The foundation of a Bell Inequality is the definition of a coincidence probability (or wave function) for correlated events. The version of this expression used by Bell is the following:
$$\begin{array}{}{\displaystyle P(a,b)=\int d\lambda \rho (\lambda )A(a,\lambda )B(b,\lambda ).}\end{array}$$(1)

Consider now the difference of two such coincident probabilities:
$$\begin{array}{}P(a,b)-P(a,{b}^{\prime})=\\ {\displaystyle \int d\lambda \rho (\lambda )[A(a,\lambda )B(b,\lambda )-A(a,\lambda )B({b}^{\prime},\lambda )].}\end{array}$$(2)

Here, zero in the form:

$$\begin{array}{}{\displaystyle 0=\int d\lambda \rho (\lambda )[A(a,\lambda )B(b,\lambda )A({a}^{\prime},\lambda )B({b}^{\prime},\lambda )}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-A(a,\lambda )B({b}^{\prime},\lambda )A({a}^{\prime},\lambda )B(b,\lambda )],\end{array}$$(3)

is added to Eq. (2) above, to get:
$$\begin{array}{c}{\displaystyle P(a,b)-P(a,{b}^{\prime})=\int d\lambda \rho (\lambda )[A(a,\lambda )B(b,\lambda )}\\ \left\{1+A({a}^{\prime},\lambda )B({b}^{\prime},\lambda )\right\}\\ {\displaystyle -\int d\lambda \rho (\lambda )[A(a,\lambda )B({b}^{\prime},\lambda )\left\{1+A({a}^{\prime},\lambda )B(b,\lambda )\right\}.}\end{array}$$(4)

Then using |*P*| ≤ 1, one can write:
$$\begin{array}{c}{\displaystyle \left|P(a,b)-P(a,{b}^{\prime})\right|\le \left|\int d\lambda \rho (\lambda )\left[\left\{1+A({a}^{\prime},\lambda )B({b}^{\prime},\lambda )\right\}\right]\right|}\\ {\displaystyle -\left|\int d\lambda \rho (\lambda )\left[\left\{1+A({a}^{\prime},\lambda )B(b,\lambda )\right\}\right]\right|,}\end{array}$$(5)

or
$$\begin{array}{}|P(a,b)-P(a,{b}^{\prime})|\le 2-P({a}^{\prime},{b}^{\prime})+P({a}^{\prime},b),\end{array}$$(6)

i.e.,
$$\begin{array}{}|P(a,b)-P(a,{b}^{\prime})|+|P({a}^{\prime},{b}^{\prime})+P({a}^{\prime},b)|\le 2,\end{array}$$(7)

which is one form of the celebrated “Bell inequalities.”

Now let us repeat these manipulations, however, starting from the explicit form of Eq. (1), namely:
$$\begin{array}{}{\displaystyle P(a,b)=\int d\lambda \rho (\lambda )A(a|\lambda )B(b|a,\lambda ).}\end{array}$$(8)

Again, the difference of two correlated probabilities:
$$\begin{array}{}{\displaystyle P(a,b)-P(a,{b}^{\prime})=\int d\lambda \rho (\lambda )[A(a|\lambda ,b)B(b,\lambda )-A(a|\lambda )B({b}^{\prime}|a,\lambda )]}\end{array}$$(9)

and now a more explicit form for what should be an expression equaling zero:
$$\begin{array}{}{\displaystyle 0=\int d\lambda \rho (\lambda )[A(a|\lambda )B(b|a,\lambda )A({a}^{\prime}|\lambda )B({b}^{\prime}|{a}^{\prime},\lambda )}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-A(a,\lambda )B({b}^{\prime}|{a}^{\prime},\lambda )A({a}^{\prime}|\lambda )B(b|a,\lambda ),\end{array}$$(10)

giving:
$$\begin{array}{c}{\displaystyle P(a,b)-P(a,{b}^{\prime})=\int d\lambda \rho (\lambda )[A(a|\lambda )B(b|a,\lambda )}\\ \{1+A({a}^{\prime}|\lambda )B({b}^{\prime}|{a}^{\prime},\lambda )\}\\ {\displaystyle -\int d\lambda \rho (\lambda )[A(a|\lambda )B({b}^{\prime}|a,\lambda )\{1+A({a}^{\prime},\lambda )B(b|a,\lambda )\};}\end{array}$$(11)

using, as above, |*P*| ≤ 1, gives:
$$\begin{array}{c}{\displaystyle \left|P(a,b)-P(a,{b}^{\prime})\right|\le \left|\int d\lambda \rho (\lambda )[\{1+A({a}^{\prime}|\lambda )B({b}^{\prime}|{a}^{\prime},\lambda )\}]\right|}\\ {\displaystyle -\left|\int d\lambda \rho (\lambda )[\{1+A({a}^{\prime}|\lambda )B(b|a,\lambda )\}]\right|.}\end{array}$$(12)

So, here we arrive at the crux of the matter insofar as Eq. (7) cannot follow because the term *∫ dλρ*(*λ*)*A*(*a*′|*λ*)*B*(*b*|*a*,*λ*) does not equal *P* (*a*′, *b*). In fact it is undefined, or nonsense, as it is the product of the absolute probability *A*(*a*′|*λ*) times the conditional probability *B*(*b*|*a*, *λ*), which is not conditioned on *a*′, but on *a*, thereby rendering the product meaningless.

The final, general conclusion is that this Bell inequality is invalid; deductions from it are void^{4}

Exceptionally, of course, when the two detections are uncorrelated, then *B*(*b*|*a*, *λ*) = *B*(*b*|*a*′, *λ*), and Bell’s result is valid.

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