of exoEarth candidates required to place 1, 2, and 3$\sigma$ constraints on $\eta_x < \eta_x'$ as a function of $\eta_x'$. Alternatively, one could use Figure \[confidence\_curves\] to interpret the possible constraint on $\eta_x'$ given an expected number of exoEarth candidates.

![The required number of exoEarth candidates, $N_{\rm EC}$, to constrain $\eta_x < \eta_x'$ at a given confidence level, where $\eta_x$ represents the statistical fraction of exoEarth candidates that exhibit a desired feature (e.g., water). \[confidence\_curves\]](Required_Ncandidates_vs_eta_for_null_result.eps){width="4in"}

We posit that a reasonably compelling limit is $\eta_x < 0.1$, whether $\eta_x = \eta_{\rm H_2 O}$ or $\eta_x = \eta_{\rm life}$. *Thus, to place a 3$\sigma$ constraint that $\eta_x < 0.1$, we require approximately 55 exoEarth candidates.*

Figure \[confidence\_curves\] equivalently states the number of exoEarth candidates required to detect the desired feature in at least one candidate at the given level of confidence. *Given 55 exoEarth candidates, a mission would have a $99.7\%$ chance of detecting at least one exoEarth with water if $\eta_{\rm H_2 O} \ge 0.1$ and $\epsilon \approx 1$*. The most probable number of detections is of course equal to $\eta_x \times N_{\rm EC}$, as shown in the sample binomial distribution in Figure \[sample\_binomial\], for which we assume $\eta_x = 0.1$ 