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Equation 15 · Part 5 · Ten Ways an Agent Evaluation Can Mislead You Even When It's Working Correctly

Numerator: hatp(1-hatp)

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

What this part means

The complete quantity above the fraction bar.

Its job in the formula

hatp(1-hatp) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

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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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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