A fraud alert on a card transaction

Question

Only 0.1\% of card transactions are fraudulent. A detector flags 98\% of true frauds (sensitivity) and clears 97\% of legitimate transactions (specificity), so it false-alarms on 3\% of legitimate ones. A transaction gets flagged. What is the probability that it is actually fraudulent?

Accepted Answer

Use natural frequencies on 100{,}000 transactions: - Fraud: 0.1\% 	imes 100{,}000 = 100; of these, 98\% = 98 are flagged (true positives). - Legitimate: 99{,}900; of these, 3\% = 2{,}997 are flagged (false positives). Among the 98 + 2{,}997 = 3{,}095 flags, only 98 are real fraud: P(	ext{fraud} \mid 	ext{flag}) = \frac{98}{3{,}095} \approx 0.032 = 3.2\%. Despite catching almost every fraud and clearing 97\% of good transactions, roughly 97\% of alerts are false. Fraud is so rare that even a small 3\% false-positive rate applied to the huge legitimate pool buries the true positives. In odds…

A fraud alert on a card transaction

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