See studies by K. M. Baker (1982), L. Rosenfield (1984), and E. Rothschild (2001).
The assumptions of the theorem are that a group wishes to reach a decision by majority vote. One of the two outcomes of the vote is correct, and each voter has an independent probability p of voting for the correct decision. The theorem asks how many voters we should include in the group. The result depends on whether p is greater than or less than 1/2:
To avoid the need for a tie-breaking rule, we assume n is odd. Essentially the same argument works for even n if ties are broken by fair coin-flips.
Now suppose we start with n voters, and let m of these voters vote correctly.
Consider what happens when we add two more voters (to keep the total number odd). The majority vote changes in only two cases:
The rest of the time, either the new votes cancel out, only increase the gap, or don't make enough of a difference. So we only care what happens when a single vote (among the first n) separates a correct from an incorrect majority.
Restricting our attention to this case, we can imagine that the first n-1 votes cancel out and that the deciding vote is cast by the n-th voter. In this case the probability of getting a correct majority is just p. Now suppose we send in the two extra voters. The probability that they change an incorrect majority to a correct majority is (1-p)p2, while the probability that they change a correct majority to an incorrect majority is p(1-p)(1-p). The first of these probabilities is greater than the second if and only if p > 1/2, proving the theorem.
The theorem is correct within its assumptions, but its assumptions do not necessarily hold in practice. Some objections that are commonly raised:
Nonetheless, Condorcet's jury theorem provides a theoretical basis for democracy, even if somewhat idealized, and as such continues to be studied by political scientists.