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  1. 076A forensic accounting screen catches 90% of the companies that are manipulating their earnings and wrongly flags 4% of the clean ones. If 2% of listed companies manipulate, what share of the companies it flags are actually manipulating?Probability and brainteasersHardSell-side equity researchBuy-side equity research

    Try it first

    Before you calculate: the screen catches 90% of manipulators. Roughly what share of its flags are real?

    Show the worked solution

    About 31%: roughly two flags in three are false alarms. Take 10,000 companies. 200 manipulate and the screen catches 180 of them. 9,800 are clean and 4% of them, 392, are flagged anyway. Of 572 flags, 180 are real, which is 31.5%. The false alarms come from the huge clean group, so a rare problem plus a small error rate swamps the true hits.

    Why is the answer not 90%?

    Think of the smoke alarm in a kitchen. It goes off for every real fire, and it also goes off for burnt toast. Fires are rare and toast burns every week, so most of the times the alarm sounds, nothing is on fire. The 90% tells you how the screen treats a manipulator; the question asks how often a flag is right, and that depends on how many clean companies are standing there to be wrongly flagged. Those are two different conditional probabilities, and swapping them is the whole trap.

    Most flags come from the big clean group, not the small guilty one10,000 companies200 manipulate2% of the population9,800 clean98% of the population180 flaggedtrue hitscaught 90%20 missedslip throughmissed 10%392 flaggedfalse alarmswrongly flagged 4%9,408 clearedcorrectly passedcleared 96%All 572 flags, drawn to scale180 real392 false alarmsA flag is right 180 / 572 = 31.5% of the time, not 90%
    Of 10,000 companies, 200 manipulate and the screen flags 180 of them, while 392 of the 9,800 clean companies are flagged by mistake, so only 180 of 572 flags, 31.5%, point at a real manipulator.

    How do you set it up so the arithmetic is easy out loud?

    Pick a round population and count people, not probabilities. With 10,000 companies the 2% who manipulate are 200, and 90% of them, 180, get flagged. The clean 9,800 produce 4% false flags, 392. Once the tree is drawn, the answer is just flagged guilty over all flagged: 180 over 572. This counting method is called natural frequenciesStating a probability problem as counts out of a round population, such as 180 out of 10,000, instead of as percentages. It makes conditional reasoning much easier to follow., and it is far harder to get wrong than juggling Bayes' formula in your head.

    The relationship
    P(M∣F)=0.02×0.900.02×0.90+0.98×0.04=0.0180.0572≈31.5%P(M \mid F) = \frac{0.02 \times 0.90}{0.02 \times 0.90 + 0.98 \times 0.04} = \frac{0.018}{0.0572} \approx 31.5\%
    P(M | F)the chance a flagged company is manipulating
    0.02the share of companies that manipulate, before any screen
    0.90the share of manipulators the screen catches
    0.04the share of clean companies it flags by mistake
    What it says in wordsTrue flags divided by all flags, where all flags include the mistakes made on the large clean group.

    What does this mean for how an analyst uses a screen?

    A flag is evidence, not a verdict. It moves the odds from 2 in 98 to 180 in 392, a jump of 22.5 times, which is exactly 90% divided by 4%. That is why a flag should start a deeper read of the notes to the accounts, never end one. If a second, independent check with the same accuracy also flags the company, the odds multiply again and the chance rises to about 91%. The limitation is the word independent: two ratio screens built on the same receivables line will fail together.

    Where candidates lose it

    Candidates answer 90%, or subtract to get 86%, because the question hands them one big accurate-sounding number and they carry it straight to the answer. The interviewer is checking whether you ask how common the thing is before trusting the test.

    The second loss is getting 31% and not saying what it means. Close with the working point: a flag is where the forensic work starts, and a short thesis resting on one screen alone is two parts false alarm to one part signal.

    What the interviewer asks next

    • If manipulation were 10% of companies, what share of flags would be real?
    • How low must the false flag rate go for half of all flags to be real?
    • Why might two forensic screens not be independent of each other?