A flagged transaction, is it really fraud?

Question

A payments system sees that 5\% of transactions are fraudulent and 95\% are legitimate. A detection rule fires on 90\% of fraudulent transactions but also on 10\% of legitimate ones. A transaction just got flagged. What is the probability it is actually fraud?

Accepted Answer

Picture 1{,}000 transactions: 50 fraud and 950 legitimate. Count the flags. - Fraud flagged: 50 	imes 0.90 = 45. - Legitimate flagged: 950 	imes 0.10 = 95. Of the 45 + 95 = 140 flagged transactions, 45 are truly fraud: P(	ext{fraud} \mid 	ext{flag}) = \frac{45}{140} = \frac{9}{28} \approx 0.321. Even with a rule that catches 90\% of fraud, most flags are false alarms because legitimate transactions vastly outnumber fraudulent ones. Takeaway. When the base rate is low, even a good detector produces mostly false positives; the rare class simply cannot outvote the common one.

A flagged transaction, is it really fraud?

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