Question

American Diabetes Association estimates that 8.3% of people in the United States have diabetes. Suppose that a medical lab has developed a simple diagnostic test for diabetes that is 98% accurate for people who have the disease and 95% accurate for people who do not have it. The medical lab gives the test to a randomly selected person. What is the probability that a person has diabeties given that the test says they do?

Answer 1

Probability that a person has diabeties given that the test says they do = 0.6526.

Explanation:

We are give that an American Diabetes Association estimates that 8.3% of people in the United States actually have diabetes.

For testing this a simple diagnostic test for diabetes has been developed which is 98% accurate for people who have the disease and 95% accurate for people who do not have it.

Let A = Probability that people in United States actually have diabetes,

P(A) = 8.3% or 0.083

P(A') = 1 - P(A) = means Probability that people in United States does not have diabetes = 1 - 0.083 = 0.917 .

Let B = Test that is accurate in detecting diabetes

P(B/A) = Probability that test says that person has diabetes given he/she actually has = 0.98

P(B/A') = Probability that test is accurate in detecting diabetes for people who actually does not have diabetes = 0.95

Probability that test says person has diabetes but in actually they don't have = 1 - 0.95 = 0.05.

Probability that test says person has diabetes = 0.083*0.98 + 0.05*(1-0.083)

= 0.12719

Now, Probability that a person has diabetes given that the test says they do = Probability that person has diabetes ÷ Probability that test says they have

= 0.083 ÷ 0.12719 = 0.6526.