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Equation 2 · Part 1 · Measuring AI Agent Architectures: Evidence, Benchmarks, and Uncertainty

Symbol p_at least one of k

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.
pat least one of kp_{\text{at least one of } k}

What this part means

pap_at least one of k is part of the quantity the equation computes from the expression on the right.

Its job in the formula

pap_at least one of k is part of the quantity the equation computes from the expression on the right.

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

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

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