Count people, not percentages

A test with 99% sensitivity and 99% specificity can still be wrong half the time. When the condition is rare, the healthy crowd is huge, so even a 1% false-alarm rate produces many false positives. Work with counts of people out of 1,000 or 10,000 instead of percentages: the base rate becomes visible instead of forgettable.

Words you need

Base rate: How common something is before you see any evidence.e.g. A disease that 1 in 100 people have has a 1% base rate.

Why it works

A positive result comes from two groups, the truly sick and the falsely alarmed. The posterior is the sick share of all positives.

P(sick | positive) = P(positive | sick) x P(sick) / P(positive)

Spot it: A question gives test accuracy plus a prevalence (base rate).

Technique reference and sources

Imagine 10,000 concrete people walking through the test. With 1% sick, that is 100 sick and 9,900 healthy. A test with 99% sensitivity finds 99 of the sick but, with 99% specificity, falsely alarms on 99 of the healthy. Among 198 positive results, only 99 are sick: exactly half. Percentages hide the 9,900; people make them impossible to ignore.

When this method fits

A question gives test accuracy plus a prevalence (base rate).
The condition is rare, so the healthy group dwarfs the sick group.
You catch yourself reading '99% accurate' as '99% chance of illness'.
An interview asks for P(condition | positive result).

Start from a concrete crowd. Pick 1,000, 10,000, or 100,000 people, whichever keeps every group whole.
Split sick and healthy. With 1% of 10,000 sick: 100 sick, 9,900 healthy.
Count positives in each group. True positives: 99% of 100 = 99. False positives: 1% of 9,900 = 99.
Divide sick positives by all positives. 99/(99 + 99) = 99/198 = 0.5. Report 50%, not 99%.

Symmetric 99/99 split: When true positives equal false positives, the posterior is exactly one half.

99 sick positives vs 99 false alarms: 0.5
9 sick positives vs 9 false alarms: 0.5

Posterior odds and likelihood ratios: Convert the prior to odds, multiply by the positive likelihood ratio (sensitivity divided by false-alarm rate), and convert back. Exact: prior odds 1:99 times 0.99/0.01 = 99 gives posterior odds 99:99 = 1:1, hence 0.5.

1% sick, 99/99 rates: prior 1:99 x 99 = 1:1, posterior 0.5

Rarer condition, lower posterior: Shrink the sick group tenfold while accuracy stays fixed and the posterior collapses.

1% sick: posterior 0.5
0.1% sick: posterior about 0.09

Common traps

Reading P(positive | sick) = 0.99 as P(sick | positive) = 0.99. The test direction and the question direction differ. Count positives in both groups and divide.

Dropping the base rate and comparing 99% against nothing. Always split the crowd first. The healthy group supplies the false positives.

Multiplying percentages in the head and losing the false positives. Convert to people out of 10,000. Two positive groups appear; add both before dividing.

Another example: Chip defect test: 2% of 1,000 chips defective, test catches 90% with a 10% false-alarm rate

Defective: 2% of 1,000 = 20. Good: 980.
True positives: 90% of 20 = 18. False positives: 10% of 980 = 98.
Total positives: 18 + 98 = 116.
Posterior: 18/116, about 0.16.

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Harvard Stat 110: Probability (Joe Blitzstein)
Free course site with videos and the open companion book Introduction to Probability; the topic spine for counting, Bayes, and linearity. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.