Bells inequality is mainly illustrated through two examples: the polarization of light, and the the spin of an electron. We will be focusing primarily on the electronic spin here. So lets imagine 100 singlet pairs of electrons, travelling somehow in two opposite directions free of any interactions. By virtue of their singlet pairing, their spins are already counter-entangled, where if the spin of one electron of a pair is clockwise, then that of the other becomes counter-clockwise and vice versa.

Now, let the 100 electrons with spin up/clockwise travel immensely far from the other 100 electrons (with spin down/counter-clockwise), such that two mutually entangled electrons cannot exchange information, locally, in a reasonable amount of time, though they are quantumly entangled. Lets try to assign a real hidden variable, , to the electrons, which defines the bias of the electrons towards getting measured along horizontal or vertical axis. Let the electrons with spin up be with Alice and those with spin down be with Bob (a naming convention). Alice and Bob, now perform reasonable measurements on the two spin states of the electrons using S-G magnets inclined inversely with each-other. Now, we follow three different cases:

The S-G magnets are inclined completely inversely with each-other; this means that two entangled-electrons would give the same outcome (as their spins are aligned inversely with each-other). Hence 100% of the electrons shall give the same outcome. Hence all 200 (or 100 pairs) electrons give the same outcome. Now, lets orient one SG magnet slightly away from the other (= , say); hence the electrons with a strong bias towards horizontal shall give different outcomes. Let the number of electrons giving different outcomes in this case be N. (Even if we orient the other SG magnet by =, keeping this SG magnet as =0, the number of electrons, N, giving different outcomes would be the same, as the relative angle between the SG magnets matters and not the absolute angle, hence N=N).

Now, lets orient both of the SG magnets by =, but in the opposite directions; this means, by convention, = for magnet-1, and = for magnet-2. Hence, the relative angle between the two SG magnets would be = = 2. Therefore, corresponding to our hidden variable, , the number of electrons giving different outcomes, N should NOT be greater than (N+N =) 2N. But, this number does not match with the predictions of quantum mechanics. According to the calculations of quantum mechanics, the number N (=N), after proper approximations, should be equal to /4. And N=(2)/4=4/4 =4N. But this contradicts our previous result that N 2N. Hence this contradiction prohibits the existence of a real and local hidden variable in quantum mechanics. This, in a nutshell, is Bells Theorem. In mathematical form, this is represented in the form of an inequality, in the original paper, as: Realism: The idea that nature exists independently of whether somebody is witnessing it or not.