  Radial velocity (RV) surveys have detected hundreds of exoplanets through
their gravitational interactions with their host stars. Some will be
transiting, but most lack sufficient follow-up observations to confidently
detect (or rule out) transits. We use published stellar, orbital, and planetary
parameters to estimate the transit probabilities for nearly all exoplanets that
have been discovered via the RV method. From these probabilities, we predict
that $25.5^{+0.7}_{-0.7}$ of the known RV exoplanets should transit their host
stars. This prediction is more than double the amount of RV exoplanets that are
currently known to transit. The Transiting Exoplanet Survey Satellite (TESS)
presents a valuable opportunity to explore the transiting nature of many of the
known RV exoplanet systems. Based on the anticipated pointing of TESS during
its two-year primary mission, we identify the known RV exoplanets that it will
observe and predict that $11.7^{+0.3}_{-0.3}$ of them will have transits
detected by TESS. However, we only expect the discovery of transits for $\sim$3
of these exoplanets to be novel (i.e., not previously known). We predict that
the TESS photometry will yield dispositive null results for the transits of
$\sim$125 RV exoplanets. This will represent a substantial increase in the
effort to refine ephemerides of known RV exoplanets. We demonstrate that these
results are robust to changes in the ecliptic longitudes of future TESS
observing sectors. Finally, we consider how several potential TESS extended
mission scenarios affect the number of transiting RV exoplanets we expect TESS
to observe.
