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Calculate Posterior Fraud Probability Using Bayes' Theorem

Last updated: Mar 29, 2026

Quick Overview

This question evaluates understanding of Bayes' theorem, conditional probability, and probabilistic reasoning in the context of classifier metrics (prior probability, true positive rate, false positive rate) for fraud detection.

Posterior Fraud Probability After a Flag

Context

You operate a fraud detection system that flags accounts as suspicious. Define:

F: account is fraudulent
+: system flags the account
Prior fraud rate p = P(F)
True positive rate (TPR) = P(+ | F)
False positive rate (FPR) = P(+ | not F)

Task

Use Bayes’ theorem to derive an expression for the posterior probability that an account is fraudulent after a flag: P(F | +).
Compute P(F | +) for the following example values: p = 1%, TPR = 90%, FPR = 5%.
Briefly interpret the result for decision thresholds (e.g., how the posterior compares to a 10%, 20%, or 50% action threshold).

Solution

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