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Equation 20 · Part 2 · Why Average Success Rate Hides the Failures That Matter Most

Symbol q_τ

nln⁡(1−qτ)≈−3n \ln(1-q_\tau) \approx -3
qτq_\tau

What this part means

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

Its job in the formula

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

The passage around this formula

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

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