Expectation Value in Bell's Theorem

We will demonstrate in this paper that Bell's theorem (Bell's inequality) does not really conflict with quantum mechanics, the controversy between them originates from the different definitions for the expectation value using the probability distribution in Bell's inequality and the expectation value in quantum mechanics. We can not use quantum mechanical expectation value measured in experiments to show the violation of Bell's inequality and then further deny the local hidden-variables theory. Considering the difference of their expectation values, a generalized Bell's inequality is presented, which is coincided with the prediction of quantum mechanics.