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 _______
HIV Test Positive: Actual Probability Only 49% Despite 95% Accuracy
Bayes Theorem reveals why test accuracy ≠ probability of disease
📊 0.49 (49%)
Patient has only 49% chance of being HIV+ despite positive test
Patient has only 49% chance of being HIV+ despite positive test
Population of 1000: Real Test Results
| Status | Count | Test+ (95%/11%) | Test- |
|---|---|---|---|
| HIV+ | 100 | 95 TP | 5 FN |
| HIV- | 900 | 99 FP | 801 TN |
| TOTAL | 1000 | 194 POS | 806 NEG |
PPV = 95/194 = 0.49 (49%)
Bayes’ Theorem Formula
P(HIV+|+) = [P(+|HIV+) × P(HIV+)] / P(+)
Step-by-Step:
P(+) = (0.95 × 0.10) + (0.11 × 0.90) = 0.095 + 0.099 = 0.194
P(HIV+|+) = 0.095 / 0.194 = 0.4897 ≈ 0.49
Diagnostic Test Metrics
Sensitivity
P(+|HIV+)
95%
Specificity
P(-|HIV-)
89%
Positive Predictive Value
P(HIV+|+)
49%
False Positive Rate
P(+|HIV-)
11%
Why 95% Accurate ≠ 95% HIV+
- Low prevalence (10%): 900 healthy vs 100 infected
- 11% false positives: 99 FP vs 95 TP
- Result: 51% of positives (99/194) are FALSE
Clinical Reality Check
⚠️ 49% PPV Means:
- 1 in 2 positive tests are false alarms
- Confirmatory testing ESSENTIAL (Western Blot, PCR)
- Low prevalence destroys PPV even with good tests
Exam Strategy: Always Use Bayes
- Construct 1000-person table (easy visualization)
- TP = Sensitivity × Prevalence
- FP = (1-Specificity) × (1-Prevalence)
- PPV = TP / (TP + FP)
- Low prevalence + imperfect test = low PPV
