Question

19. Consider the problem of diagnosing a rare but fatal disease, which is present in only 1 out of 100,000 people. Suppose that a test for the disease is 90% accurate in detecting its presence when a person actually has the disease. Furthermore, the test is (100 – alpha) % accurate in yielding a negative result when a person does not have the disease. How small does alpha have to be in order for the probability to be greater than ½ that a person who tests positive for the disease actually has the disease?

Answer 1

We are given here that:
P(disease) = 1/100,000 = 0.00001
Therefore, P(no disease) = 1- 0.00001 = 0.99999

Also, we are given here that:
P( + | disease) = 0.9
P( - | no disease) = 1 - ()

Using law of total probability, we get:

P(+) = P( + | disease)P(disease) + P( + | no disease) P(no disease)
P(+) = 0.9*0.00001 + 0.99999

Given that the test is +, probability that the person actually has the disease is computed using Bayes theorem here as:

P( disease | + ) = P( + | disease)P(disease) / P(+ ) > 0.5

0.9*0.00001 > 0.5(0.9*0.00001 + 0.99999)

0.5*0.9*0.00001 > 0.5*0.99999

< 0.5*0.9*0.00001 / 0.5*0.99999 = 0.000009000090001

Therefore 0.0009000090001 is the required value here.