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Equation 12 · Part 7 · Reliable AI Agents Are Control Systems, Not Chatbots

Probability operator

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

What this part means

The probability operator gives the chance of the event named inside its brackets or parentheses.

Its job in the formula

Pr appears in the conditional probability being evaluated. The vertical bar identifies the information or condition supplied to that probability.

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

Probability assigns a number from 0 to 1 to an event under a stated model. Zero means impossible within that model; one means certain.

Open the illustrated probability: a quantified chance guide →

Sources cited in the article section

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