Lyme disease is the leading tick borne disease in the United States and Europe. Diagnosis of the disease is difficult and is aided by a test that detects particular antibodies in the blood. The article “Laboratory Considerations in the Diagnosis and Management of Lyme Borreliosis” (American Journal of Clinical Pathology [1993]: 168– 174) used the following notation: 1 represents a positive result on the blood test 2 represents a negative result on the blood test L represents the event that the patient actually has Lyme disease LC represents the event that the patient actually does not have Lyme disease The following probabilities were reported in the article:
P(L) = 0.00207
P(LC) = 0.99793
P(1| L) = 0.937
P(2| L) = 0.063
P(1| LC) = 0.03
P(2| LC) = 0.97
a. For each of the given probabilities, write a sentence giving an interpretation of the probability in the context of this problem.
b. Use the given probabilities to construct a “hypothetical 1000” table with columns corresponding to whether or not a person has Lyme disease and rows corresponding to whether the blood test is positive or negative.
c. Notice the form of the known conditional probabilities; for example, P(1| L) is the probability of a positive test given that a person selected at random from the population actually has Lyme disease. Of more interest is the probability that a person has Lyme disease, given that the test result is positive. Use the table constructed in Part (b) to compute this probability.