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Equation 3 · Part 5 · Measuring Tool Protocols and the Model Context Protocol: Evidence, Benchmarks, and Uncertainty

Probability operator

Pr⁡[all k attempts succeed]=pk.\Pr[\text{all } k \text{ attempts succeed}] = p^{k}.
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 part of the quantity the equation computes from the expression on the right.

The passage around this formula

That gap is large enough to be worth writing down explicitly. If a task’s outcome were an independent Bernoulli draw with a fixed success probability p equal to the reported average, the probability that all k independent attempts at the same task succeed would be Pr⁡[all k attempts succeed]=pk\Pr[\text{all } k \text{ attempts succeed}] = p^{k}. At p = 0.60 and k = 8 , that model predicts roughly 1.7%. The reported figure — under 25% — sits well above that naive prediction, and the direction of the gap is informative on its own: it is only possible if outcomes are not independent draws from one fixed probability, but rather reflect a task population that splits into instances the agent reliably solves and instances it reliably does not, with the…

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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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