Test for a rare disease and Bayes' formula

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A disease affects $0.5\%$ of the population. A test has sensitivity $0.98$ and specificity $0.95$.

(a) Build the tree and compute the probability that a random person tests positive.

(b) Compute the probability that a positive person is actually sick.

(c) Draw a Venn diagram.

Solution

Data: $P(M)=0.005$, $P(\overline M)=0.995$, $P(+\mid M)=0.98$ (sensitivity), $P(+\mid\overline M)=1-0.95=0.05$ (complement of specificity).

(a) Using the tree (disease first, then test outcome) and the total probability formula: $P(+) = P(M)\cdot P(+\mid M) + P(\overline M)\cdot P(+\mid\overline M) = 0.005\cdot0.98 + 0.995\cdot0.05 = 0.0049 + 0.04975 = 0.05465$

(b) By Bayes’ theorem: $P(M\mid+) = \frac{P(+\mid M)\cdot P(M)}{P(+)} = \frac{0.0049}{0.05465} \approx 0.0897$ Only about $9\%$ of positive people are actually sick: because of the disease’s low prevalence, even a highly accurate test produces a large number of false positives compared to true positives.

(c) The Venn diagram shows the small circle $M$ ($0.005$) and the circle $+$ ($0.05465$), with the intersection $M\cap +$ ($0.0049$) very small compared to $+\setminus M$ ($0.04975$).

$\boxed{P(+)\approx0.0547;\ P(M\mid+)\approx0.0897}$