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$(1-q_\tau)^n$

Why this formula appears here

Suppose an evaluation runs n independent trials of a consequential task and observes zero failures above the severity threshold $\tau$ . 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 $q_\tau$ , the probability of observing zero events in n independent trials is (1- $q_\tau$ )^n ; setting that probability equal to 0.05 and solving gives n $\ln(1-q_\tau)$ $\approx$ -3 , and for small $q_\tau$ , where $\ln(1-q_\tau)$ $\approx$ - $q_\tau$ , this reduces to the boundary

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q_\tau

Symbol q_τ

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n

Symbol n

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(1-q_\tau)^n

Equation 19 · 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 $\tau$ . 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 $q_\tau$ , the probability of observing zero events in n independent trials is (1- $q_\tau$ )^n ; setting that probability equal to 0.05 and solving gives n $\ln(1-q_\tau)$ $\approx$ -3 , and for small $q_\tau$ , where $\ln(1-q_\tau)$ $\approx$ - $q_\tau$ , this reduces to the boundary

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(1−qτ)n(1-q_\tau)^n(1−qτ​)n