False Positive Rate

Problem

Suppose there is a quick blood test for an infectious disease, which only 0.5% of the population is infected. It always generates a positive result if you are infected, but it also mistakenly returns positive at the rate of 2%. What is the probability that you are tested positive but not infected?

Answer

Let \(A\) be the event of being infected and \(B\) be the event of tested positive.

\[\begin{align*}  P(A\mid B) &= \frac{P(A\cap B)}{P(B)} \\
           &= \frac{P(A\cap B)}{P(B\cap A) + P(B\cap \bar A)}\\
            &= \frac{P(B\mid A)P(A)}{P(B\mid A)P(A) + P(B\mid \bar A)P(\bar A)} \\
           &= \frac{1\times 0.005}{1\times 0.005 + 0.02\times 0.995} \\
           &\approx 0.20\end{align*}
\]

From the calculation, it follows that even if you are tested positive, you are not infected with a probability of almost 80%.