Bell's inequalities are mathematical inequalities that demonstrate the limitations of classical physics in explaining certain quantum phenomena. They were derived by physicist John Bell in 1964 to test the predictions of quantum mechanics against those of local hidden variable theories, which suggest that particles have predetermined properties. When experiments are conducted to measure entangled particles, the results often violate Bell's inequalities, supporting the predictions of quantum mechanics. This violation indicates that the behavior of entangled particles cannot be explained by classical physics, suggesting that particles can be interconnected in ways that defy traditional notions of locality and realism.