= Detection probability
{title2=$P_D$}
The \b[detection probability] $P_D$ is the <conditional probability> that a specified procedure declares a detection when the target or signal condition is present. With $D$ the detection event and $H_1$ the signal-present condition,
$$
P_D=P(D\mid H_1).
$$
It depends on the signal, <noise> model, measurement conditions and decision rule. In a <statistical hypothesis test>, it is the <statistical power> at the specified alternative. Its complement is the probability of missing that signal; the <false positive rate> instead conditions on signal absence.<NIST and SEMATECH e-Handbook of Statistical Methods>[§7.1.3, What are statistical tests?: significance level and errors of the second kind]{ref}
High $P_D$ does not by itself imply that most detections are real. Suppose, in an invented collection of10,000 independent trial conditions,100 contain a target. A procedure detecting90% of targets and falsely alarming in5% of the9,900 target-absent conditions has expected counts90 real detections and495 false alarms. Only $90/(90+495)\approx15.4\%$ of its alarms then correspond to targets. These are expected counts under the example model, not measured detector performance.
More generally, if $\pi=P(H_1)$ and $P_{FA}=P(D\mid H_0)$ with $H_0$ meaning absence, <Bayes' theorem> gives
$$
P(H_1\mid D)=\frac{P_D\pi}{P_D\pi+P_{FA}(1-\pi)}.
$$
The order of conditioning is therefore essential. Changing a decision threshold commonly trades detection against false alarms; a <receiver operating characteristic> displays that tradeoff.
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