Question

Please show your work and explain your answer. Do not use common sense.

Consider the example we did in class: about 1 in every thousand people has disease X, and suppose that there is a diagnostic test that is correct 98% of the time when given to a person without the disease, and is correct 99% of the time when given to a person with the disease. Now, by looking for certain symptoms, a doctor will be able to tell whether a person has disease X or not with a 84% accuracy. For simplicity, we will assume the diagnostic test and the symptoms that a doctor looks for are independent. If the test for David is positive, and a doctor observes the symptom of X on David, what is the chance that David actually has X?

Answer 1

P(tested positive and doctor observe symptom) =P(has X and tested positive and doctor observe symptom) +P(not has X and tested positive and doctor observe symptom)

=(1/1000)*0.99*0.84+(1-1/10000)*(1-0.98)*(1-0.84)

=0.004031

therefore P(has X given tested positive and doctor observe symptom)

=P(has X and tested positive and doctor observe symptom)/P(tested positive and doctor observe symptom)

=(1/1000)*0.99*0.84/0.004031

=0.2063