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Equation 14 · Part 5 · The Token Tax of Giving a Model More Tools

Probability operator

error(m)  =  Pr⁡[tool⋆∉Sm]⏟retrieval gap, shrinks as m grows  +  Pr⁡[miscall∣tool⋆∈Sm]⏟confusion gap, grows as m grows\mathrm{error}(m) \;=\; \underbrace{\Pr[\mathrm{tool}^\star \notin S_m]}_{\text{retrieval gap, shrinks as } m \text{ grows}} \;+\; \underbrace{\Pr[\text{miscall} \mid \mathrm{tool}^\star \in S_m]}_{\text{confusion gap, grows as } m \text{ grows}}
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 is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

A separate, applied study on enterprise agent routing gives a way to decompose exactly what breaks as a real tool catalogue scales, rather than only that something does. Scaling from 10 to 110 candidate agents or tools, the authors report routing F1 on under-specified requests dropping 16 to 23 percentage points across the models tested, and split that drop with an oracle analysis into two distinct components: a retrieval gap , the model’s failure to surface the correct tool at all, and a confusion gap , a roughly 10-percentage-point reduction in the theoretical best-case score that persists even when retrieval is assumed perfect [ 11 ] . Written as a decomposition of the error a shortlist…

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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 surrounding passage

These citations provide research context; check each source for the exact claim it supports.