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Equation 8 · Part 4 · Ten Failure Modes That Define Production AI Agent Architectures

=

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

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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 Pat least one=1−(1−q)kP_{\text{at least one}} = 1 - (1 - q)^{k}. This is…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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