Bayes' Theorem: Medical Testing

Probability · Axiom Academy

EXAMPLE Bayes' Theorem: Medical Testing Understanding false positives and why rare diseases are tricky Excellent work! You've completed this example. Here's what we learned: Counterintuitive result: Even with a 99% accurate test, a positive result only means about 2% chance of actually having the disease! This happens because the disease is so rare. Base rate matters: When prevalence is low (0.1%), false positives vastly outnumber true positives. Out of 100,000 people, only 100 have the disease, but 4,995 healthy people test positive. Bayes' Theorem structure: P(D|+) = P(+|D) × P(D) / P(+). The denominator includes both true positives AND false positives. Law of Total Probability: P(+) = P(+|D)P(D) + P(+|D c )P(D c ) accounts for all ways to get a positive test. Clinical implication: This is why doctors often order follow-up tests before diagnosing rare diseases, even with highly accurate initial tests. Bayes' Theorem helps us update probabilities based on new evidence. Practice with different prevalence rates to see how it affects the results!

This is the written version of the interactive lesson above. See the full Probability course.