@imbAF When the RHS is zero or one, the square is equal to the square root.

@imbAF you misunderstand what is happening. If you are finding a basis, it is convenient to have orthonormal bases, so we will want to impose the condition that basis vectors be of unit length. That is one application of that integral. Each basis vector in QM is a particular quantum state, and each of them needs to be normalisable to unity. A different notion is that a general quantum state must also be normalisable; it is this one that, if you did not consider enough basis vectors, then

it can fail to be normalised.

It is this, that "less than the total amount of possible states, then it can happen that the probability is less than one" could apply.

@bolbteppa XD

what is a good resource to learn about bell's and bell-like inequalities and this whole sphere of things related to EPR paradox?

it would be nice if such things were spoken of in math (not physics-ey) probability theoretic terms

perhaps Semiclassical knows...

There are good textbooks on quantum information that talk about the details of Bell's inequalities.