$$P_{\text{at least one}} = 1 - (1 - q)^{k}$$

Equation 8 · AI Agents & Systems

Ten Failure Modes That Define Production AI Agent Architectures

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

The trade-off has a clean, if deliberately simplified, shape. Suppose a supervisor dispatches a task to k subagents running in parallel specifically to shorten wall-clock time, and each subagent independently produces a silently wrong or incomplete result — one that does not raise an error, only a bad answer the supervisor’s synthesis step has to catch — with probability q . Under an independence assumption that will not hold exactly in practice, because subagents sharing a planner, a toolset, and a training distribution are correlated rather than independent, the probability that at least one branch’s bad output reaches the synthesis step unflagged is $P_{\text{at least one}} = 1 - (1 - q)^{k}$. This is…

Meanings in this article

• $P_{\text{at least one}}$: the probability that at least one branch’s bad output reaches the synthesis step unflagged.
• $q$: the probability.

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