Rare disease, positive test

Understanding posterior probability from diagnostic test results

This is an easy probability question that tests your ability to apply Bayes' theorem to a real-world medical screening scenario. It's a common warm-up in quant interviews because it catches candidates who confuse sensitivity and specificity with the actual probability of disease given a positive result.

The key insight is that base rate matters. Even with a reasonably accurate test, a positive result on a rare disease is far less conclusive than intuition suggests. You'll need to reason carefully about the relationship between the prevalence of the disease, the test's true-positive and false-positive rates, and the posterior probability. This is a foundational skill for anyone working in risk assessment or Bayesian inference.

• Sensitivity (true-positive rate) vs. specificity (true-negative rate)
• Base rates and their impact on posterior inference
• Bayes' theorem and conditional probability