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 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...
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Apply Bayes' theorem with prevalence 1 percent, sensitivity 98 percent and false positive rate 5 percent.
Updated On: Oct 1, 2026
- \(98\%\)
- \(95\%\)
- \(16.5\%\)
- \(1\%\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
Let \(D\) be having the disease and \(+\) a positive test. \(P(D) = 0.01\), \(P(+|D) = 0.98\) (sensitivity), and \(P(+|D^c) = 1 - 0.95 = 0.05\) (specificity is \(0.95\)).
Step 2: Bayes' theorem:
\[ P(D|+) = \frac{P(+|D)P(D)}{P(+|D)P(D) + P(+|D^c)P(D^c)} \]
\[ = \frac{0.98\times0.01}{0.98\times0.01 + 0.05\times0.99} = \frac{0.0098}{0.0098+0.0495} = \frac{0.0098}{0.0593} \approx 0.165 \]
Step 3: Interpretation:
The answer is about \(16.5\%\). It is small because the disease is rare, so false positives from the healthy majority outnumber true positives. Option A (\(98\%\)) is the sensitivity, and B (\(95\%\)) is the specificity. D (\(1\%\)) is the prevalence.
Final Answer:
The probability that the person has the disease is about \(16.5\%\), option (C). \[ \boxed{16.5\%} \]
Let \(D\) be having the disease and \(+\) a positive test. \(P(D) = 0.01\), \(P(+|D) = 0.98\) (sensitivity), and \(P(+|D^c) = 1 - 0.95 = 0.05\) (specificity is \(0.95\)).
Step 2: Bayes' theorem:
\[ P(D|+) = \frac{P(+|D)P(D)}{P(+|D)P(D) + P(+|D^c)P(D^c)} \]
\[ = \frac{0.98\times0.01}{0.98\times0.01 + 0.05\times0.99} = \frac{0.0098}{0.0098+0.0495} = \frac{0.0098}{0.0593} \approx 0.165 \]
Step 3: Interpretation:
The answer is about \(16.5\%\). It is small because the disease is rare, so false positives from the healthy majority outnumber true positives. Option A (\(98\%\)) is the sensitivity, and B (\(95\%\)) is the specificity. D (\(1\%\)) is the prevalence.
Final Answer:
The probability that the person has the disease is about \(16.5\%\), option (C). \[ \boxed{16.5\%} \]
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