proof of Bell's theorem (quantum physics)

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Assume we have some source of electrons with entangled spins. This means that if A and B are a pair, if I choose to measure them both in the same axis, one will be up and other down.
 Now, A and B are sent off to two boxes very far from each other, which we will call Trump and Biden. Trump and Biden can each measure the spin of their electron in three different axes.
These are the 3 axes and we choose them so that they are 120 degrees apart from each other. When A arrives at Trump, Trump randomly chooses an axis to measure A's spin in, and Biden does the same thing with B. But they do it independently from each other, so Trump could measure spin around 2 and Biden around 3, for example. If the boxes measure their electron to be spinning, then they flash red, or if it spins down they flash white.
Someone notes if they're the same colour or not. This experiment has been repeated many times. The question is, what fraction of the time will their colour be different? The answer is actually half, let's say Trump measured around axis 1, and happened to get spin down. Let's see how likely Biden is to measure spin down as well.The reason I can just pick a particular case like this is because all the other cases will work in the same way, to the same result.
 Well, 1/3 of the time, Biden will choose axis 1 as well, since A and B are entangled, Biden's particle will definitely have spin up in that direction so the colours don't match
What about if Biden measured around axis 2? Because A is down in the first axis, so we just have to figure out how much of this spin is pointing in the 2nd direction. Actually as you can see, this vector is partly pointing in the negative direction of axis 2. What does this mean? It means that if you measure the spin in axis 2,it is either going to be up or down, but more likely to be down. So in this case A and B are more likely than not going to have the same spin and the colours are more likely to be the same. 

What about axis 3? This is really similar to 2. As you can say, this vector is pointing in the negative direction of axis 3,so most of the time B will be down if it is measured in axis 3, so again they A and B will most likely have the same colour. 

Let's review those 3 cases again: If Biden is on axis 1, Trump and Biden definitely won't have the same colour. If Biden is on axis 2 or 3 though, more often than not Trump and Biden will have the same colour. It works out so that overall there's an exactly 50 percent chance that Trump and Biden have the same colour and 50 percent chance they don't.

Local theory 
Firstly, let's think about what happens when Trump and Biden happen to pick the same axis. A and B have to be opposite spins. We know that if nothing goes faster than light than light, the only way this can happen is if the particles decided their spin while they were in contact. We don't know how they decide who gets which spin, but the awesome thing about this argument is that it doesn't matter. All that matters is that each particle A and B decide what spins there are in each of the 3 directions beforehand.

There are 8 possibilities: we will look at the case where A is spun up in axis 1 and 3 and spun down in 2 and the case where A is spun up in every axis. Why can we get this? Because we can get the other cases either by swapping which particle we call A and B, which doesn't matter, or swapping the names of the axes, which also doesn't matter.

Let's consider the case that A is all spin up. If so, what's the chance that Trump and Biden will have the same colour? No matter which axes get picked, A is always up, and B is down, so they're always different spins whenever they adopt this plan. What about the other plan? This is A's spin in each and underneath is B's.

Trump and Biden randomly pick axes, and there are a few possibilities for what happens: Trump could pick then Biden could pick as well, so they are different colours or he could pick 2, which means that they would have the same colour etc, and we just go through and tally up all 9 possibilities and find that they are different colours.

So that means that whether or not A and B picked a plan like this or like the previous type, they will always be more likely to have the same colour. But as we already know, this isn't the right result. So you see, local theories just don't have the right tools to explain I think that's because spin is a crazy phenomena that is nothing like the classical properties of matter that we are used to on the other hand any local theory is quite classical because it assume particle can't ffect each other instantly over distances which really limits. How weird it can be and in this case, it's just too classical to explain something like spin. So, that's Bell's theorem. 

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