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 Quote by IsometricPion I don't think this is quite correct. Rather, if X correlates with Bertlmann wearing a pink sock then P1(pink|X)=(P1(pink)/P(X))>0.5. Instead, $\int{}P_1(pink|X)P(X)dX=P_1(pink)=0.5$ (obviously if X is a causal factor it must correlate with P1(pink)). [..]).
Thanks for that clarification! I had not looked at it that way. However, X is like EPR's hidden function: Bertlmann's unknown and unpredictable mood determines what socks he will wear. X stands for the physical model, which is here an invisible random function (indeed, it happens in his head) that delivers one of {pink, not pink}. Obviously the chance to observe a Bertlmann pair of socks on Bertlmann's feet is simply 1. Then we must have, for the case that half of the time a pink sock is observed on the left foot:
P1(pink|X)=P1(pink)/P(X) = P1(pink)/1 =0.5

It's exactly the same as for a fair coin: P(head | fair coin) = 0.5.

I can imagine that someone would like to split the probability estimation up into unknown "knowns": then we can separate it into the cases that Bertlmann decides to put a pink sock on his left leg, and the cases that he decides to put another colour on his left leg. However, what we are interested in the result over many times, and then we are necessarily back at where we were here above. Thus, I don't see any use for that.