Abstract: The majority preference rule is one of the most important and widely used rule in solving group optimization problems. However, in the process of using this rule to find the optimal solution for a given group optimization problem, sometimes there will be a "paradox of voting" phenomenon of group preference cyclic ordering. Therefore, the probability calculations of "paradox of voting" has become a fundamental research topic in group optimization. In this paper, the concepts of "sorting profile of the paradox of voting" and "selection profile of the paradox of voting" of a group on scheme set are introduced. With the help of them, a fundamental theorem for the probability calculations of the "paradox of voting" is established under the general condition that each individual in a group has different probability distribution for the preference ranking of all schemes, and when a group chooses the best scheme for a problem. Thus, the fundamental problem that has not been solved thoroughly for a long time in the study for the probability calculations of the "paradox of voting" is solved.

Key words: group optimization, majority preference rule, paradox of voting, probability