Bayes' Theorem
Bayes' theorem updates what you believe when new evidence arrives. See why a positive result from a 99% accurate test can still probably be wrong.
Real lesson card · Page 1 of 3
Why a positive test can mean 17%
Prior probability
The base rate of something before you see any test result — how common it is across the whole population.Example
If a disease affects 1 in 100 people, the prior probability that a random person has it is 1%.Setup
A disease affects 1% of people. A test catches 99% of real cases and gives a false positive 5% of the time. You test positive. What’s the chance you actually have it?- 1Imagine 10,000 people: 100 are sick, 9,900 are healthy.WhyWorking in counts makes the base rate visible.
- 2True positives: 99 of the 100 sick test positive. False positives: 5% of 9,900 = 495 healthy people test positive.WhyBoth groups produce positives, and the healthy group is huge.
- 3Chance you’re sick = 99 divided by (99 + 495) = 99/594.WhyOnly the true positives, out of all positives, are real cases.
Takeaway
About 17%. The test is accurate, but the disease is so rare that false positives outnumber true ones five to one.Myth
A 99% accurate test means a positive result is 99% likely to be right.Reality
Accuracy interacts with the base rate. When the condition is rare, most positives come from the large healthy group, so a single positive can still be probably wrong.Recall check from the same lesson
If a disease is rare, a positive result from a highly accurate test can still mean you probably don't have it.
Review the explanation
Answer: True. The many healthy people generate enough false positives to outnumber the few true positives, so most positive results are false when the base rate is low.
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