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Published equation contexts

p^±1.96p^(1−p^)n\hat{p} \pm 1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Why this formula appears here

4. Insufficient trial counts producing false confidence. A success rate reported from a small number of trials carries a much wider margin of error than the single reported number suggests, and the interval widens fast as the trial count shrinks. For a naive binomial estimate — treating each trial as an independent coin flip with unknown success probability p — the 95 percent confidence interval around an observed rate p^\hat{p} from n trials is approximately p^±1.96p^(1−p^)n\hat{p} \pm 1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. At ten trials and an observed 90 percent success rate, that formula gives an interval of roughly plus or minus nineteen percentage points — an agent genuinely succeeding anywhere from about 71 percent to essentially…

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nn

Symbol n

n occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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With a fixed numerator, increasing a nonzero denominator reduces the fraction.

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p^±1.96p^(1−p^)n.\hat{p} \pm 1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

Equation 15 · Model Evaluation

Ten Ways an Agent Evaluation Can Mislead You Even When It's Working Correctly

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

4. Insufficient trial counts producing false confidence. A success rate reported from a small number of trials carries a much wider margin of error than the single reported number suggests, and the interval widens fast as the trial count shrinks. For a naive binomial estimate — treating each trial as an independent coin flip with unknown success probability p — the 95 percent confidence interval around an observed rate p^\hat{p} from n trials is approximately p^±1.96p^(1−p^)n\hat{p} \pm 1.96\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. At ten trials and an observed 90 percent success rate, that formula gives an interval of roughly plus or minus nineteen percentage points — an agent genuinely succeeding anywhere from about 71 percent to essentially…

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