Business Statistics
Bayes and default flags
A screening tool flags 80% of future defaults. If a borrower is flagged, are they likely to default?
Compare true and false alerts in the population. When defaults are uncommon, false alerts can outnumber true ones.
P(D|F) = P(D)P(F|D) ÷ [P(D)P(F|D) + P(not D)P(F|not D)]
Out of 1,000 borrowers, 50 × 80% gives 40 true flags and 950 × 10% gives 95 false flags. Divide 40 by the 135 total flags.
Of 1,000 similar borrowers, expect 50 defaults at a 5% base rate. At 80% sensitivity, expect 40 true flags. At a 10% false-positive rate, expect 95 false flags. The share of flagged borrowers expected to default is 40 ÷ 135 ≈ 29.63%.
Out of 1,000 borrowers, 50 × 80% gives 40 true flags and 950 × 10% gives 95 false flags. Divide 40 by the 135 total flags.
If the false-positive rate falls to zero and the true-positive rate stays positive, what share of flagged borrowers default in the model?