probability

Question

A diagnostic test has a probability0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities:

(a) that the test result will be positive;

(b) that, given a positive result, the person is a sufferer;

(c) that, given a negative result, the person is a non-sufferer;

(d) that the person will be misclassified.

Answer 1

Answer:

Let T ≡ “Test positive”, S ≡ “Sufferer”, M ≡ “Misclassified.”

Then P(T|S)=0.95, P(T|S' )=0.10, P(S)=0.005.

Hence (a) P(T) = P(T|S)P(S) + P(T|S' )P(S' ) = (0.95 × 0.005) + (0.1 × 0.995) = 0.10425.

(b) P(S|T) = P(T|S)P(S) P(T|S)P(S) + P(T|S')P(S' ) = 0.95 × 0.005 (0.95 × 0.005) + (0.1 × 0.995) = 0.0455.

(c) P(S' |T' ) = P(T' |S' )P(S' ) P(T' ) = 0.9 × 0.995 1 − 0.10425 = 0.9997.

(d) P(M) = P(T ∩ S' ) + P(T' ∩ S) = P(T|S')P(S' ) + P(T' |S)P(S)=0.09975.