I noticed the question whether Bell's inequalities are based on a false premise, (https://physics.stackexchange.com/questions/219904/could-bells-theorem-be-based-on-a-false-premise) but I bring another argumentation of the question. Also I intend here to close an issue that I left open in the past, namely, whether quantum entanglements are a testimony of nonlocality. These two issues have a common ground, see below.
Bell based his inequalities on a probabilistic calculus, with positive probabilities. Is it plausible to believe that the standard formalism of QM, based on complex amplitudes, can be replaced by a formalism with real and positive numbers?
An example: the sum of two complex numbers of equal absolute value, but phases differing by π, mutually cancel out. But the sum of two probabilities never gives zero.
Proposed exercise: it is known that given a spin 1 particle one cannot construct a 101 function (see A. Peres' example with 33 directions in space - Journal of Physics A, vol. 24, Number 4, L175). Try to explain this impossibility in base of Bell's hidden variables governed by probabilities. Note that for single particles there is no question of locality or non-locality.
Then what Bell proved? Did he prove that QM is nonlocal? Or did he prove that the primary items in the quantum world are not probabilities?
This question is immediately related to another one formulated in the title, about nonlocality.