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Equation 12 · Part 5 · Reliable AI Agents Are Control Systems, Not Chatbots

subscript

Pr⁡(task success)=∏i=1npi.\Pr(\text{task success}) = \prod_{i=1}^{n} p_i.
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

Single-turn model quality can hide long-horizon fragility. Suppose a task has n dependent stages and each stage succeeds with conditional probability pip_i given that all previous stages succeeded. Then Pr⁡(task success)=∏i=1npi\Pr(\text{task success}) = \prod_{i=1}^{n} p_i. If one makes the deliberately crude assumption pip_i=p , a 98% reliable stage repeated 50 times yields 0.98^{50}≈\approx0.364 . Real agent steps are neither independent nor identically distributed: an early mistake can corrupt later observations, while a test can expose and reverse it. The simple product is useful because it reveals the architecture’s burden. Long tasks require mechanisms that change conditional probabilities after observing evidence , not merely a model with…

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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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