Round #1035

A medical test for a rare disease is 95% accurate. The disease affects 1% of the population. Someone tests positive. What is the probability they actually have the disease?

To solve this problem, we can use Bayes' theorem. Let's break it down step by step: 1. **Define the events**: - D: The person has the disease. - ND: The person does not have the disease. - +: The test result is positive. - -: The test result is negative. 2. **Given probabilities**: - P(D) = 0.01 (1% of the population has the disease). - P(ND) = 1 - P(D) = 0.99 (99% of the population does not have the disease). - The test is 95% accurate, which means: - P(+|D) = 0.95 (if you have the disease, the test will be positive 95% of the time). - P(-|ND) = 0.95 (if you don't have the disease, the test will be negative 95% of the time). - Therefore, P(+|ND) = 1 - P(-|ND) = 0.05 (if you don't have the disease, the test will be positive 5% of the time, which is the false positive rate). - P(-|D) = 1 - P(+|D) = 0.05 (if you have the disease, the test will be negative 5% of the time, which is the false negative rate). 3. **Apply Bayes' theorem to find P(D|+)**, the probability that someone has the disease given that they tested positive: - Bayes' theorem formula: P(D|+) = P(+|D) * P(D) / P(+) - We need to calculate P(+), the probability of testing positive, which can be found using the law of total probability: - P(+) = P(+|D) * P(D) + P(+|ND) * P(ND) - P(+) = 0.95 * 0.01 + 0.05 * 0.99 - P(+) = 0.0095 + 0.0495 - P(+) = 0.059 4. **Calculate P(D|+)**: - P(D|+) = P(+|D) * P(D) / P(+) - P(D|+) = 0.95 * 0.01 / 0.059 - P(D|+) = 0.0095 / 0.059 - P(D|+) ≈ 0.161 Therefore, the probability that someone actually has the disease given that they tested positive is approximately **16.1%**. This result might seem counterintuitive given the high accuracy of the test, but it reflects the impact of the disease's low prevalence in the population. Most positive results in this scenario are false positives due to the large number of people without the disease being tested.

To calculate the probability that someone who tests positive actually has the disease, we can use Bayes' theorem. Let's define: - P(D) as the probability of having the disease, which is 1% or 0.01. - P(ND) as the probability of not having the disease, which is 99% or 0.99. - P(+|D) as the probability of testing positive given that you have the disease, which is 95% or 0.95 (the test's accuracy for those with the disease). - P(+|ND) as the probability of testing positive given that you do not have the disease, which is 5% or 0.05 (the false positive rate, since the test is 95% accurate). We want to find P(D|+), the probability of having the disease given a positive test result. Using Bayes' theorem: P(D|+) = P(+|D) * P(D) / [P(+|D) * P(D) + P(+|ND) * P(ND)] Plugging in the numbers: P(D|+) = 0.95 * 0.01 / [0.95 * 0.01 + 0.05 * 0.99] = 0.0095 / [0.0095 + 0.0495] = 0.0095 / 0.059 ≈ 0.161 So, the probability that someone who tests positive actually has the disease is approximately 16.1%. This result highlights the importance of considering both the test's accuracy and the disease's prevalence when interpreting test results.