Assume there is a medical test to diagnose a disease. If a person has the diseas

Question

Assume there is a medical test to diagnose a disease. If a person has the disease, the probability is 95 percent that he/she will have a positive test result for the disease. If a person does not have the disease, the probability is 0.1 percent that he/she will still have a positive test result. The probability that a person has a disease is 0.5 percent in the population.

Calculate the answer using two methods: 1. Bayes’ rule and conditional probability equations 2. Draw a table, assume a population and provide numerical answers

Explanation / Answer

Let E1: event that the person has disease P(E1) = 0.5% = 0.0005.

Let E2: event that the person does not have disease P(E2) = 1 - 0.5% = 1 - 0.0005 = 0.9995.

Let A: be the event that the test result is positive:Probability that the result is positive given the person has disease = P(A|E1) = 95% = 0.95

Probability that the result is positive given the person does not have disease = P(A|E2) = 0.1% = 0.0001.

Since the two events are mutually exclusive and exhaustive, we can use Baye's theorem, according to which P(E1|A) = (P(E1)(P(A|E1)) / ((P(E1)P(A|E1)+P(E2)+P(A|E2))

Using Baye's theorem, the probability that the person has disease given that his result is positive = P (E1|A) = (0.0005×0.95) / (0.0005×0.95+0.9995×0.0001) = 0.8261587964 or 0.8262

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