Round #1054

A disease affects 0.5% of the population. A test is 99% sensitive and 99% 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) = 0.005 (0.5% of the population has the disease) - P(ND) = 1 - P(D) = 1 - 0.005 = 0.995 - Sensitivity = P(TP|D) = 0.99 (99% sensitive, meaning 99% of people with the disease test positive) - Specificity = P(TN|ND) = 0.99 (99% specific, meaning 99% of people without the disease test negative) - Therefore, P(TP|ND) = 1 - P(TN|ND) = 1 - 0.99 = 0.01 (1% of people without the disease test positive, which is the false positive rate) We want to find P(D|TP), the probability that someone who tests positive actually has the disease. 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 be found using the law of total probability: \[ P(TP) = P(TP|D) \cdot P(D) + P(TP|ND) \cdot P(ND) \] \[ P(TP) = 0.99 \cdot 0.005 + 0.01 \cdot 0.995 \] \[ P(TP) = 0.00495 + 0.00995 \] \[ P(TP) = 0.0149 \] Now, we can calculate P(D|TP): \[ P(D|TP) = \frac{0.99 \cdot 0.005}{0.0149} \] \[ P(D|TP) = \frac{0.00495}{0.0149} \] \[ P(D|TP) = \frac{495}{1490} \] \[ P(D|TP) = \frac{99}{298} \] So, the exact value of the probability that someone who tests positive actually has the disease is \(\frac{99}{298}\).

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 0.005 (0.5% of the population). - P(ND) as the probability of not having the disease, which is 1 - P(D) = 0.995. - P(+|D) as the sensitivity of the test, which is 0.99 (99% sensitive). - P(+|ND) as the false positive rate, which is 1 - specificity = 1 - 0.99 = 0.01. - P(D|+) as the probability of having the disease given a positive test result, which we are trying to find. Using Bayes' theorem: P(D|+) = P(+|D) * P(D) / [P(+|D) * P(D) + P(+|ND) * P(ND)] Substituting the given values: P(D|+) = 0.99 * 0.005 / [0.99 * 0.005 + 0.01 * 0.995] = 0.00495 / [0.00495 + 0.00995] = 0.00495 / 0.0149 = 0.3322... So, the exact value of the probability that someone who tests positive actually has the disease is 0.00495 / 0.0149. This calculation requires more explanation due to the need for a step-by-step breakdown of Bayes' theorem application.