Relativistic spin-orbit and spin-spin couplings has been shown to modify the gravitational waveforms expected from inspiraling binaries with a black hole and a neutron star. As a result inspiral signals may be missed due to significant losses in signal-to-noise ratio, if precession effects are ignored in gravitational-wave searches. We examine the sensitivity of the anticipated loss of signal-to-noise ratio on two factors: the accuracy of the precessing waveforms adopted as the true signals and the expected distributions of spin-orbit tilt angles, given the current understanding of their physical origin. We find that the results obtained using signals generated by approximate techniques are in good agreement with the ones obtained by integrating the 2PN equations. This shows that a complete account of all high-order post-Newtonian effects is usually not necessary for the determination of detection efficiencies. Based on our current astrophysical expectations, large tilt angles are not favored and as a result the decrease in detection rate varies rather slowly with respect to the black hole spin magnitude and is within 20--30\% of the maximum possible values.