  The study of binary stars in the Galactic halo provides crucial insights into
the dynamical history and formation processes of the Milky Way. In this work,
we aim to investigate the binary fraction in a sample of accreted and in-situ
halo stars, focusing on short-period binaries. Utilising data from Gaia DR3, we
analysed the radial velocity (RV) uncertainty $\sigma_{\mathrm{RV}}$
distribution of a sample of main-sequence stars. We used a novel Bayesian
framework to model the dependence in $\sigma_{\mathrm{RV}}$ of single and
binary systems allowing us to estimate binary fractions $F$ in a sample of
bright ($G_{\mathrm{RVS}}$ < 12) Gaia sources. We selected the samples of
in-situ and accreted halo stars based on estimating the 6D phase space
information and affiliating the stars to the different samples on an
action-angle vs energy ($L_{\mathrm{z}}-E$) diagram. Our results indicate a
higher, though not significant, binary fraction in accreted stars compared to
the in-situ halo sample. We further explore binary fractions using cuts in $E$
and $L_z$ and find a higher binary fraction in both high-energy and prograde
orbits that might be explained by differences in metallicity. By cross-matching
our Gaia sample with APOGEE DR17 catalogue, we confirm the results of previous
studies on higher binary fractions in metal-poor stars and find the fractions
of accreted and in-situ halo stars consistent with this trend. Our finding
provides new insights into binary stars' formation processes and dynamical
evolution in the primordial Milky Way Galaxy and its accreted dwarf Galaxies.
