Explaining entanglement, the Einstein–Podolsky–Rosen (EPR) paradox, and Bell's inequality

Based the experiment described by N. David Mermin in Physics Today April 1985 pages 38-47 

“This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does, not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.

While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.” Einstein, Podolsky, and Rosen 1935

The experiment consists of two detectors, A and B, and one source, C (see figure below). The source produces 2 identical "particles" one received by A and the other received by B. The detectors have 3 measurement settings and flash red (R) or green (G). If the detector settings are the same, then the detectors will flash the same colors. If the detector settings differ, they may or may not flash the same colors. The setting for each detector is selected at random and independent of the other detector. If the measurements from the detectors are determined by characteristics of the “particles” when they are created, they can be represented by a set of instructions for the detectors that describe the result, flashing red or green, for each detector setting [1,2,3]. The same instruction is sent to both detectors. The full set of 8 possible instructions is

[RRR] [RRG] [RGR] [RGG] [GRR] [GRG] [GGR] [GGG]

Clearly,

a)      if the detectors have the same settings, they flash the same colors

b)      if the instructions are [RRR] or [GGG], the detectors will flash the same colors

Note that if the instructions are NOT [RRR] or [GGG], then the remaining 6 instructions have two of the unequal settings that will produce the same colors (e.g., [RRG] will produce the same colors if the settings of the detectors A and B are either [1,2] or [2,1]) in addition to when the settings are the same ([1,1], [2,2],[3,3]). Therefore, there are 5 out of the 9 settings that will produce the same colors for these instructions.

When the detector settings are set at random and the instructions are set at random, then the probability that the detectors flash the same colors is the probability that the detectors flash the same color when the colors in the instructions are all the same (p = 1) times the probability that the colors in the instructions are all the same (2/8, two of the eight instructions) plus the probability that the detectors flash the same color when the colors in the instructions are NOT all the same (5/9) times the probability that the colors in the instructions are NOT all the same (6/8, the other 6 instructions).  

1 x 2/8 + 5/9 x 6/8 =  2/3

The probability will be different if not all the instructions are used or if the probability of each instruction occurring is different. However, the minimum probability that the same colors flash occurs when instructions sent are those that are NOT all the same color (i.e., NOT [RRR] nor [GGG]), which is 5/9 (Bell's inequality), and notably greater than 1/2.    

The problem occurs when this experiment is conducted in the real world (e.g., measuring spins of electrons or polarization of photons). The overall probability (not considering the detector setting) of the lights flashing the same color is 1/2. But as illustrated above, this is not possible if the same colors flash when the settings of the detectors are the same. The probability has to be 5/9 or higher. This is inconsistent with the characteristics of the “particle” measured by the detectors being set when the “particles” are created (i.e., instructions are used). This implies that when one “particle” is measured then the other “particle” knows the result and changes its own characteristic (instruction), the detector setting, or something else reducing the probability of getting the same color to make the overall probability 1/2. Spooky action at a distance.   

Obviously, it is a little more complex than this, but it illustrates that entanglement is not just the fact that the two "particles" were created at the same time and have the same characteristics and therefore if you know the characteristics of one you know the characteristics of the other.  

R code to simulate this experiment

Diagram from N. David Mermin in Physics Today April 1985 pages 38-47