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Published equation contexts

pi∗=pi+(1−pi)dicirip_i^*=p_i+(1-p_i)d_i c_i r_i

Why this formula appears here

An agent can recover only if the system generates a discriminating observation. Let stage i have direct success probability pip_i . If failure occurs, let did_i be the probability it is detected, cic_i the probability the cause is diagnosed correctly, and rir_i the probability repair succeeds without creating another material defect. A simplified effective stage reliability is pi∗=pi+(1−pi)dicirip_i^*=p_i+(1-p_i)d_i c_i r_i. The formula is intentionally conservative and incomplete. Multiple retries, repair costs, correlated defects, and false alarms require richer models. Its value is conceptual: “the agent can iterate” improves reliability only through detection, diagnosis, and repair.

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Published contexts (1)

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pi∗=pi+(1−pi)diciri.p_i^*=p_i+(1-p_i)d_i c_i r_i.

Equation 20 · AI Agents & Systems

Why Coding Agents Fail: Long-Horizon Reliability in OpenAI Codex

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

An agent can recover only if the system generates a discriminating observation. Let stage i have direct success probability pip_i . If failure occurs, let did_i be the probability it is detected, cic_i the probability the cause is diagnosed correctly, and rir_i the probability repair succeeds without creating another material defect. A simplified effective stage reliability is pi∗=pi+(1−pi)dicirip_i^*=p_i+(1-p_i)d_i c_i r_i. The formula is intentionally conservative and incomplete. Multiple retries, repair costs, correlated defects, and false alarms require richer models. Its value is conceptual: “the agent can iterate” improves reliability only through detection, diagnosis, and repair.

Meanings in this article

  • pi∗p_i^*: the simplified effective stage reliability.
  • pip_i: the direct success probability.
  • rir_i: the probability repair succeeds without creating another material defect.
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