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Equation 12 · Part 6 · Reliable AI Agents Are Control Systems, Not Chatbots

superscript

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

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the article section

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