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Equation 33 · Part 5 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

Probability operator

Pr⁡(at least one success)=1−(1−q)k.\Pr(\text{at least one success})=1-(1-q)^k.
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 appears in the conditional probability being evaluated. The vertical bar identifies the information or condition supplied to that probability.

The passage around this formula

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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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.