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Pr⁡(at least one success)=1−(1−q)k\Pr(\text{at least one success})=1-(1-q)^k

Why this formula appears here

Code-generation research often reports pass@\mathrm{pass@}k : the probability that at least one of k sampled solutions passes. If independent attempts each succeed with probability q , then Pr⁡(at least one success)=1−(1−q)k\Pr(\text{at least one success})=1-(1-q)^k. This is valuable for search: more attempts increase the chance of finding one acceptable candidate. Production consistency asks a different question. If k required deployments must all succeed, the same independence simplification gives

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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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Pr⁡(at least one success)=1−(1−q)k.\Pr(\text{at least one success})=1-(1-q)^k.

Equation 33 · AI Agents & Systems

Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

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

Code-generation research often reports pass@\mathrm{pass@}k : the probability that at least one of k sampled solutions passes. If independent attempts each succeed with probability q , then Pr⁡(at least one success)=1−(1−q)k\Pr(\text{at least one success})=1-(1-q)^k. This is valuable for search: more attempts increase the chance of finding one acceptable candidate. Production consistency asks a different question. If k required deployments must all succeed, the same independence simplification gives

Meanings in this article

  • qq: the probability.
  • kk: the first rises with.
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