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qτ≲3nq_\tau \lesssim \frac{3}{n}

Why this formula appears here

Suppose an evaluation runs n independent trials of a consequential task and observes zero failures above the severity threshold τ\tau . It is tempting to read that as evidence the risk is negligible. It is not, and the size of the gap has a name: the rule of three. If the true failure probability is qτq_\tau , the probability of observing zero events in n independent trials is (1-qτq_\tau)^n ; setting that probability equal to 0.05 and solving gives n ln⁡(1−qτ)\ln(1-q_\tau) ≈\approx -3 , and for small qτq_\tau , where ln⁡(1−qτ)\ln(1-q_\tau) ≈\approx -qτq_\tau , this reduces to the boundary qτ≲3nq_\tau \lesssim \frac{3}{n}. as the 95 percent upper confidence bound on the true rate, a result formalised by Hanley and Lippman-Hand’s…

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qτq_\tau

Symbol q_τ

q_τ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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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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How to interpret it

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

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Published contexts (1)

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qτ≲3nq_\tau \lesssim \frac{3}{n}

Equation 23 · Model Evaluation

Why Average Success Rate Hides the Failures That Matter Most

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

Suppose an evaluation runs n independent trials of a consequential task and observes zero failures above the severity threshold τ\tau . It is tempting to read that as evidence the risk is negligible. It is not, and the size of the gap has a name: the rule of three. If the true failure probability is qτq_\tau , the probability of observing zero events in n independent trials is (1-qτq_\tau)^n ; setting that probability equal to 0.05 and solving gives n ln⁡(1−qτ)\ln(1-q_\tau) ≈\approx -3 , and for small qτq_\tau , where ln⁡(1−qτ)\ln(1-q_\tau) ≈\approx -qτq_\tau , this reduces to the boundary qτ≲3nq_\tau \lesssim \frac{3}{n}. as the 95 percent upper confidence bound on the true rate, a result formalised by Hanley and Lippman-Hand’s…

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