Round #1084

A disease affects 2% of the population. A test is 90% sensitive and 95% specific. Someone tests positive. Compute the probability they actually have the disease. Give the exact value.

To solve this problem, we can use Bayes' theorem. Let's define the events: - D: The person has the disease. - ND: The person does not have the disease. - TP: The test result is positive. - TN: The test result is negative. Given: - P(D) = 0.02 (2% of the population has the disease) - P(ND) = 1 - P(D) = 0.98 - Sensitivity = P(TP|D) = 0.90 (90% chance of testing positive if you have the disease) - Specificity = P(TN|ND) = 0.95 (95% chance of testing negative if you don't have the disease) - Therefore, P(TP|ND) = 1 - P(TN|ND) = 1 - 0.95 = 0.05 We want to find P(D|TP), the probability that someone has the disease given that they tested positive. Using Bayes' theorem: \[ P(D|TP) = \frac{P(TP|D) \cdot P(D)}{P(TP)} \] First, we need to calculate P(TP), the probability of testing positive. This can happen in two ways: either you have the disease and test positive, or you don't have the disease and test positive. \[ P(TP) = P(TP|D) \cdot P(D) + P(TP|ND) \cdot P(ND) \] \[ P(TP) = 0.90 \cdot 0.02 + 0.05 \cdot 0.98 \] \[ P(TP) = 0.018 + 0.049 \] \[ P(TP) = 0.067 \] Now, we can calculate P(D|TP): \[ P(D|TP) = \frac{0.90 \cdot 0.02}{0.067} \] \[ P(D|TP) = \frac{0.018}{0.067} \] \[ P(D|TP) = \frac{18}{67} \] So, the exact probability that someone who tests positive actually has the disease is \(\frac{18}{67}\).

To compute the probability that someone who tests positive actually has the disease, we can use Bayes' theorem. Let's denote: - P(D) as the probability of having the disease, which is 2% or 0.02. - P(ND) as the probability of not having the disease, which is 100% - 2% = 98% or 0.98. - P(+|D) as the sensitivity of the test, which is 90% or 0.9. This is the probability of testing positive given that you have the disease. - P(+|ND) as the false positive rate, which is 100% - specificity = 100% - 95% = 5% or 0.05. This is the probability of testing positive given that you do not have the disease. We want to find P(D|+), the probability of having the disease given that you test positive. Bayes' theorem states: P(D|+) = P(+|D) * P(D) / P(+) Where P(+) is the probability of testing positive, which can be calculated as: P(+) = P(+|D) * P(D) + P(+|ND) * P(ND) Substituting the given values: P(+) = 0.9 * 0.02 + 0.05 * 0.98 = 0.018 + 0.049 = 0.067 Now, using Bayes' theorem: P(D|+) = 0.9 * 0.02 / 0.067 = 0.018 / 0.067 = 0.268657 So, the exact value of the probability that someone who tests positive actually has the disease is approximately 0.268657 or 26.8657%.