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To check the correlation between Hidden Variable Theory and Quantum Mechanics, Bell calculated the expectation value

$<\sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)> = \int d^n V \rho(\vec V) \sigma_{e}(\vec a,\vec V) \sigma_{p}(\vec b,\vec V)$

Here I am assuming that "Alice" is measuring the spin of an electron e along $\vec a$ and "Bob" is measuring the spin of the positron p along $\vec b$. Then $\sigma_{e}(\vec a,\vec V)$ and $\sigma_{p}(\vec b,\vec V)$ are the resulting spin values ($\pm \frac{1}{2}$) of the electron and positron, respectively. The vector $\vec V$ is an n-dimensional vector containing the hidden variables and $\rho(\vec V)$ is a probability distribution for the hidden variables.

But does this not assume QM is probabilistic? I thought Einstein disagreed with the probabilistc nature of Quantum Mechanics (God does not throw dice).

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I'm no historian (so won't post an answer) but I believe Einstein accepted well enough that quantum mechanics makes probabilistic predictions. What he objected to was the view that these probabilities are "fundamental". In statistical mechanics you also have probabilities, but these don't reflect any underlying stochasticity in the microscopic dynamics. I think Einstein wanted something similar for quantum mechanics. – Nathaniel Aug 8 '12 at 14:58
I would also point out that it doesn't matter what Einstein thought about a particular issue. What matters is what the experimental result actually is. – Colin McFaul Aug 8 '12 at 15:01
Einstein was a coauthor on the EPR paradox paper, so I'm sure he was well aware of these sorts of issues. – Jerry Schirmer Aug 8 '12 at 15:02

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