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Equation 12 · Part 1 · Reliable AI Agents Are Control Systems, Not Chatbots

Symbol i

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

What this part means

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

Its job in the formula

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

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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