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Equation 5 · Part 7 · Why Average Success Rate Hides the Failures That Matter Most

Probability operator

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})
Pr⁡\Pr

What this part means

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

Its job in the formula

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

The passage around this formula

…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…

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Learn the underlying idea

Probability assigns a number from 0 to 1 to an event under a stated model. Zero means impossible within that model; one means certain.

Open the illustrated probability: a quantified chance guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.