(\[Rdet\]) is transformed into an integral over merger time via $dz_m
= \frac{dz_m}{dt_m} dt_m = H_0 (1+z)E(z) dt_m$ [@hogg]. Thus the detection rate integral can be represented as a Monte Carlo sum over all simulated binaries: \[eq:rate\] R\_[det]{}= p\_[det]{} t , where $\Delta M_f$ is the total star-forming mass that was simulated in the Monte Carlo to represent the time bin $\Delta t$, all terms but the first are computed at the merger redshift of the simulated source.

The detection probability for a given source at its merger redshift $p_{\rm det} (z, m_1, m_2)$ is simply the fraction of sources of a given mass located at the given redshift that exceed the detectability threshold in SNR, assuming that sources are uniformly distributed in sky location and orbital orientation. If a single detector with an SNR threshold (e.g., $\rho_{\rm thr}=8$) is used as a proxy for detectability, the detection probability can be expressed as a cumulative distribution function on the projection parameter $w$. In the Appendix, $w$ is defined such that $w=0$ when the detector has no response to the gravitational wave, and $w=1$ for an optimally located and oriented (face-on and directly overhead) binary. The detection probability is p\_[det]{} = 