A screening test for a condition present in 1 in 1000 people has 99% sensitivity and 95% specificity. A randomly chosen person tests positive. What is the probability they have the condition?

A screening test for a condition present in 1 in 1000 people has 99% sensitivity and 95% specificity. A randomly chosen person tests positive. What is the probability they have the condition?

Approach: Write Bayes with the two branches, a positive from a true case and a positive from a false alarm, then compare the sizes of the two terms rather than trusting the headline accuracy.

11/566. Take 100000 people. The base rate puts 100 of them in the condition group and the 99% sensitivity gives 99 true positives. The other 99900 produce 4995 false positives at a 5% false alarm rate. The posterior probability is 99/(99 + 4995) = 11/566. That is about 1.94%, so the false positives outnumber the true positives 50 to 1: the prevalence is 1/1000 while the false alarm rate is 1/20, and a test's headline accuracy carries no information until it is weighed against the prevalence.

Follow-up: How large must the specificity be for a single positive result to make the condition more likely than its absence?

Key concepts: base rate, sensitivity, false positives, posterior probability.