Level 2: Probability · Problem 27 of 87

Bayes: a negative screening result

A statistics problem from Statisticsdrill. Solve it with one answer box, with a fresh sample on every attempt and the worked steps and spreadsheet function in the answer, and it is checked the moment you submit.

The task

Suppose a screening test checks patients for a condition. 2% of patients are affected. The screen flags 76% of affected patients (sensitivity) and correctly clears 99% of the others (specificity). A patient comes back clear (not flagged). What is the probability that this patient is affected anyway? Round to 4 decimal places.

What this problem teaches

The formula: P(D | -) = P(- | D) P(D) / [ P(- | D) P(D) + P(- | not D) P(not D) ]. A clear result is reassuring when the condition is rare and the sensitivity is high. 1 minus this answer is the negative predictive value.

A hint if you get stuck

P(true | clear) = missed true cases / all clears. All clears = prevalence × (1 - sensitivity) + (1 - prevalence) × specificity.