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Message: <blockquote data-quote="topkollek" data-source="post: 31282895" data-attributes="member: 510150">We can precisely quantify the bias introduced by the second preference count on Condorcet’s Jury Theorem (and therefore on the Law of Large Numbers guarantee that the majority picks the “best” candidate with probability → 1).Here are the official numbers (Election Commission + Wikipedia consensus):<ul> <li data-xf-list-type="ul">First-preference total: 13,319,616 valid votes</li> <li data-xf-list-type="ul">Final decisive jury (second count): n = 10,271,081 (only 77.11% of the electorate)</li> <li data-xf-list-type="ul">Exhausted ballots: 3,048,535 (22.89%)</li> <li data-xf-list-type="ul">Usable transfers from eliminated candidates: 273,131 votes only (just 2.66% of the final jury)</li> <li data-xf-list-type="ul">Of those transfers: Sajith received 61.5% (≈168,000), Anura 38.5% (≈105,000)</li> <li data-xf-list-type="ul">Anura’s final share: 55.8868% (5,740,179 votes)</li> </ul><h3>Quantified Bias Components</h3><table style='width: 100%'><tr><th>Bias Type</th><th>Exact Impact</th><th>Effect on Theorem (Condorcet + LLN)</th></tr><tr><td>Jury size reduction</td><td>n reduced by 22.89% (13.32M → 10.27M)</td><td>Convergence slower by factor 0.771 in the exponent of error probability. Still irrelevant at this scale.</td></tr><tr><td>Directional transfer bias</td><td>Sajith got 61.5% of transfers → 0.30 percentage-point drag against Anura (final 55.89% vs hypothetical 56.19% if transfers split 50-50)</td><td>Slightly lowers effective voter competence p by ~0.3 pp.</td></tr><tr><td>Selection / filtering bias</td><td>Only engaged voters who ranked a top-two candidate participate; exhausted are disproportionately low-engagement third-party supporters</td><td>Mild heterogeneous p (core voters likely higher competence). Non-random jury.</td></tr><tr><td>Overall net bias</td><td>~0.3–0.5 pp downward shift in observed proportion + 23% smaller jury</td><td>Weakens the “P(correct) → 1” guarantee marginally</td></tr></table><h3>Impact on Probability of “Wrong” Winner (Scenarios)</h3>We use the normal approximation to the binomial (extremely accurate at n > 10 million). Assume Anura was the “objectively best” candidate; we calculate P(majority picks wrong) under different bias assumptions.<table style='width: 100%'><tr><th>Scenario</th><th>Effective p (Anura competence)</th><th>z-score vs 50%</th><th>P(majority wrong)</th><th>How much worse than ideal?</th></tr><tr><td>Ideal Condorcet (no bias, full n, p=0.5589)</td><td>0.5589</td><td>377.3</td><td>≈ 0 (<< 10⁻³⁰⁰)</td><td>Baseline</td></tr><tr><td>Observed (real second count)</td><td>0.5589</td><td>377.3</td><td>≈ 0 (<< 10⁻³⁰⁰)</td><td>Identical</td></tr><tr><td>Core supporters only (no transfers)</td><td>0.5636</td><td>402</td><td>Even smaller</td><td>Actually stronger</td></tr><tr><td>Pessimistic: transfers were pure coin-flips (p=0.50)</td><td>≈0.557</td><td>360</td><td>Still ≈ 0 (<< 10⁻²⁸⁰)</td><td>Negligible degradation</td></tr><tr><td>Extreme: transferred voters had p=0.51 only</td><td>≈0.556</td><td>355</td><td>Still ≈ 0</td><td>Tiny</td></tr></table>Even in the worst plausible case (transferred voters barely better than random), the probability of the “wrong” winner remains astronomically smaller than 1 in 10²⁵⁰. You would need a ~8–10 percentage-point systematic bias in the transferred group (or exhaustion overwhelmingly one-sided) to start making the outcome uncertain — far beyond what actually occurred.<h3>Mathematical Intuition (Chernoff/Hoeffding bound)</h3>The theorem’s error probability decays exponentially:P(wrong)≤2exp⁡(−2n(p^−0.5)2)P(\text{wrong}) \leq 2 \exp(-2 n (\hat{p} - 0.5)^2)P(wrong)≤2exp(−2n(p^−0.5)2)<ul> <li data-xf-list-type="ul">With full n and observed p^  \hat{p}  p^: exponent ≈ –69,400</li> <li data-xf-list-type="ul">With 23% smaller jury + 0.3 pp drag: exponent ≈ –53,400 Both give numbers so small they are zero for any practical purpose (smaller than the chance of a proton decaying in the next 10²⁰⁰ years).</li> </ul>Bottom line The second preference count does introduce measurable bias (mainly 23% jury shrinkage + 0.3 pp directional drag against Anura), but its impact on Condorcet’s Jury Theorem is tiny in practice. The Law of Large Numbers with over 10 million voters completely overwhelms the bias.The guarantee that “the majority picked the best of the two finalists with near-certainty” remains essentially intact — the probability of error is still effectively zero under any reasonable model of voter competence.Sri Lanka’s 2024 result is still one of the cleanest real-world demonstrations of the theorem at massive scale. The bias exists mathematically, but it is negligible.</blockquote>

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