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Equation 33 · Part 3 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

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

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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