  Until recently, black holes (BHs) could be discovered only through accretion
from other stars in X-ray binaries, or in merging double compact objects.
Improvements in astrometric and spectroscopic measurements have made it
possible to detect BHs also in non-interacting BH binaries (nBHB) through a
precise analysis of the companion's motion. In this study, using an updated
version of the Startrack binary-star population modelling code and a detailed
model of the Milky Way (MW) galaxy we calculate the expected number of
detections for Gaia and LAMOST surveys. We develop a formalism to convolve the
binary population synthesis output with a realistic stellar density
distribution, star-formation history (SFH), and chemical evolution for the MW,
which produces a probability distribution function of the predicted
compact-binary population over the MW. This avoids the additional statistical
uncertainty which is introduced by methods which Monte Carlo sample from binary
population synthesis output to produce one potential specific realisation of
the MW compact-binary distribution, and our method is also comparatively fast
to such Monte Carlo realisations. Specifically, we predict $\sim41$-$340$ nBHBs
to be observed by Gaia, although the numbers may drop to $\sim10$-$70$ if the
recent ($\lesssim100\;$ Myr) star formation is low ($\sim1\;M_\odot$/yr ). For
LAMOST we predict $\lesssim14$ detectable nBHBs, which is lower partially
because its field-of-view covers just $\sim6\%$ of the Galaxy.
