If each juror is even slightly better than a coin flip at reaching the truth, then the MAJORITY of a large jury is almost certainly right — and the bigger the jury, the surer. It’s the mathematical case for democracy and trial by jury: competence, pooled, converges on truth. But flip the coin the other way — jurors worse than chance — and the crowd converges on being WRONG. Slide the jury size.
Condorcet’s Jury Theorem (1785): if each of N independent voters is correct with probability p and decisions are by majority, then for p > 0.5 the probability the MAJORITY is correct exceeds p and rises toward 1 as N grows — competence, aggregated, amplifies. (For p < 0.5 it runs the other way, converging on error — the tyranny of the incompetent majority.) With p = 0.6, three jurors are right ~65% of the time but fifteen ~79%, climbing toward certainty. It is a foundational argument for juries and for democracy, with the honest caveat that it needs INDEPENDENCE and competence. A fail-loud self-check throws unless majority-correct probability rises with N when p > 0.5. ◆ real social-choice theory, node-verified.
Independent, identically-competent jurors (the exact binomial result); real juries have correlated errors, shared misinformation and unequal competence that can break the convergence — the p>0.5-amplifies, p<0.5-collapses dichotomy is exact.