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qτ=10−5q_\tau = 10^{-5}

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trials in total. This is where the arithmetic becomes unforgiving in a way the mean-difference case never does. Detecting a fixed three-percentage-point difference between two average success rates takes roughly the same, bounded number of trials — on the order of a thousand — regardless of how good either system actually is, because that calculation is about the spread of a difference, not the rarity of an event. Bounding or estimating a rare catastrophic-failure rate is different in kind: the required sample size is inversely proportional to the rate itself. At qτq_\tau = 10^{-3} , getting even a rough twenty-event estimate needs on the order of twenty thousand trials. At qτq_\tau = 10^{-5} —…

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qτ=10−5q_\tau = 10^{-5}

Equation 28 · Model Evaluation

Why Average Success Rate Hides the Failures That Matter Most

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

trials in total. This is where the arithmetic becomes unforgiving in a way the mean-difference case never does. Detecting a fixed three-percentage-point difference between two average success rates takes roughly the same, bounded number of trials — on the order of a thousand — regardless of how good either system actually is, because that calculation is about the spread of a difference, not the rarity of an event. Bounding or estimating a rare catastrophic-failure rate is different in kind: the required sample size is inversely proportional to the rate itself. At qτq_\tau = 10^{-3} , getting even a rough twenty-event estimate needs on the order of twenty thousand trials. At qτq_\tau = 10^{-5} —…

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