A Positive Test

A disease occurs with probability $0.5\%$ in the population. A test returns positive for sure if you have the disease, and gives a false positive $7\%$ of the time if you do not. A random stranger tests positive. What is the probability that the stranger has the disease?

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#Bayes with a rare disease

\PP(D \mid +) = \frac{\PP(+ \mid D)\,\PP(D)}{\PP(+ \mid D)\,\PP(D) + \PP(+ \mid \neg D)\,\PP(\neg D)} = \frac{1 \cdot 0.005}{1 \cdot 0.005 + 0.07 \cdot 0.995}. \tag{1}

#Crunch the numbers

= \frac{0.005}{0.005 + 0.06965} = \frac{0.005}{0.07465} \approx 0.067. \tag{2}

Out of ten thousand people, fifty carry the disease and all test positive, but seven percent of the 9950 healthy add about 697 false alarms. The genuinely sick are a thin sliver of all positives, so a positive stranger is sick only about 6.7 percent of the time.

#Result

The probability is about $6.7\%$ . Even a test that never misses a true case leaves a positive stranger far more likely healthy than sick, because the $7\%$ false positives drawn from the $99.5\%$ healthy majority swamp the tiny $0.5\%$ who are genuinely ill. This is the base-rate fallacy.