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in Bell's theorem we consider the measurement of correlation [itex]A\otimes B[/itex] and [itex]A\otimes B'[/itex]

Then the theorem states that the "result" for A in both quantities is the same. But does QM not only gives probabilities of the result, hence the result could be in principle different for the measurement with the A operator in those 2 correlation, so that no factorization is possible for the result of measurement ?