Bayesian Disease Probability

A test for a rare disease (prevalence 1 in 10,000) has 99% sensitivity and 99% specificity. A randomly selected person tests positive. The approximate probability they actually have the disease is:

This is a classic **base-rate fallacy** / Bayesian reasoning problem. We apply entity["scientific_concept","Bayes' theorem","probability updating rule"]. --- ### Step 1: Set up the probabilities - Prevalence (prior): \( P(D) = 1/10{,}000 = 0.0001 \) - Sensitivity: \( P(+ \mid D) = 0.99 \) - Specificity: \( P(- \mid \text{no }D) = 0.99 \Rightarrow P(+ \mid \text{no }D) = 0.01 \) --- ### Step 2: Apply Bayes' theorem genui{"math_block_widget_always_prefetch_v2":{"content":"P(D\\mid +)=\\frac{P(+\\mid D)P(D)}{P(+\\mid D)P(D)+P(+\\mid \\neg D)P(\\neg D)}"}} Substitute values: \[ P(D \mid +) = \frac{0.99 \cdot 0.0001}{0.99 \cdot 0.0001 + 0.01 \cdot 0.9999} \] --- ### Step 3: Compute approximately - Numerator ≈ \( 0.000099 \) - Denominator ≈ \( 0.000099 + 0.009999 \approx 0.010098 \) \[ P(D \mid +) \approx \frac{0.000099}{0.010098} \approx 0.0098 \] --- ### Final Answer **≈ 1% (about 0.98%)** --- ### Key Insight Even with a highly accurate test (99% sensitivity & specificity), the disease is so rare that **false positives vastly outnumber true positives**, making the actual probability of having the disease after a positive test surprisingly low.