It is estimated that approximately 8.18% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 96% of all adults over $40$ with diabetes as having the disease and incorrectly diagnoses 2% of all adults over $40$ without diabetes as having the disease.

    a) Find the probability that a randomly selected adult over $40$ 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 $40$ is diagnosed as not having diabetes.

           

    c) Find the probability that a randomly selected adult over $40$ actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives").

           

(Note: it will be helpful to first draw an appropriate tree diagram modeling the situation)