Natural frequencies
Brainteasers & GuesstimatesWorking a probability problem by counting people out of a round population instead of multiplying decimals together.
Also written: frequency format, natural frequency
Natural frequencies are the same mathematics presented differently. Instead of multiplying 0.04 by 0.90, you take 10,000 borrowers, say that 400 default, and say that 360 of those are flagged. Nothing has changed except that the numbers are now countable objects rather than decimals.
It matters because decimals hide errors and counts do not. A misplaced decimal point in 0.008 is invisible; 76 people where there should be 768 is obvious. Under time pressure and while talking, that difference is worth more than any arithmetic shortcut.
Choose the population so that every intermediate figure is a whole number. If the base rate is 4% and the false positive rate is 8%, start at 10,000 rather than 1,000, because 9,600 times 8% gives 768 rather than 76.8, and a fractional person is a distraction you do not need mid answer.
It also makes the answer explainable. Saying that 1,128 borrowers get flagged and 360 of them default lands with a listener in a way that a posterior probability does not, which is why the format is worth using in a client conversation and not only in an interview.
Worked example
Decimal route: 0.04 times 0.90 is 0.036, and 0.96 times 0.08 is 0.0768, so 0.036 over 0.1128, which is 31.9%.
Frequency route: of 10,000 borrowers, 400 default and 360 are flagged, while 9,600 are sound and 768 are flagged anyway. That is 360 out of 1,128.
Same answer, but the second version can be said out loud, checked by a listener, and corrected mid sentence. The first cannot.