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  1. 032A loan scoring model catches 90% of applicants who will go on to default, but it also flags 15% of good borrowers. If 4% of applicants default, what share of flagged applicants actually default?Statistics and forecastingCoreBlackRockWilmington · 2025

    Try it first

    Instinct first: what share of flagged applicants will default?

    Show the worked solution

    Only 20%. Take 1,000 applicants. 40 will default and the model flags 36 of them. 960 are good and the model wrongly flags 15% of them, 144 people. The flagged pile holds 180, and 36 of those default: one in five. The model is good at catching defaulters, but defaulters are so rare that false alarms outnumber them four to one.

    Why is 90% the wrong answer when the model catches 90% of defaulters?

    Picture a smoke alarm that always sounds when there is a fire and also sounds now and then for burnt toast. Because toast burns far more often than houses do, most alarms in a year are toast. A flag's meaning depends on how common the thing it looks for is: when defaulters are 4% of applicants, even a modest false alarm rate on the other 96% produces more wrong flags than right ones. The 90% is the chance a defaulter gets flagged; the question asks the reverse, the chance a flag is a defaulter.

    Count 1,000 applicants through the model: the flagged pile is mostly good borrowers1,000applicants40will default, 4%960good borrowers, 96%36flagged, 90%4missed144flagged, 15%816clearedFlagged pile36 of 180= 20% defaultCleared pile: 4 defaulters in 820, about 0.5%, so a clear is far more informative than a flag.
    Of 1,000 applicants, 36 defaulters and 144 good borrowers are flagged, so only 36 of the 180 flagged applicants, 20%, actually default. The cleared pile is much cleaner: 4 defaulters in 820.

    Why count people instead of using the formula?

    Bayes' rule gives the same answer, but natural frequencies are faster to say and harder to get wrong under pressure. Turn every percentage into a count of people out of a round number, and the answer is simply the flagged defaulters over everyone flagged. The formula version is 0.04 x 0.9 over (0.04 x 0.9 plus 0.96 x 0.15), which is 0.036 over 0.18, or 20%. Offer it as the check after the counts.

    The relationship
    P(D∣F)=P(F∣D) P(D)P(F∣D) P(D)+P(F∣G) P(G)=0.9×0.040.9×0.04+0.15×0.96=0.20P(D \mid F) = \frac{P(F \mid D)\,P(D)}{P(F \mid D)\,P(D) + P(F \mid G)\,P(G)} = \frac{0.9 \times 0.04}{0.9 \times 0.04 + 0.15 \times 0.96} = 0.20
    Dthe applicant will default
    Gthe applicant is a good borrower
    Fthe model flags the applicant
    P(F|D)the catch rate, 90%
    P(F|G)the false alarm rate, 15%
    What it says in wordsThe chance a flag is real is the true flags divided by all flags, true and false.

    Then say what a lender does with it. A flag at 20% is a reason for a closer look, not a rejection. The cleared pile, by contrast, holds only 4 defaulters in 820, about 0.5%, against 4% before the model, so the model is most useful for waving through the safe majority. Cutting the false alarm rate from 15% to 5% would lift the flagged default share to about 43%.

    Where candidates lose it

    Most candidates answer 90% or something close, swapping the chance of a flag given default for the chance of default given a flag. It is the same slip as reading a medical test's accuracy as the chance you are ill.

    Say the base rate first, then count 1,000 people through the tree out loud. The interviewer mostly wants to hear that you know the base rate drives the answer.

    What the interviewer asks next

    • What false alarm rate would make half of all flags real defaulters?
    • If the lender rejects every flagged applicant, how many good borrowers does it turn away per defaulter avoided?
    • How does the answer change for a riskier segment where 15% of applicants default?

    Asked at BlackRock, Generalist, Wilmington, 2025 (Wall Street Oasis): Questions are pretty straightforward and test about statistics models about loan application and loan origination.