theorem.R

setwd("~/Desktop/Fred")
set.seed(112233)
rm(list = ls())

N <- 1e03
## Input your own N x 4 matrix containing elements -1 and +1 here
## ... or create a random one, anyway you like
spreadsheet <- matrix(sample(c(-1, +1), N * 4, replace = TRUE), N, 4)

M <- 1e06
results <- rep(0, M)
for(i in 1:M) {
     Achoices <- sample(c(1, 2), N, replace = TRUE)
     Bchoices <- sample(c(1, 2), N, replace = TRUE)
     rows11 <- (Achoices == 1) & (Bchoices ==1)
     corr11 <- mean(spreadsheet[rows11, 1]*spreadsheet[rows11, 3])
     rows12 <- (Achoices == 1) & (Bchoices == 2)
     corr12 <- mean(spreadsheet[rows12, 1]*spreadsheet[rows12, 4])
     rows21 <- (Achoices == 2) & (Bchoices == 1)
     corr21 <- mean(spreadsheet[rows21, 2]*spreadsheet[rows21, 3])
     rows22 <- (Achoices == 2) & (Bchoices == 2)
     corr22 <- mean(spreadsheet[rows22, 2]*spreadsheet[rows22, 4])
results[i] <- corr11 + corr12 + corr21 - corr22
}

library(MASS)
truehist(results)

sum(results > 2)

## [1] 0