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E[cost to first success]=cˉp\mathbb{E}[\text{cost to first success}] = \frac{\bar{c}}{p}

Why this formula appears here

There is also a structural reason cost cannot be separated from reliability. If a run costs cˉ\bar{c} on average and succeeds with probability p , and the operator simply retries until success, the expected spend to first success is E[cost to first success]=cˉp\mathbb{E}[\text{cost to first success}] = \frac{\bar{c}}{p}. which means a cheap unreliable agent and an expensive reliable one can occupy the same operating point. Reporting either number alone conceals which one you have bought. The honest artefact is a curve — quality against spend, produced by sweeping the effort level and the retry policy — rather than a point, and the honest comparison places two systems at matched spend and asks which reaches further.

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With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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E[cost to first success]=cˉp,\mathbb{E}[\text{cost to first success}] = \frac{\bar{c}}{p},

Equation 4 · Agent Evaluation

Evaluating Trajectories, Not Answers

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

There is also a structural reason cost cannot be separated from reliability. If a run costs cˉ\bar{c} on average and succeeds with probability p , and the operator simply retries until success, the expected spend to first success is E[cost to first success]=cˉp\mathbb{E}[\text{cost to first success}] = \frac{\bar{c}}{p}. which means a cheap unreliable agent and an expensive reliable one can occupy the same operating point. Reporting either number alone conceals which one you have bought. The honest artefact is a curve — quality against spend, produced by sweeping the effort level and the retry policy — rather than a point, and the honest comparison places two systems at matched spend and asks which reaches further.

Meanings in this article

  • E\mathbb{E}: The expected value operator: the probability-weighted average of the quantity inside its brackets.
  • pp: the probability.
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