Equation 5 · Why Average Success Rate Hides the Failures That Matter Most
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
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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Symbol q_τ
q_τ is part of the quantity the equation computes from the expression on the right.
Symbol τ
τ is part of the quantity the equation computes from the expression on the right.
Symbol p
p is one of the signed contributions combined to compute the quantity on the left.
Symbol C
C is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →Probability operator
The probability operator gives the chance of the event named inside its brackets or parentheses.
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 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 — a deleted database rather than a clumsy sentence — is . 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, , differ by…
Read the full surrounding passage
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 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 — a deleted database rather than a clumsy sentence — is . 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, , differ by orders of magnitude — one agent’s failures cluster near the harmless end of C , the other’s failures put meaningful mass past . 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.
Sources cited in the article section
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