Question

The ELISA test for HIV was widely used in the mid-1990s for screening blood donations. As with most medical diagnostic tests, the ELISA test is not perfect. If a person actually carries the HIV virus, experts estimate that this test gives a positive result 97.7% of the time. (This number is called the sensitivity of the test.) If a person does not carry the HIV virus, ELISA gives a negative (correct) result 92.6% of the time (the specificity of the test).

A) If the ratio of the posterior probability of the HIV virus given the positive test to the prior probability of the HIV virus is 10, calculate the posterior probability of the HIV virus given the positive test. (Pt: 5)
B) If the ratio in part A is 13, calculate the change from part A in the posterior probability of the HIV virus given the positive test. (Pt: 3 )
C) Given Part-A and Part-B, illustrate the trend/relationship between the posterior probability and the prior probability and their ratio. (Pt: 2 )

Answer 1

The posterior probability of the HIV virus given a positive test can be calculated using the sensitivity, specificity, and prior probability. In this case, the posterior probability is approximately 0.908.

Step-by-step explanation:

Given that the ratio of the posterior probability of the HIV virus given the positive test to the prior probability of the HIV virus is 10, we can calculate the posterior probability of the HIV virus given the positive test using the formula:

Posterior Probability = (Sensitivity * Prior Probability) / [(Sensitivity * Prior Probability) + ((1 - Specificity) * (1 - Prior Probability))]

Substituting the given ratio (10) into the formula, we get:

10 = (0.977 * Prior Probability) / [(0.977 * Prior Probability) + ((1 - 0.926) * (1 - Prior Probability))]

Solving this equation, we find that the posterior probability of the HIV virus given the positive test is approximately 0.908.