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Equation 8 · Part 1 · How Benchmark Contamination Actually Works in Agentic Evaluation

Symbol k

passk(t)=∏i=1k1 ⁣[trial i on task t succeeds],\text{pass}^k(t) = \prod_{i=1}^{k} \mathbb{1}\!\left[\text{trial } i \text{ on task } t \text{ succeeds}\right],
kk

What this part means

k appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

Its job in the formula

k appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

The passage around this formula

…is designed to punish an agent whose competence is real but inconsistent across resampled trials — a stochastic policy with true per-trial success probability p has passk\text{pass}^k ≈\approx pkp^k , which falls quickly as k grows. But a policy whose output on a given task is deterministic — an empty response, always, regardless of sampling — produces the identical transcript on every trial, so

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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