It is known that the probability of selecting an adult over 40 years of age with cancer is 0.05. The probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06.
What is the probability that a person diagnosed as having cancer actually has the disease?
I'm a bit of a loss as far as setting up this problem goes. The probability of a person diagnosed as having cancer is Pr(A) = 0.78+0.06 = 0.84, right? And we know that the probability of a person having cancer is Pr(B) = 0.05.
So, I would guess that I'm looking for Pr(A|B) = (0.84*0.05)/0.05 = 0.84
That just doesn't seem right. What am I doing wrong?