I've been trying to wrap my head around the issues with Bell's Inequalities (while following the three related threads in this forum) and I think I finally have it figured out well enough to ask a question that's been bothering me. In particular, I'll start with http://drchinese.com/David/Bell_Theorem_Easy_Math.htm" [Broken] for complete background) My understanding of Bell's Inequalities for this case is that it states that any local realistic hidden variable theories must have a minimum probability of 0.3333 which is higher than the prediction of QM of 0.25. Finally, actual experiment verifies the QM prediction rather than the "realistic" prediction of 0.333. First, do I have this basically right? Second, consider the following potential scenario: 1) There are two separate local hidden variables for a photon, let's call them polarization angle and phase. 2) Detection of the photon through a measurement apparatus is a function of both polarization angle with respect to the measurement apparatus (an angle we'll call θ), and phase with respect to the measurement apparatus. 3) The polarization angle and the phase of the photon interacts with the quantum state of the measurement apparatus. Thus without knowledge of the quantum state of the measurement apparatus, it cannot be known what a given measurement will be. If you knew what the quantum state of the measurement apparatus would be at the time of measurement and you knew of the values of the hidden variables for the photon, then you could determine a priori whether or not the photon would be detected. Thus there are four total hidden variables for the system, two for the photon and two for the measurement apparatus (one with respect to the polarization angle and one with respect to the phase). 4) Fortunately, it is possible to understand the statistical characteristics of the system to model the probability of a measurement even when the quantum state of the measurement apparatus remains hidden. 5) Lets call the function which defines the probability of a photon passing through the measurement apparatus for the polarization hidden variable Alpha. Alpha is a function of θ. 6) Let's call the function that defines the statistical probability for the phase hidden variable Beta. Beta is also a function of θ. So the probability of a detection at a given angle is defined by: ρ = Alpha(θ)Beta(θ) The probability is doubly dependent on θ. Is there anything about this scenario which is not realistic? Not local? Could a theory with these characteristics not be a local hidden variable theory? Now what if: Alpha(θ) = Cos(θ) and Beta(θ) = Cos(θ) Do we not get a combined probability function which matches QM? Namely: ρ = Cos^2(θ) Returning to the 0.3333 probability of Dr. Chinese's example. It seems to me that the potential for a double-dependency on θ, makes the Bell's Inequality not work out. You'd have to multiply the minimum probability (i.e. 0.333 * 0.333 = 0.111). With this new Bell's Inequality of > 0.1111 we don't get a violation with a measurement of 0.25. What Bell seems to be missing (at least to my naive mind) is the idea that there could be hidden variables lurking in the measurement apparatus in addition to those in the photon or particle being measured in such a way as to make it possible that there are double dependencies on the same hidden variable. Clearly, lots of people have been studying Bell's Inequalities and their relation to QM for almost 50 years, so I must be missing something obvious. What am I missing?