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Equation 2 · Part 8 · Measuring AI Agent Architectures: Evidence, Benchmarks, and Uncertainty

subscript

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.
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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.

Open the illustrated subscripts: which member of a family? guide →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.