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Equation 12 · Part 9 · Measuring AI Agent Reliability: What the Evidence Actually Supports

Numerator: binomck

passk  :=  Etasks ⁣[ (ck)(nk) ]\text{pass}^{k} \;:=\; \mathbb{E}_{\text{tasks}}\!\left[\, \frac{\binom{c}{k}}{\binom{n}{k}} \,\right]
(ck)\binom{c}{k}

What this part means

The complete quantity above the fraction bar.

Its job in the formula

binomck occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

Pass@k answers a specific question: given a budget of k independent attempts, what is the chance that at least one succeeds? That is the right question for a search-and-verify workflow, where a cheap checker can identify the one attempt that worked among several candidates. It is the wrong question for almost everything else an agent does, because most agentic tasks do not offer a free, cheap oracle that can pick the winning attempt out of a pile of candidates after the fact — the “attempt” is the deployment. For that setting, Yao and colleagues, building the tau-bench benchmark for tool-using agents interacting with simulated customers under domain policies, proposed the complementary…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

Open the illustrated fractions: division written vertically guide →

Sources cited in the surrounding passage

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