For any finite number of parts, measurements, and outcomes in a Bell scenario, we estimate the probability of random N-qudit pure states to substantially violate any Bell inequality with uniformly bounded coefficients. We prove that under some conditions on the local dimension, the probability to find any significant amount of violation goes to zero exponentially fast as the number of parts goes to infinity. In addition, we also prove that if the number of parts is at least 3, this probability also goes to zero as the local Hilbert space dimension goes to infinity.
Notice that there is no loss of generality in supposing that every box has access to the same number m of inputs as well as to the same number v of outputs. This abstraction could be realized considering extra buttons that are never pushed and additional light bulbs that are never turned on.
We say that two Bell inequalities are equivalent if a behavior violates one of them if and only if it also violates the other.
An apparently very drastic behavior like is already enough for our purposes.