Question

* The 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 the diagnosis is correct? Explain each step.​

Answer 1

The probability that the diagnosis is correct is 0.95249.

Explanation:

We are given that the 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.

Let the probability that people in the United States have diabetes = P(D) = 0.083.

So, the probability that people in the United States do not have diabetes = P(D') = 1 - P(D) = 1 - 0.083 = 0.917

Also, let A = event that the diagnostic test is accurate

So, the probability that a simple diagnostic test for diabetes is accurate for people who have the disease = P(A/D) = 0.98

And the probability that a simple diagnostic test for diabetes is accurate for people who do not have the disease = P(A/D') = 0.95

Now, the probability that the diagnosis is correct is given by;

Probability = P(D)
`*` P(A/D) + P(D')
`*` P(A/D')

= (0.083
`*` 0.98) + (0.917
`*`0.95)

= 0.08134 + 0.87115

= 0.95249

Hence, the probability that the diagnosis is correct is 0.95249.