← All parts of this equation

Equation 33 · Part 4 · Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

superscript

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

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

Read this part in the article →

Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

Open the illustrated exponents: repeated multiplication and powers guide →

Sources cited in the article section

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