  The detection of two NS--BH mergers by LIGO-Virgo provided the first direct
confirmation of the existence of this type of system in the Universe. These
detections also imply the existence of pulsar--black hole (PSR--BH) systems. In
this analysis, we use the non-detection of any PSR--BH systems in current radio
surveys to estimate a 95\% upper limit of $\sim$150 PSR--BH binary systems that
are beaming towards the Earth in the Milky Way. This corresponds to a 95\%
upper limit of $\mathcal{R}_{\rm LIGO} = 7.6$~yr$^{-1}$ on the merger detection
rate for the LIGO-Virgo network scaled to a range distance of 100~Mpc, which is
consistent with the rates derived by LIGO-Virgo. In addition, for the first
time, we use the merger detection rates estimate by LIGO-Virgo to predict the
number of detectable PSR--BH systems in the Milky Way. We find there to be
$\left< N_{\rm obs, NSBH, e} \right> = 2^{+5}_{-1}$ and $\left< N_{\rm obs,
NSBH, p} \right> = 6^{+7}_{-4}$ detectable PSR--BH systems in the Milky Way
corresponding to the event-based and population-based merger detection rates
estimated by LIGO-Virgo respectively. We estimate the probability of detecting
these PSR--BH systems with current radio pulsar surveys, showing that the
Arecibo PALFA survey has the highest probability of detecting a PSR--BH system,
while surveys with recently commissioned and planned telescopes are almost
guaranteed to detect one of these systems. Finally, we discuss the hurdles in
detecting PSR--BH systems and how these can be overcome in the future.
