Interactive Supplement · Conditional Probability

The Cashier's Dilemma

The detector pen reacted — but its accuracy is only half the story. The other half is how rare counterfeits truly are. Adjust the three dials and watch the probability you actually care about respond.

How rare are counterfeits? 1 in 10,000
The base rate — how common fake bills actually are in circulation.
Pen sensitivity (catches real fakes) 99%
Of every 100 genuine counterfeits, how many make the pen react. (True positive rate.)
False-alarm rate (reacts to good bills) 2%
Of every 100 perfectly legitimate bills, how many trip the pen by mistake.
If the pen reacts, probability the bill is actually fake
—
200 bills that made the pen react
Truly fake (a real catch) Genuine bill (a false alarm)
Group (per 1,000,000 bills)Count
Genuine bills—
Counterfeit bills—
Pen reacts — and bill is fake (true positive)—
Pen reacts — but bill is genuine (false positive)—
Total pen reactions—
The trap: when fakes are vanishingly rare, the false alarms from the enormous pile of genuine bills can swamp the true catches — even with an excellent pen. This is the same mathematics behind medical screening tests, fraud alerts, and spam filters. The reverse probability you actually care about, P(fake | pen reacted), depends just as much on how rare fakes are as on how good the pen is.
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