narrative might be more useful for you.