17 / 18 · Reason beyond certainty
Count before you trust the percentage
Distinguish P(flag | defect) from P(defect | flag) using natural frequencies.
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Go to practice ↓A question to keep in mind
A detector finds 90% of defects. Is a flagged item 90% likely to be defective?
A conditional probability P(A | B) describes the share of A cases within the B group. Reversing A and B changes the denominator. A detector's sensitivity is about flagged items among defects; what a user wants after a flag is about defects among flagged items.
Teaching model: exactly 1,000 items, 10 defective. The detector flags 90% of the defective items and 10% of the 990 nondefective items. It flags 9 real defects and 99 nondefects, for 108 flags total. Only 9/108, about 8.33%, of flags are defects.
The base rate matters because a small false-positive rate applied to a large group can produce many flags. These stipulated counts are not measured performance of a real product. In real use the rates and prevalence are estimates that can shift between populations.
Work through an example
- Split the population first: 10 defective + 990 nondefective = 1,000. Apply each rate only to its own group.
- True flags: 10 × 0.9 = 9. False flags: 990 × 0.1 = 99. The post-flag denominator is 108, not 10 or 1,000.
- Bayes' rule records the same accounting: P(D | F) = P(F | D)P(D) / P(F). Counting groups makes the denominator visible.
Your turn
0 / 3A new batch: 1,000 items, 100 defective, 90% sensitivity, 10% false-positive rate. Calculate the flag groups.
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True flags / false flags / all flags / percentage: 90 / 90 / 180 / 50%
There are 90 true flags and 90 false flags, 180 flags total. Half of the flags are defective: 50%. The detector rates stayed fixed; the base rate changed.
To calculate P(defect | flag), which group belongs in the denominator?
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All flagged items
Always name the population after the conditioning bar before doing arithmetic.
A low-prevalence deployment has many false flags. What is a reasonable response?
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Use a confirmation step and compare error costs and deployment-specific rates.
A probability does not decide an action by itself. Costs, benefits, and further evidence matter.
Bring it back to your own work
Find a percentage in a claim you care about. Write its numerator, denominator, and whether either group was selected before measurement.
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