# Solution: Stats and Probability – Interpreting Posterior Probability for a Rare Disease ## **Step 1: Understanding the Problem** The question relates to interpreting the **posterior probability** that an individual actually has a rare disease, given a positive result from a diagnostic test. The test has known **sensitivity** (true positive rate) and **specificity** (true negative rate), but the disease is rare (very low base rate or prevalence in the population). --- ## **Step 2: Key Concepts** - **Sensitivity:** Probability that the test is positive given the person has the disease, i.e., \( P(\text{Test} + | \text{Disease}) \). - **Specificity:** Probability that the test is negative given the person does not have the disease, i.e., \( P(\text{Test} - | \text{No Disease}) \). - **Base Rate (Prevalence):** Probability that a randomly selected person from the population has the disease, i.e., \( P(\text{Disease}) \). - **Posterior Probability (Positive Predictive Value):** Probability that a person has the disease given a positive test result, i.e., \( P(\text{Disease} | \text{Test} +) \). --- ## **Step 3: Applying Bayes' Theorem** Bayes' theorem gives: \[ P(\text{Disease} | \text{Test} +) = \frac{P(\text{Test} + | \text{Disease}) \cdot P(\text{Disease})}{P(\text{Test} +)} \] Where \[ P(\text{Test} +) = P(\text{Test} + | \text{Disease}) \cdot P(\text{Disease}) + P(\text{Test} + | \text{No Disease}) \cdot P(\text{No Disease}) \] - \( P(\text{Test} + | \text{No Disease}) = 1 - \text{Specificity} \) (false positive rate) --- ## **Step 4: Reasoning with a Very Low Base Rate** When the base rate \( P(\text{Disease}) \) is very low: - The numerator \( P(\text{Test} + | \text{Disease}) \cdot P(\text{Disease}) \) is very small. - The denominator is dominated by \( P(\text{Test} + | \text{No Disease}) \cdot P(\text{No Disease}) \), since \( P(\text{No Disease}) \) is close to 1 and even a small false positive rate can create a large number of false positives. **Interpretation:** Even with a highly sensitive and specific test, a positive result in a population with very low prevalence will often be a false positive. Thus, the posterior probability \( P(\text{Disease} | \text{Test} +) \) is much lower than intuition might suggest. --- ## **Step 5: Evaluating the Answer Choices** - **a) It will be close to 100%.** This is incorrect. A low base rate means most positives are false positives. - **b) It will be close to 50%.** This is not generally true for very rare diseases; the probability is usually much lower. - **c) It will be less than 50%.** This is correct for very low disease prevalence, even with a good test. - **d) It depends on the specificity but not the sensitivity.** Both sensitivity and specificity influence the result, so this is not accurate. --- ## **Step 6: Explanation and Final Answer** **Explanation:** For rare diseases, even a good diagnostic test will yield a low probability of actually having the disease after a positive test due to the overwhelming number of false positives from the much larger non-diseased population. Both specificity and sensitivity matter, but the rarity of the disease dominates the calculation. --- ## **Final Answer** **The most accurate statement is:** > **c) It will be less than 50%.** **Summary:** The posterior probability of disease given a positive test result is typically less than 50% when the disease is very rare, even with a highly accurate test, because false positives outnumber true positives in such scenarios.

# Solution: Stats and Probability – Interpreting Posterior Probability for a Rare Disease ## **Step 1: Understanding the Problem** The problem involves determining the posterior probability that an individual has a rare disease based on a positive result from a diagnostic test. This test is characterized by known **sensitivity** and **specificity**, while the disease itself has a very low prevalence in the population. --- ## **Step 2: Key Concepts** - **Sensitivity**: The probability that the test correctly identifies a diseased individual, denoted as \( P(\text{Test} + | \text{Disease}) \). - **Specificity**: The probability that the test correctly identifies a non-diseased individual, denoted as \( P(\text{Test} - | \text{No Disease}) \). - **Base Rate (Prevalence)**: The probability that a randomly selected individual has the disease, denoted as \( P(\text{Disease}) \). - **Posterior Probability (Positive Predictive Value)**: The probability that an individual has the disease given a positive test result, denoted as \( P(\text{Disease} | \text{Test} +) \). --- ## **Step 3: Applying Bayes' Theorem** Bayes' theorem aids in calculating the posterior probability: \[ P(\text{Disease} | \text{Test} +) = \frac{P(\text{Test} + | \text{Disease}) \cdot P(\text{Disease})}{P(\text{Test} +)} \] Where the total probability of a positive test result is calculated as: \[ P(\text{Test} +) = P(\text{Test} + | \text{Disease}) \cdot P(\text{Disease}) + P(\text{Test} + | \text{No Disease}) \cdot P(\text{No Disease}) \] In this formula: - \( P(\text{Test} + | \text{No Disease}) = 1 - \text{Specificity} \) represents the false positive rate. --- ## **Step 4: Reasoning with a Very Low Base Rate** In situations where the base rate \( P(\text{Disease}) \) is very low: - The numerator \( P(\text{Test} + | \text{Disease}) \cdot P(\text{Disease}) \) becomes very small. - Conversely, the denominator is largely influenced by \( P(\text{Test} + | \text{No Disease}) \cdot P(\text{No Disease}) \), which is significant since \( P(\text{No Disease}) \) is close to 1, leading to a considerable number of false positives. **Interpretation:** Despite a test's high sensitivity and specificity, the number of false positives in a population with very low disease prevalence can dominate results, leading to a much lower posterior probability than expected. --- ## **Step 5: Evaluating the Answer Choices** - **a) It will be close to 100%.** Incorrect. The low base rate leads to a high rate of false positives. - **b) It will be close to 50%.** Not generally true for rare diseases; the probability is generally much lower. - **c) It will be less than 50%.** Correct. For very low disease prevalence, this is typically true. - **d) It depends on the specificity but not the sensitivity.** Both sensitivity and specificity impact the result, rendering this statement inaccurate. --- ## **Step 6: Explanation and Final Answer** **Explanation:** The rarity of the disease means that even a positive test result often indicates a false positive, leading to a posterior probability that is significantly lower than might be intuitively expected. Both sensitivity and specificity play roles, but the low prevalence is the key factor. --- ## **Final Answer** **The most accurate statement is:** > **c) It will be less than 50%.** **Summary:** In the context of rare diseases, the posterior probability of having the disease following a positive test result is typically less than 50%, primarily due to the prevalence of false positives compared to true positives.