These aren't descriptions of individual experiments, but rather conclusions that you got from many experiments. Let's say you do a million experiments, choosing

These aren't descriptions of individual experiments, but rather conclusions that you got from many experiments. Let's say you do a million experiments, choosing randomly which coins to ask Alice and Bob about. In every experiment where you asked them about the same numbered coin, their answers were always the same. Among experiments where you asked Alice about coin 1 and Bob about coin 2, their answers were different 5% of the time. Etc. Here's a quick proof why the results are impossible with hidden variables. Assume that before each experiment, Alice and Bob secretly agree on the answers they will give to you for each coin. At this stage they can conspire with each other as much as they want, flip actual coins, etc. Let's designate their pre-agreed answers as A1,A2,A3,B1,B2,B3 (each is either "heads" or "tails"). Then we know these facts: 1) P(A1 = B1) = 1, likewise for A2 and A3. So we can just use A1,A2,A3 and forget about B1,B2,B3. 2) P(A1 ≠ A2) = 0.05 3) P(A2 ≠ A3) = 0.05 4) P(A1 ≠ A3) = 0.2 But whenever A1 ≠ A3 is true, one of A1 ≠ A2 or A2 ≠ A3 must be true. So P(A1 ≠ A3) must be less than or equal to P(A1 ≠ A2) + P(A2 ≠ A3). However 0.2 is greater than 0.05 + 0.05, so we've reached a contradiction. That means Alice and Bob couldn't have agreed on all three coins in advance. In fact, if you do the whole thing in a lab (with photons), it will be physically impossible to measure all three coins, only two of them. That's exactly how quantum mechanics says "you're forbidden from knowing some variables at the same time".