The SPIRou near infrared spectro-polarimeter is destined to begin science operations at the Canada-France-Hawaii Telescope in mid-2018. One of the instruments primary science goals is to discover the closest exoplanets to the Solar System by conducting a 3-5 year long radial velocity survey of nearby M dwarfs at an expected precision of $sim 1$ m s$^{-1}$; the SPIRou Legacy Survey-Planet Search (SLS-PS). In this study we conduct a detailed Monte-Carlo simulation of the SLS-PS using our current understanding of the occurrence rate of M dwarf planetary systems and physical models of stellar activity. From simultaneous modelling of planetary signals and activity, we predict the population of planets detected in the SLS-PS. With our fiducial survey strategy and expected instrument performance over a nominal survey length of $sim 3$ years, we expect SPIRou to detect $85.3^{+29.3}_{-12.4}$ planets including $20.0^{+16.8}_{-7.2}$ habitable zone planets and $8.1^{+7.6}_{-3.2}$ Earth-like planets from a sample of 100 M1-M8.5 dwarfs out to 11 pc. By studying mid-to-late M dwarfs previously inaccessible to existing optical velocimeters, SPIRou will put meaningful constraints on the occurrence rate of planets around those stars including the value of $eta_{oplus}$ at an expected level of precision of $lesssim 45$%. We also predict a subset of $46.7^{+16.0}_{-6.0}$ planets may be accessible with dedicated high-contrast imagers on the next generation of ELTs including $4.9^{+4.7}_{-2.0}$ potentially imagable Earth-like planets. Lastly, we compare the results of our fiducial survey strategy to other foreseeable surv
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