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Equation 5 · Why Average Success Rate Hides the Failures That Matter Most

What does this equation mean?

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})

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Inputs and operationsPr(failure and severity > τ) = (1-p) × Pr(C > τ mid failure)
Result or conditionq_τ
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This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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qτq_\tau

Symbol q_τ

q_τ is part of the quantity the equation computes from the expression on the right.

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τ\tau

Symbol τ

τ is part of the quantity the equation computes from the expression on the right.

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pp

Symbol p

p is one of the signed contributions combined to compute the quantity on the left.

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CC

Symbol C

C is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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multiplication

multiplication

Multiply the quantities on either side.

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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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How to interpret it

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What the article says around this equation

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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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 orders of magnitude — one agent’s failures cluster near the harmless end of C , the other’s failures put meaningful mass past τ\tau . A leaderboard built entirely on p ranks these two agents identically. An operator who deploys the higher-severity one because its accuracy number looked the same has made a real decision without the information that decision required.

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