In a certain region of the country it is known from
pastexperience that theprobability of selecting an adult over 40
yearsof age with cancer is 0.05. If the probability of a
doctorcorrectly diagnosing a person with cancer as having the
disease is0.78 and the probability of incorrectly diagnosing a
person withoutcancer as having the disease is .06, what is the
probability that aperson is diagnosed as having cancer?

There is a 95% probability that an adult over 40 does not have cancer. (Since either the adult has cancer or does not have cancer, and the sum of the probabilities is 100%).

A 78% probability of a person that has cancer being diagnosed,

A 6% probability of a person that does not have cancer being diagnosed.

What is the probability that a person is diagnosed as having cancer?

[tex]P = P_{1} + P_{2}[/tex]

[tex]P_{1}[/tex] is the probability of those who have cancer being diagnosed. So it is 78% of 5%. So

[tex]P_{1} = 0.05*0.78 = 0.039[/tex]

[tex]P_{2}[/tex] is the probability of those who do not have cancer being diagnosed. So it is 6% of 95%. So

[tex]P_{1} = 0.06*0.95 = 0.057[/tex]

So

[tex]P = P_{1} + P_{2} = 0.039 + 0.057 = 0.096[/tex]

There is a 9.6% probability that a person is diagnosed as having cancer.