  Eighty years ago Einstein demonstrated that a particular interpretation of
the reduction of wave function led to a paradox and that this paradox
disappeared if statistical interpretation of quantum mechanics was adopted.
According to the statistical interpretation a wave function describes only an
ensemble of identically prepared physical systems. Searching for an intuitive
explanation of long range correlations between outcomes of distant
measurements, performed on pairs of physical systems prepared in a spin singlet
state, John Bell analysed local realistic hidden variable models and proved
that correlations consistent with these models satisfy Bell inequalities which
are violated by some predictions of quantum mechanics. Several different local
models were constructed, various inequalities proven and shown to be violated
by experimental data. Some physicists concluded that Nature is definitely not
local. We strongly disagree with this conclusion and we critically analyze some
influential finite sample proofs of various inequalities and so called quantum
Randi challenges. We also show how one can win so called Bell game.The
violation of inequalities does not prove that local and causal explanation of
correlations is impossible. The violation of inequalities gives only a strong
argument against counterfactual definiteness and against a point of view
according to which experimental outcomes are produced in irreducible random
way. We also explain the meaning of sample homogeneity loophole and show how it
can invalidate statistical significance tests. We point out that this loophole
was not closed in several recent experiments testing local realism.
