Here we will use Quantum Mechanics to prove that the probability is 1/2. We will need only 3 facts here, all of which are a consequence of QM:
1. The polarization in each photon is decided only when we measure them. Let me explain, since this is the most crucial difference: Until we measure any of the photons, we do not have any clue as to what their polarization is. In fact, the polarization is not determined until we make the measurement (hence it is not decided at the birth but only when we measure them). Also, since (as before in part 2) the photons are born correlated (identical) once we find out the polarization in one of them the other one will instantly acquire (at that exact moment!) the same polarization.
This is the spooky action at a distance that Einstein objected to. The main difference with part 2 is that the polarization is not decided at birth but the measurement itself specifies it.
2. Collapse of the wavefunction. This is a standard result of QM (Einstein was ok with that). When you make a measurement and find the polarization along a certain direction, subsequent measurements will always yield the same result with probability 1. This is because once you measure something, there is no uncertainty thereafter as to what the value is. For example, if a detector is along direction 3 and it flashes green, then any other measurement will find the same photon to be along direction 3.
Here is the critical step though: In addition, from assumption 1, the other photon will instantly be forced to have polarization along direction 3 too, even though we measured the first photon! Hence not only the measured photon will always yield again the same result with probability 1, but the other photon (due to the correlation they had from birth) will always yield the same result with probability 1 also...
At this point you should go back and read the last paragraph again since here lies the difference between the classical common sense description and the Quantum Mechanical one. In QM the measurement of the first photon will INSTANTLY force BOTH photons to align in the same direction, which is the direction of the axis of the detector. In the classic view this does not happen since the direction of the photons is predetermined at birth and it is fixed since then. Keep also in mind that the direction of the axis of the detectors is decided right before the photons hits the detector, while the other photon is far far away.
For example, we decide to put the detector along direction 1 and then we get a green light from photon A, which means that photon B is instantly also aligned with along direction 1. Then in the next shot we decide to put the detector along direction 2, and we get a red light from photon A, which means that photon B is instantly aligned opposite of direction 2 (same with photon A).
3. If the polarization of a photon forms an angle θ with respect to the orientation detector, the probability that the detector will flash green is P=cos^2(θ/2). This is a Quantum Mechanical result that I will not prove here, but I will explain how it makes a lot of sense to be that way.
Here, the photon is pointing up and the detector is along some direction A that forms and angle θ with the polarization of the photon. First, observe that cos^2(θ/2) is always positive and between 0 and 1 (like any probability!). Second, for θ=0 the detector is exactly along the polarization of the photon and hence it will always flash green (P=1). Third, for θ=180 degrees the detector is aligned opposite of the polarization and hence it will always flash red (P=0). Fourth, for θ=90 degrees, the detector is placed horizontally while the polarization is vertical, in which case there is a 50-50 chance of getting a green light. This just means that if we do the experiment with this last alignment 100 times, about 50 of them we will get a green light and about 50 of them red (This is just probabilities however; we may as well get 100 times the same light in actuality).
Now let's calculate again the probability that both detectors will flash green under this new Quantum view. For simplification let's assume that the axes that the detectors can be aligned are 120 degrees apart.
Suppose we align detector A along direction 1. When the photon is detected, let's assume it generates a green light. According to what we said in assumption 2, we instantly know now that the other photon is also pointing along direction 1. So, what is the probability that the other detector will also flash green?
Well, the detector will either be pointing along direction 1, or direction 2, or direction 3. If it is pointing along direction 1 it will flash green with probability cos^2(0)=1. If it is pointing along direction 2 it will flash green with probability cos^2(-120)=1/4, and if it is pointing along direction 3 it will flash green with probability cos^2(120)=1/4. Hence the total probability is
P = (1 + 1/4 + 1/4)/3 = 1/2
In this case we see that if Quantum Mechanics is right, Bell's inequality is violated since it says that this probability should be always greater than 5/9 > 1/2 .
Why is this result different than the common-sense answer 5/9? Because we allow for instant communication between the 2 photons, so that when the first is measured the other is instantly aligned in the same direction. Note that if you only measured instead of 3 directions just 2 (say up and down) there is no violation of the inequality (in that case
Before making any measurement, we predict that the probability is 5/9. After doing the measurement, the probability drops to 1/2. The action of measuring affects the probabilities of the outcomes - that concept lies at the heart of this story and transcends Quantum Mechanics.
Concluding, all these are just predictions until they are tested by experiment. In 1983 Alain Aspect did the first thorough experiment and proved that Bell's inequality was violated by 5 standard deviations. However Bell himself had pointed out that ideally the detectors had to be separated far away, so that there is no way for a signal to travel between them. Otherwise maybe after you set the detector's direction there can be a way for the photons to have predetermined at birth polarization and still get the violation answers (for example, if the detectors somehow communicate with each other). To prove the instant action from a distance, you have to have the detectors far far away.
In 1998 at team at Innsbruck generated 2 photons, fed them into optical fibers and sent them 400m apart allowing for 1.6μs of time to decide for the orientation of the detectors after any light speed signal could travel between them. Then they sat down in their computers and waited to observe a posteriori the data from the detectors. Low and behold, each time they changed the orientation of the detectors they saw a change in the statistics - that somehow the photons where communicating instantly when measured. Or better, that they are a single entity and not 2 individual items. The experiment violated the inequality with unprecedented accuracy. The paper was published in Nature the next year and Alain Aspect reviewed it.
I will end the story by quoting Aspect who mostly quotes Feynman at the last paragraph of his review:
"It has not yet become obvious to me that there is no real problem... I have entertained myself always by squeezing the difficulty of Quantum Mechanics into a smaller and smaller place, so as to get more and more worried about that particular item. It seems almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than the another. But there you are - it is bigger..." Yes, it is bigger by 30 standard deviations.
Thanks to Gary Felder and his excellent article on Bell's theorem, located here.