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\paragraph{}  It is straightforward to show the probability of an eclipse at any depth is $P_e_o $P_e  = r_s + r_g$ but the authors wish to study the probability of an eclipse of \textbf{any specified} magnitude depth. In order to calculate this probability, the function $i(\Delta m)$ must be derived. Although the maximum primary eclipse depth $\Delta m$ is a well defined function of the inclination angle i, iteration is needed to find this inverse function $i(\Delta m)$ 
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The mean values for eclipse probabilities are meaningful only for similar sized systems (radii, luminosities and orbit sizes) so the authors have divided the HR diagram into a number of bins, and each bin is further subdivided according to orbital period and $\Delta m$. The final results are thus histograms giving the mean probability $P_e(\Delta m, P)$ for each HR bin.  The authors have also included another type of probability which is useful when comparing the synthetic population with obserrvations: observations:  namely the fraction of eclipsing binaries relative to all stars (single and binary) in the same HR diagram bin. The authors derive this 'normalized probability' $O_e(\Delta m, P)$