Night-side chorus waves are often observed during plasma sheet injections, typically confined around the equator and thus potentially responsible for precipitation of $\lesssim 100 keV$ electrons. However, recent low-altitude observations have revealed the critical role of chorus waves in scattering relativistic electrons on the night-side. This study presents a night-side relativistic electron precipitation event induced by chorus waves at the strong diffusion regime, as observed by the ELFIN CubeSats. Through event-based modeling of wave propagation under ducted or unducted regimes, we show that a density duct is essential for guiding chorus waves to high latitudes with minimal damping, thus enabling the strong night-side relativistic electron precipitation. These findings underline both the existence and the important role of density ducts in facilitating night-side relativistic electron precipitation.