Abstract: The famous Condorcet jury theorem provides the theoretical foundation for voting theory. Based on this theorem, it is only limited that all voters must have the same probability in their preference for the alternatives, which is impossible to happen in real voting, so in fact it only gives a special case of the general situation. This paper substantively extends the Condorcet jury theorem, and establishes a proper jury theorem for the relationship between the probability of the voting group using the majority preference rule to make a strict preference choice for the alternatives when each voter has its own different preference probabilities for the alternatives. At the same time, some important properties of the group strict preference probability determined by the established theorem are given. Finally, it is also proved that when the number of voters increases infinitely, the group strict preference probability determined by this theorem will tend to its maximum limit value of 1.

Key words: group decision making, voting theory, probability, strict preference, majority preference rule, Condorcet jury theorem, proper jury theorem