Talk:A card game for Bell's theorem and its loopholes/Guy vandegrift

My 5/6 formula doesn't exactly match Anupam and Mermin 1987
5/6 = .8333333333 But 2(\sqrt 2 -1) .828427...

Trying to sort out the CHSH inequality
We begin with Bell's inequality (equation 16 from Bell's EPR paper 1964):
 * $$1+P(\vec b, \vec c) \ge |P(\vec a, \vec b)-P(\vec a, \vec c)|$$

Henceforth drop the vector symbols:
 * $$1+P(b c) \ge |P(ab)-P(ac)|$$

Since the LHS is positive, Bell's inequality becomes:
 * $$|LHS|\ge |RHS|$$

From Wikipedia:
 * $$P(ab)=\frac{N_{++} - N_{+-} - N_{-+} + N_{--} }{N_{total}}=P_{A=B}-P_{A\ne B}=2P_{A=B}-1$$

Hence:
 * $$P(ac)=\frac{N_{++} - N_{+-} - N_{-+} + N_{--} }{N_{total}}\quad =\quad P_{A=C}-P_{A\ne C}\quad =\quad 2P_{C=A}-1$$
 * $$P(bc)=\frac{N_{++} - N_{+-} - N_{-+} + N_{--} }{N_{total}}\quad =\quad P_{B=C}-P_{B\ne C}\quad =\quad 2P_{B=C}-1$$

Substituting:
 * $$LHS\quad =\quad 2P_{B=C}$$
 * $$RHS\quad =\quad 2P_{A=B}\;-\;\quad 2P_{A=C}$$

Something is wrong. It's late. I seem to have found something useful at http://www.scholarpedia.org/article/Bell%27s_theorem#Bell.27s_inequality_theorem