1,000 cases through the detector
Filled dots are real issues, rings are benign. They stream through the gate: over the threshold rings the alarm, under it stays quiet. Watch the alarm bin — at a low base rate it fills with rings.
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Why good detectors still cry wolf
Base-rate neglect is the most durable statistical blind spot there is: we judge an alarm by how good the detector is and forget how rare the thing it's hunting for actually is. A test that's "99% accurate" sounds unimpeachable — but if one case in a thousand is real, then out of a thousand you catch the one real case and raise about ten false alarms alongside it. Nine or ten of every ten alarms are wrong, and no amount of detector quality fixes that; only a higher base rate does. This is the maths behind alert fatigue, vendor detection claims, medical screening, and fraud flags.
The picture makes it concrete. Two bell curves are the detector's score for benign cases and for real ones; the more they overlap, the worse the detector (that's d′, and its AUC). The threshold you drag is the line above which the detector shouts "alarm". Move it left and you catch more real issues but flood the alarm bin with false ones; move it right and the bin goes quiet but real issues sail through. The thousand dots below carry the base rate the curves can't show — that's the number doing the damage.
So the useful move on your own alert channel isn't "buy a better detector". It's: know your base rate first, then set the threshold for the precision you can actually live with — and if nine in ten alarms are false, expect people to stop looking, because they will. Pin a vendor's claimed sensitivity and specificity, then drag the base rate down to yours and watch the promised precision collapse. Sources: Kahneman on base-rate neglect; Gigerenzer's natural-frequency framing (Reckoning with Risk); signal-detection theory.