Q.10
10% of the population in a town is HIV+.
A new diagnostic kit for HIV detection is available: this kit correctly
identifies HIV+ individuals 95% of the time, and HIV–
individuals 89% of the time.
A particular patient is tested using this kit and is found to be positive.
The probability that the individual is actually positive is ________.
This classic probability problem demonstrates how diagnostic test accuracy doesn’t equal the actual chance of having a condition due to prevalence rates.
Problem Breakdown
10% of the town population is HIV+ (P(HIV+) = 0.10, so P(HIV-) = 0.90). The kit has:
- 95% sensitivity (true positive rate: P(+|HIV+) = 0.95)
- 89% specificity (true negative rate: P(-|HIV-) = 0.89, so false positive rate P(+|HIV-) = 0.11)
We need to find P(HIV+|+) for a positive test result.
Bayes’ Theorem Formula
P(HIV+|+) = P(+|HIV+)×P(HIV+)/P(+)
where P(+) = total probability of positive test = true positives + false positives
Step-by-Step Calculation
Assume 100-person population for clarity:
- HIV+ individuals: 10 × 0.95 = 9.5 true positives
- HIV- individuals: 90 × 0.11 = 9.9 false positives
- Total positives: 9.5 + 9.9 = 19.4
Common Mistakes Explained
This problem has no multiple-choice options listed, but students often make these errors:
- ❌ Mistaking sensitivity (95%) for P(HIV+|+) – ignores false positives
- ❌ Ignoring base rate (10% prevalence) – base rate fallacy
- ❌ Overlooking false positive rate (11%) – doesn’t calculate total P(+)
- ✅ Correct approach: Full Bayes theorem application gives 49%
Key Takeaway
A 95% accurate test doesn’t mean 95% chance of disease. When prevalence is low (10%), false positives nearly equal true positives, making the actual probability closer to 49%. Always use Bayes’ theorem for diagnostic test interpretation.
