Where you sit on the detection curve is decided by consequences, not by preference. Write both costs down. A false alarm costs an evacuation, a dispatched crew, lost production in a plant — and, far worse and much slower to show up, it teaches people to ignore the system or to disable the device, at which point the detection rate you paid for becomes zero. A miss costs a fire that grows unattended through the period when a single person with an extinguisher could have ended it, and in a sleeping occupancy or a remote landscape it can cost everything. Weighed by how often each case arises, the comparison is an expected cost, \( C = p_{\text{miss}} C_{\text{miss}} + p_{\text{fa}} C_{\text{fa}} \), where \( p_{\text{miss}} \) and \( p_{\text{fa}} \) are how often a missed fire and a false alarm actually occur at that threshold (each error rate multiplied by how often fires and non-fires arise, not the bare rates), and the threshold you want is the one that makes it smallest. In practice the numbers are rough, which is fine: the useful output is usually the direction — that a wildland camera network watching an unpopulated valley should sit far more sensitive than a smoke detector above a stove. After this Concept you can justify an alarm threshold from the costs on both sides rather than accepting a default.
The ROC (receiver operating characteristic) curve is a tool for evaluating a binary classifier's tradeoff between true positive rate and false positive rate across all possible decision thresholds, w…