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Published equation contexts

qτ=Pr⁡(failure and severity>τ)=(1−p)⋅Pr⁡(C>τ∣failure)q_\tau = \Pr(\text{failure and severity} > \tau) = (1-p)\cdot \Pr(C > \tau \mid \text{failure})

Why this formula appears here

It helps to state the structural point precisely rather than just by example. Let p be an agent’s probability of completing a task successfully, so 1-p is its failure probability, and let severity be a random quantity C ≥\geq 0 describing how bad a given failure turns out to be, conditional on failure occurring. The probability that a given run produces a failure whose severity clears some consequential threshold τ\tau — a deleted database rather than a clumsy sentence — is qτ=Pr⁡(failure and severity>τ)=(1−p)⋅Pr⁡(C>τ∣failure)q_\tau = \Pr(\text{failure and severity} > \tau) = (1-p)\cdot \Pr(C > \tau \mid \text{failure}). The success rate p says nothing at all about the second factor. Two agents can share an identical p of 0.99 while their conditional severity distributions, Pr⁡(C>τ∣failure)\Pr(C > \tau \mid \text{failure}) , differ by…

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Pr⁡\Pr

Probability operator

The probability operator gives the chance of the event named inside its brackets or parentheses.

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Published contexts (1)

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qτ=Pr⁡(failure and severity>τ)=(1−p)⋅Pr⁡(C>τ∣failure)q_\tau = \Pr(\text{failure and severity} > \tau) = (1-p)\cdot \Pr(C > \tau \mid \text{failure})

Equation 5 · Model Evaluation

Why Average Success Rate Hides the Failures That Matter Most

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

It helps to state the structural point precisely rather than just by example. Let p be an agent’s probability of completing a task successfully, so 1-p is its failure probability, and let severity be a random quantity C ≥\geq 0 describing how bad a given failure turns out to be, conditional on failure occurring. The probability that a given run produces a failure whose severity clears some consequential threshold τ\tau — a deleted database rather than a clumsy sentence — is qτ=Pr⁡(failure and severity>τ)=(1−p)⋅Pr⁡(C>τ∣failure)q_\tau = \Pr(\text{failure and severity} > \tau) = (1-p)\cdot \Pr(C > \tau \mid \text{failure}). The success rate p says nothing at all about the second factor. Two agents can share an identical p of 0.99 while their conditional severity distributions, Pr⁡(C>τ∣failure)\Pr(C > \tau \mid \text{failure}) , differ by…

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