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.
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