Statistics.com published a problem of the week: The rate of residential insurance fraud is 10% (one out of ten claims is fraudulent). A consultant has proposed a machine learning system to review claims and classify them as fraud or no-fraud. The system is 90% effective in detecting the fraudulent claims, but only 80% effective in correctly classifying the non-fraud claims (it mistakenly labels one in five as “fraud”). If the system classifies a claim as fraudulent, what is the probability that it really is fraudulent?
My peer and I both came up with the same answer independently and it doesn't match the published 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 72 (80%) as “fraud.” So a total of 81 claims have been labeled as fraudulent, but only 9 of them, 11%, are actually fraudulent.
Who was right