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Data Release 3 (DR3) from the Gaia Mission includes radial velocitymeasurements of over 33 million targets. Among many scientific applications,the overlap of this stellar sample with targeted exoplanet transit survey starspresents an opportunity to understand planet occurrence in the context ofstellar multiplicity on a large scale. Yet, any interpretation of occurrencerelies upon an understanding of survey sensitivity. While the sensitivity toplanets in transit surveys is well understood, a characterization of thesensitivity of Gaia to unresolved binaries is also critical. We describe here astatistical framework called paired, which we developed to enable the forwardmodeling of Gaia radial velocity observables for large samples of stars. Thepaired machinery links the reported radial velocity noise for a given star fromGaia to the probability of a spatially unresolved stellar companion. Wedemonstrate how this enables the user, given an observed distribution ofindividual binary ``probabilities" for a set of stars, to understand thisdistribution within the sensitivity limits of Gaia. For the subset of starswith the highest probability of excess radial velocity noise, we describe theability of paired to constrain the semi-amplitude of the stellar binary. Wherepossible, we benchmark our inferred radial velocity semi-amplitudes againstthose from ground-based radial velocity surveys, and the subset published byGaia DR3 itself. We aim for paired to be a community tool for the explorationof the effects of binarity on planets at a population level, but also for anyuser interested in stellar populations.

paired: A Statistical Framework for Detecting Stellar Binarity with Gaia RVs. I. Sensitivity to Unresolved Binaries