SOLUTION

This is a problem in conditional probability. (It?s also a Bayesian problem, but applying the formula in Bayes Rule only helps to obscure what?s going on.) Consider 100 claims. 10 will be fraudulent, and the system will correctly label 9 of them as “fraud.” 90 claims will be OK, but the system will incorrectly classify 18 (20%) as “fraud.” So a total of 27 claims have been labeled as fraudulent, but only 9 of them, 33%, are actually fraudulent. Translating into probability terms:

Given that a claim has been labeled as fraudulent, the probability is 0.33 that it is actually fraudulent.

Or, in symbols:

P(fraud|”fraud” label) = 0.33 (the vertical bar means “given that.”)