Suppose that there is a new test for chronic kidney disease

Suppose that there is a new test for chronic kidney disease. For patients that actually have chronic kidney disease, the test gives a positive result 92 % of the time. For patients that do not have chronic kidney disease, the test gives a positive result 12 % of the time. It is known that 6.5 % of the population has chronic kidney disease. Suppose a patient is tested for chronic kidney disease and the test comes back positive. What is the probability that the person actually has chronic kidney disease?

Solution

1.

Let

K = with kidney problem
P = positive
N = negative

As

P(P) = P(K) P(P|K) + P(K\') P(P|K\')

P(P) = 0.065*0.92 + (1-0.065)*0.12 = 0.172

Hence,

P(K|P) = P(K) P(P|K) / P(P) = 0.065*0.92/0.172 = 0.347674419 [ANSWER]