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I've been working through the paper *Bell nonlocality* by Brunner et al. after seeing it in user glS' answer here. Early on in the paper, the standard Bell experimental setup is defined:

Where $x, y \in \{0,1\}$, $a, b \in \{-1, 1\}$, and the two people (Alice & Bob) measure a shared quantum system generated by $S$ according to their indepedent inputs $x$ and $y$, outputting the results as $a$ and $b$.

The paper then has the following equation:

$P(ab|xy) \ne P(a|x)P(b|y)$

And claims the fact this is an *inequality* means the two sides are *not* statistically independent. It's been a long time since I took probability & statistics in university, so I'm interested in this equation, what it means, and why it is a test for statistical independence. Why is this equation used, and what is the intuitive meaning of each side? I have basic knowledge of conditional probability and Bayes' theorem.

@ Danylo Y found a major error in my answer, can you please remove your acceptance, so I can delete it – David Bar Moshe – 2020-12-01T14:42:14.903