Question

A screening test for a disease shows a positive test result in 95% of all cases when the disease is actually present and in 15% of all cases when it is not. When the test was administered, 17.4% of the results were positive. What is the prevalence of the disease?

Answer 1

Let \(T\) = test is positive
\(D \) = disease present
\(\overline{D}\)= disease absent

You are given:
\(P(T|D)=0.95\\
P(T|\overline{D})=0.15 \\
P(T)=0.174\)

You are asked: to find \(P(D)\), which is the prelevance of the disease

So:
\(P(T)=P(T \cap D) +P(T \cap \overline{D})\), by law of total probability since \(D \) and \(\overline{D}\) form a partition

\(P(T)=P(T|D)P(D) +P(T|\overline{D})P(\overline{D})\\
0.174=0.95(P(D))+0.15(1-P(D))\\
0.174=0.95P(D)+0.15 - 0.15P(D)\\
0.024 =0.8P(D)\\
P(D)=0.024/0.8=0.03=3\%\)