It is estimated that approximately 8.46% Americans are afflicted with diabetes.

Question

It is estimated that approximately 8.46% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 95% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 4% of all adults over 40 without diabetes as having the disease. 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"). (the answer is not 0.040 0r 0.00423)

Explanation / Answer

probability of  diagnosed as not having diabetes =P( affected by diabetes and  diagnosed as not having diabetes+ not affected by diabetes and  diagnosed as not having diabetes) =0.0846*(1-0.95)+(1-0.0846)*(1-0.04) =0.8830

therefore probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes =P(affected by diabetes and  diagnosed not having diabetes)/P(iagnosed not having diabetes)

=0.0846*(1-0.95)/0.8830=0.0048

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