Asked about Probability, and reading an experiment honestly
A test for a condition affecting 1 in 1000 people is 99 percent accurate. You test positive. What is the chance you have it?
An answer that works
About 9 per cent. Take 100,000 people: 100 have the condition and 99 of them test positive, while 99,900 do not and 1 per cent of those, 999 people, test positive anyway. That is 1,098 positives of which 99 are real, so 99 divided by 1,098, about 0.09.
Then they push further
Why is the answer so far from 99 percent?
Because the healthy group is a thousand times larger, so even a 1 per cent error rate on it produces ten times more positives than the ill group can supply. The test did not change; the base rate dominates. At a prevalence of 1 per cent the same test gives exactly 50 per cent, and at 10 per cent it gives 92 per cent.
You test positive a second time, with an independent test. Now what?
Work in odds. Prior odds are 1 to 999, and each positive multiplies by the likelihood ratio, which is sensitivity over false positive rate, 0.99 over 0.01, so 99. After one positive the odds are 0.099, about 9 per cent; after two they are 9.81, about 91 per cent.
Where does this show up in your work?
Any classifier for a rare event. A fraud or abuse model with a 1 per cent false positive rate on a 0.1 per cent base rate sends an analyst ten false alarms per real case, so the queue gets ignored. Precision at the operating point is this same calculation, and it is what users actually experience.
What gets marked up
Reasoning in counts of people out loud is the clearest route under pressure and is marked up over reciting the formula. Saying 99 per cent, or failing to notice that 99 per cent accurate hides two separate rates, is the failure.
Reading an answer is not being able to give one. The section that teaches this is Probability, and reading an experiment honestly, 6 hours of Probability & statistics.
Learn this properlyAlso asked about Probability & statistics