Question

A certain disease has a prevalence of \(1\%\) in the population. A diagnostic test for the disease has a sensitivity of \(98\%\) and a specificity of \(95\%\). If a person from this population tests positive, then the probability that they actually have the disease is...

• A. \(98\%\)
• B. \(95\%\)
• C. \(16.5\%\)
• D. \(1\%\)

Hint:

Apply Bayes' theorem with prevalence 1 percent, sensitivity 98 percent and false positive rate 5 percent.

Answer 1

The Correct Option is C

Step 1: Plan:
Use a population of $10{,}000$ people and count.

Step 2: Count:
Diseased: $100$. Of these, $98$ test positive. Healthy: $9900$. Of these, $5\%$ test positive, i.e. $495$.
Total positives: $98 + 495 = 593$. The share who really have the disease: $\frac{98}{593} \approx 0.165 = 16.5\%$.

Final Answer:
The probability that the person has the disease is about $16.5\%$, option (C). \[ \boxed{16.5\%} \]