Each doctor sees an ambiguous case — an "explainable phenomenon vs. genuine anomaly" call — and is individually correct with probability p > 0.5: better than a coin flip, but far from certain. The Condorcet Jury Theorem (1785) says that if N such doctors judge independently and the panel goes with the majority vote, the panel's accuracy rises toward 100% as N grows — even though no single doctor is anywhere near that reliable.
P(majority correct) = Σ (k = ⌈N/2⌉ to N) C(N,k) · p^k · (1-p)^(N-k)
as N → ∞, P(majority correct) → 1 (for any fixed p > 0.5)
as N → ∞, P(majority correct) → 0 (for any fixed p < 0.5)
- Why it works — individual errors are random noise that point in different directions; averaging many independent noisy judgments cancels the noise while the shared signal (the true diagnosis) reinforces itself. This is the same mathematics behind forecaster/expert-panel aggregation and ensemble machine-learning classifiers.
- The independence assumption is load-bearing — switch to "Correlated" and each doctor has a 55% chance of simply copying one shared, case-wide judgment instead of forming their own. The panel's real accuracy then stalls well below the Condorcet curve, because copied errors don't cancel — they stack. Independence, not panel size alone, is what makes crowds wise.
- Distinct from a single forecaster — this is not calibration of one expert's confidence (see
ufo-experts-pollution); it's about how many independently-erring judges combine, regardless of how confident any one of them feels.