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Equation 2 · Part 6 · Measuring AI Agent Architectures: Evidence, Benchmarks, and Uncertainty

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pat least one of k=1−(1−p)k,pall k succeed=pk.p_{\text{at least one of } k} = 1 - (1-p)^k, \qquad p_{\text{all } k \text{ succeed}} = p^k.
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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 distinction matters because the two metrics behave very differently even under the most favourable possible assumption — that trials are statistically independent. Write p for the probability that a single trial on a given task succeeds. Under independence, pat least one of k=1−(1−p)k,pall k succeed=pkp_{\text{at least one of } k} = 1 - (1-p)^k, \qquad p_{\text{all } k \text{ succeed}} = p^k. The first quantity rises quickly toward one as k grows, which is the intuitive but misleading sense in which “just try again” seems to fix unreliability. The second quantity falls quickly toward zero, and it is the quantity that matters if a task must be completed correctly on a specific occasion rather than merely being solvable in principle across several attempts.

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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.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

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