Equation 12 · Part 8 · Reliable AI Agents Are Control Systems, Not Chatbots

Starting index or lower bound: i=1

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

i=1

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

i=1 appears in the bound of this product. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionStarting index or lower bound: i=1Article meaning

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 $p_i$ given that all previous stages succeeded. Then $\Pr(\text{task success}) = \prod_{i=1}^{n} p_i$ . If one makes the deliberately crude assumption $p_i$ =p , a 98% reliable stage repeated 50 times yields 0.98^{50} $\approx$ 0.364 . 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

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the article section

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