Step 1: Apply Bayes’ theorem.
Calculate the Positive Predictive Value (PPV), which is $P(\text{Disease}|\text{Test Positive})$.
Bayes’ theorem states:
\[P(D|+) = \frac{P(+|D) \cdot P(D)}{P(+|D) \cdot P(D) + P(+|\overline{D}) \cdot P(\overline{D})}\]
Step 2: Substitute values.
- $P(+|D) =$ Sensitivity $= 0.89$
- $P(+|\overline{D}) =$ False Positive Rate $= 0.01$
- $P(D) =$ Prevalence $= 0.002$
- $P(\overline{D}) = 1 - 0.002 = 0.998$
Step 3: Calculate numerator and denominator.
Numerator = $0.89 \times 0.002 = 0.00178$
Denominator = $0.00178 + (0.01 \times 0.998) = 0.00178 + 0.00998 = 0.01176$
Step 4: Final probability.
\[P(D|+) = \frac{0.00178}{0.01176} \approx 0.1514 = 15.14%\]
Step 5: Conclusion.
The probability of a person having the disease given a positive test result is 15.14%.