Question

It is estimated that approximately 8.25% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 96% of all adults over 4040 with diabetes as having the disease and incorrectly diagnoses 3% of all adults over 4040 without diabetes as having the disease.

A) Find the probability that a randomly selected adult over 4040 does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called "false positives").

B) Find the probability that a randomly selected adult of 4040 is diagnosed as not having diabetes.

C) Find the probability that a randomly selected adult over 4040 actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives").

Answer 1

a)

probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes =(1-0.0825)*0.03=0.027525

b)

probability that a randomly selected adult of 40 is diagnosed as not having diabetes

=P(have diabetes and diagnosed as not having diabetes )+P(not have diabetes and diagnosed as not having diabetes )

=0.0825*(1-0.96)+(1-0.0825)*(1-0.03)=0.893275

c)

probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes

=P(have diabetes and diagnosed as not having diabetes )/P(diagnosed as not having diabetes )

=0.0825*(1-0.96)/0.893275=0.003694