Say I have entangled particles A and B. Assuming we are talking about photons, we have 50/% chance of particle A being spin up.
Is it possible to affect the probability of particle A to be spin up greater or less than 50% of the time?
Can I do some kind of rotation/operation on the particle A by itself to affect it's probability to be something like 51/49 chance of being spin up?
The simple answer to your question is Yes. You can create 51/49 polarization, or 52/48 polarization, or whatever mixture you might care to produce.
This can easily be seen in production of Type I entangled photon pairs. For this type, 2 BBo crystals are used. Each thin crystal generates VV pairs that are not polarization entangled. They are placed in line with each other, but one is rotated 90 degrees relative to the other so it is producing HH pairs. When down conversion occurs, it can have occurred from either crystal. You won’t know which though. This places the output in a 50/50 superposition of VV>+HH> which is fully entangled. See figure 1.a. in either of the following papers by Kwiat et al:
https://arxiv.org/pdf/quant-ph/9810003
https://arxiv.org/pdf/1001.4182
So far so good. Now, if the 2 crystals are oriented say 75 degrees relative to each other (instead of 90 degrees), they will produce pairs that are distinctly more VV> than HH>. They will be polarization entangled, but not maximally so. By suitable rotation, you can achieve the desired mixture.
However: as you dial the mixture away from 50/50, the entanglement quality changes accordingly.
Edited to add/clarify: Alice cannot do anything to A to change the marginal probability that Bob sees for his B photons. That is strictly a function of how the initial pairs are created.
I think two cases have to be distinguished. Assume that the entangled particles $A$ and $B$ are sent to Alice and Bob, respectively. We assume that Alice and Bob are far away, i.e., the time for light to travel from Alice to Bob or vice versa is longer than the time our experiments take.
Can Alice do anything to change the marginal probabilities of Bob? E.g., change the frequencies of Bob from $50/50$ to $45/56$. The answer to this is no. If Alice makes a "spin measurement" in the same direction as Bob, and she measures down than with $100?%$ certainty Bob will measure up. So, the conditional probability is affected by Alice. But Alice has no control over the outcome of her measurement. Therefore, in the long run, Bob's probability is still $50/50$.
Can Bob do something to change the probability for spin measurements on particle $B$? The answer is yes. One may use, for example, a liquid crystal to change the polarization of the particle. This may rotate the state from a superposition to an eigenstate of the spin operator. This is more or less the way LCDs work. By changing the polarization, they change the probability of light going through the second polarization.
On the EPR-argument: The point of the EPR argument was that Quantum mechanics violates locality. The measurement done by Alice instantaneously changes the state of Bob's particle from a superposition to either spin up or spin down with $50/50$ probability. This is not the same as if you put a pair of shoes in two boxes and send one to Alice and one to Bob (this was also explained by John Bell). The difference is that according to standard quantum mechanics, there exists no fact about spin up or down prior to the "measurement." Einstein's hope was to replace quantum mechanics with a local theory in which the outcome of the spin measurement is already decided (as for the shoes in the boxes) so one can get rid of the spooky action at a distance.
John Bell showed that this is not possible. I.e., even if you add "hidden-variables" to the description, you cannot get a local theory.
Forget about the term "entangled" and just focus on correlated particles. For electrons, the terms "spin up" or "spin down" are common. When it comes to photons, though, the main parameter for correlation is polarization. But perfectly correlated photons need more than polarization. There are other variables, including frequency, trajectory, and timing. All of these factors ensure that later measurements line up accurately.
As for your question about the 51/49% split, it doesn’t really apply here because, in electron experiments, measurements only register as spin up or spin down. Before the measurement the electron could have been at some arbitrary angle, but if it registers as "up," it’s because its angle skewed more towards "up." Same goes for down if the spin already leaned toward the down. Keep in mind that electron experiments like the Stern-Gerlach experiment create a magnetic field that pulls the electrons up or down depending on which way the electrons orientation was already skewed.
You would want these correlations—or anti-correlations—to be as close to perfect as possible. That’s the whole point of EPR: perfect correlation. Photon pairs need to have all variable perfectly correlated and if their polarizations are off by 1 degree as you suggest, then the whole experiment is faulty and later measurement would mean nothing. The whole point of PERFECTLY correlating them is to have something to later measure against, or to compare to.
