Condorcet's jury theorem

A result showing that, under specified conditions, majority voting becomes more likely to reach the correct decision as the number of independent, individually competent voters increases.

Binomial distribution: A probability distribution for the number of successes in a fixed number of independent trials with the same success probability. It models how many voters choose correctly when each votes independently with equal accuracy.
Jury: A group of people assembled to deliberate or make a decision, especially in a legal proceeding. The theorem’s name comes from its model of jurors deciding a question with a correct answer.
Jury theorem paradox: The tension between Condorcet’s jury theorem and cases where additional voters can reduce collective accuracy. It highlights that larger groups need not improve decisions once the theorem’s assumptions fail.
Marquis de Condorcet: A French Enlightenment philosopher, mathematician, and political theorist who lived from 1743 to 1794. He formulated the jury result while studying how collective decisions can track truth.
Law of large numbers: A theorem stating that sample averages converge toward their expected value as the number of observations grows. It explains why the correct-choice share tends to exceed one half when individual accuracy does.
Probability: A mathematical framework for quantifying uncertainty about events. The theorem compares the probabilities of correct and incorrect collective decisions.
Condorcet's paradox: A possibility in which majority preferences cycle, even when each individual’s preferences are transitive. It shows that aggregating judgments across multiple alternatives can be incoherent despite accurate voters.
Essay on the Application of Analysis to the Probability of Majority Decisions: Condorcet’s 1785 work applying probability theory to voting, juries, and collective judgment. This is the work in which the jury theorem appeared.
Chernoff bound: An inequality that bounds the probability that a sum of independent random variables deviates substantially from its expectation. It quantifies how quickly the chance of a mistaken majority can shrink.
Statistical independence: A condition in which knowledge of one variable gives no information about another. It is the key restriction preventing shared influences from making votes move together.

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