Consider flipping 3 coins. The sides of the coins are labeled: (A,a), (B,b) and (C,c). There are 8 possible results of flipping the 3 coins:
and the probability of getting any specific one (p(A,B,C) , p(a,b,c), etc,) is 1/8. The letter p stands for probability or chance.
If we dont care what the result of flipping coin {Bb} is, the probability of getting A,C on the other 2 coins is:
Similarly, the probability of getting the B,c combination while ignoring coin {Aa} is:
Therefore the sum of the probabilities is:
Inserting:
We are left with:
And since all of these probabilities are positive numbes,
Finally since the probability p(B,c) is identical to the probability of p(B,C), we arrive at the Bell inequality.
The Bell Inequality for a locally realistic system describing the results of a measurement of the number density of the X and Y components of spin of the particle streams.
In this equation, " n[X+,Y+]" signifies the number of particles with spin up along both the X and Y axes.
In a Quantum Mechanical system, the measurments are always correlated and measuring the spin along 1 axis totally randomizes the spin along the other 2 axes.