The base rate fallacy happens when people ignore how common something is before looking at new evidence. In medical testing, this can make a positive result seem more alarming than it really is. Even a highly accurate test can produce many false positives when the disease is rare.
Understanding base rates helps students interpret test results, screening programs, and risk claims more accurately.
Bayes' theorem gives a clear way to combine the base rate with test accuracy. A positive test result comes from two sources: true positives from people who really have the condition and false positives from people who do not. The probability of having the disease after a positive result depends on both groups, not just on the test's sensitivity.
This is why a test that is 99% accurate can still give a surprisingly low chance of disease when the base rate is very small.
Key Facts
- Base rate = the prior probability that an event is true before new evidence is considered.
- Sensitivity = P(positive test | disease), the chance a test correctly detects disease.
- Specificity = P(negative test | no disease), the chance a test correctly clears a healthy person.
- False positive rate = 1 - specificity.
- Bayes' theorem: P(disease | positive) = P(positive | disease)P(disease) / P(positive).
- For testing: P(disease | positive) = sensitivity x base rate / [sensitivity x base rate + false positive rate x (1 - base rate)].
Vocabulary
- Base rate
- The overall frequency or probability of a condition in the population before considering a specific test result.
- Prior probability
- The probability assigned to an event before using new evidence or data.
- Sensitivity
- The probability that a test gives a positive result when the person truly has the condition.
- Specificity
- The probability that a test gives a negative result when the person truly does not have the condition.
- Posterior probability
- The updated probability of an event after combining the prior probability with new evidence.
Common Mistakes to Avoid
- Confusing P(positive | disease) with P(disease | positive). These are different conditional probabilities, and Bayes' theorem is needed to convert one into the other.
- Ignoring the number of healthy people in the population. When a disease is rare, even a small false positive rate can create many false positives.
- Calling a 99% accurate test a 99% chance of disease after a positive result. Test accuracy alone does not determine the probability because the base rate also matters.
- Using percentages without converting them consistently. Mixing 1%, 0.01, and 1 in the same calculation can lead to errors by factors of 100.
Practice Questions
- 1 A disease affects 1% of a population. A test has 95% sensitivity and 90% specificity. If a person tests positive, what is P(disease | positive)?
- 2 In a group of 10,000 people, 2% have a condition. A test has 99% sensitivity and a 5% false positive rate. How many true positives and false positives do you expect, and what fraction of positive tests are true positives?
- 3 A rare disease screening test has very high sensitivity but only moderate specificity. Explain why most positive results might still come from healthy people.