In deriving his inequalities, Bell starts his argument by stating the following: a)- that according to QM, if Alice measures +1 then Bob must measure -1. b)- if Alice and Bob are remote from each other such that Alices measurement does not influce Bob's measurement, then the results must be predetermined c)- Since the QM does not predict individual results, it implies that the QM wavefunction is not complete and can be supplemented with "hidden variables" to obtain a more complete state. He then goes on to calculate what might be expected if such hidden variables are introduced leading to his inequalities. From the above and what I understand so far, the following argument results 1) Bell's ansatz (equation 2 in his paper) correctly represent those local-causal hidden variables 2). Bell's ansatz necessarily lead to Bell's inequalities 3). Experiments violate Bell's inequalities Conclusion: Therefore the real physical situation of the experiments is not Locally causal. There is no doubt in my mind that statement (2) has been proven mathematically since I do not know of any mathematical errors in Bells derivation. Similarly, there is very little doubt in my mind that experiments have effectively demonstrated that Bell's inequalities are violated. I say little doubt because no loophole-free experiments have yet been performed but for the sake of this discussion we can assume that loopholes do not matter. Now the issue I have difficulty understanding is statement (1) and it is fair to say if statement (1) fails, the argument fails with it. Bell represents local reality by stating the joint probability of the outcome at A and B by as the product of the individual probabilities at each station, essentially the following P(AB|H) = P(A|H)P(B|H) However, in probability theory, P(AB|H) = P(A|H)P(B|AH) according to the chain rule, and in the case in which knowledge of A gives us no information about B, P(B|AH) = P(B|H) and we can then reduce the the equation P(AB|H) = P(A|H)P(B|H). In the situation Bell is trying to model, he says if Alice gets +1 then Bob MUST get -1. Therefore if we know that Alice already got +1, we therefore now know that Bob MUST have gotten -1. In other words, knowledge of A changes the hypothesis space for calculating the probability of B and P(B|AH) is not equal to P(B|H). So it appears to me that Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails. What am I missing?