A positive signal does not tell you its own probability of being correct. You need the event's base rate and the signal's behavior in both possible states.
Consider an original teaching example. Ten percent of components have a fault. A detector flags 80% of faulty components and 20% of sound components. Given a flag, what is the chance of a fault?
Count everyone who could produce the signal
Use 1,000 proportionally allocated components to organize the probabilities. This is a frequency representation, not a claim that a random batch has exactly these counts.
| Component | Flagged | Not flagged | Total |
|---|---|---|---|
| Faulty | 80 | 20 | 100 |
| Sound | 180 | 720 | 900 |
| Total | 260 | 740 | 1,000 |
Among the 260 flags, 80 are faulty. The answer is 80/260 = 4/13, about 30.8%. The detector's 80% sensitivity answers a different question: the probability of a flag given a fault.
In probability form:
P(fault | flag) = (0.10 × 0.80) / (0.10 × 0.80 + 0.90 × 0.20)
The denominator includes both true and false flags. Omitting the sound group is the most consequential mistake in this calculation.
Change the population
Keep the detector's conditional rates fixed but reduce the fault rate to 1%. Among 10,000 proportional cases, 80 faulty and 1,980 sound components are flagged. The posterior becomes 80/2,060 = 4/103, about 3.9%.
The detector has the same conditional behavior. The flag is less persuasive because it is now produced against a much larger sound population. This conclusion assumes those conditional rates really remain unchanged.
A check you can explain aloud
In the original population, a negative signal gives fault probability 20/740 = 1/37. Weighting both posteriors by their signal frequencies recovers the prior:
(0.26)(4/13) + (0.74)(1/37) = 0.08 + 0.02 = 0.10.
That is a useful consistency check. It does not replace defining the observation mechanism. Two reports copied from the same detector do not become independent evidence simply because they appear on separate screens.
Try a variation
Suppose the fault rate is 20%, sensitivity remains 80% and the false-positive rate falls to 10%. The posterior after a flag is 0.16/(0.16+0.08) = 2/3. Explain which changed input raised the answer before quoting the decimal.
For a structured sequence of original problems and solutions, read the Probability & Trading Interview Workbook preview. If your difficulty is choosing the model rather than calculating it, use the probability error log.
Read before choosing
Open the actual pages.
11 sample pages, including complete explanations. No email address or account required.
Open the PDF previewPreview page 4 of 11. Use Enlarge page for a closer view. When the page is focused, use left and right arrows to change pages.
