Question

12. 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

a)P( test result will be positive)=P(have disease and tested positive)+P(not have disease and tested positive)

=0.005*0.95+0.995*0.10=0.10425

b)

P(person is a sufferer given tested positive)=P(have disease and tested positive)/P(tested positive)

=0.005*0.95/0.10425=0.0456

c)

P(person is a nonsufferer given tested negative)=P(person is a nonsufferer )/P(tested negative)

=0.995*(1-0.10)/(1-0.10425)=0.9997

d)

P(person will be misclassified)=P(have disease and tested negative)+P(not have disease and tested positive)

=0.005*0.05+0.995*0.1=0.09975