Probability: Real-World Scenarios (8)

Question

A medical test for a rare disease has 95% accuracy: it correctly identifies 95% of diseased patients (true positive rate) and correctly identifies 95% of healthy patients (true negative rate). The disease affects 1 in 1000 people. If a randomly selected person tests positive, what is the probability that they actually have the disease?

Solution — Step by Step

Why This Works

This is the classic “base rate fallacy” example. Most people guess that a 95%-accurate test gives 95% confidence in the result. But when the disease is rare, false positives swamp true positives. Out of 1000 people, 1 has the disease and tests positive correctly. But 50 healthy people (5% of 999) also test positive falsely. So out of 51 positive results, only 1 is a true case.

This is why screening for rare conditions can be misleading without follow-up tests, and why understanding conditional probability matters for medicine, security screening, and even spam filters.

Alternative Method

Use a “natural frequency tree” — easier for many students. Imagine 100,000 people:

• 100 have the disease, 99,900 don’t.
• Of 100 diseased: 95 test positive, 5 test negative.
• Of 99,900 healthy: 4995 test positive (5% false positive), 94,905 test negative.

Total positives: $95 + 4995 = 5090$. Of these, only 95 actually have the disease.

$P(D \mid T) = 95/5090 \approx 0.0187$. Same answer, no formula needed.

Final answer: $P(D \mid T) \approx 1.87\%$.