Place a +/-1, however you like, in every single one of the 16 000 positions.

Give the columns names: A1, A2, B1, B2.

Independently of one another, and independently for each row, toss two fair coins.

Define two new columns S and T containing the outcomes of the coin tosses, encoded as follows: heads = 1, tails = 2.

Define two new columns A and B defined (rowwise) as follows: A = A1 if S = 1, otherwise A = A2; B = B1 if T = 1, otherwise B = B2.

Our spreadsheet now has eight columns, named: A1, A2, B1, B2, S, T, A, B.

Define four "correlations" as follows:

rho11 is the average of the product of A and B, over just those rows with S = 1 and T = 1.

rho12 is the average of the product of A and B, over just those rows with S = 1 and T = 2.

rho21 is the average of the product of A and B, over just those rows with S = 2 and T = 1.

rho22 is the average of the product of A and B, over just those rows with S = 2 and T = 2.

I claim that the probability that rho11 - rho12 - rho21 - rho22 is larger than 2.5, is smaller than 0.005 (5 pro mille, or half of one percent).

Proof: in the last displayed formula

(in the appendix) of http://arxiv.org/abs/1207.5103,

set N = 4000, delta = 0.035, and epsilon = 1/16.

Notation: is defined to be equal to the average of A times B over those rows for which S = 1 and T = 1. Similarly for the other three averages. The four averages are taken over four disjoint subsets of rows of the whole table.

- Code: Select all
`delta <- 0.035`

N <- 4000

epsilon <- 1/16

pb <- 4 * ( exp(-2 * N * delta^2) + exp(-2 * (0.25 - delta) * N * epsilon^2) )

pb

I want to know if Michel Fodje, Fred Diether, or anyone else, disagrees with this claim. If so, why?

If not, I am happy to go on from here and elaborate on the relevance of this little bit of elementary probability theory for the analysis of computer simulation programs of local hidden variable models.

After that, we could possibly go on to discuss the possible relevance of all that to the metaphysics of quantum mechanics and EPR-B type experiments.

PS. Notice that I am just getting a bound to a probability, using some convenient and simple probability inequalities (due to W. Hoeffding) for tails of the binomial and hypergeometric distributions, together with two applications of Boole's inequality. Replacing the Hoeffding bounds with exact computations for the binomial and the hypergeometric gives a sharper bound: the probability in question is actually smaller than 1 in 10 thousand. And this is before optimizing the choice of delta. Obviously, what the probability is exactly, depends on exactly how the spreadsheet is filled with +/- 1 's. Suffice it to say, the probability is so small that I'm willing to bet several hundred Euro's against the event's occurrence. If someone wants me to wager several thousand Euro's against the event occuring, I would ask for a bit larger N (e.g. 10 000 or 40 000).