Question

1. CPK and SGOT tests are used in the diagnosis of myocardial infarction (MI). When the CPK test is given to a patient who does not have a MI, the probability of a negative finding (i.e. its specificity) is 0.6. The probability that the SGOT test will be negative for a non-MI patient is 0.7. When both tests are given to a non-MI patient the probability that at least one is negative is 0.9. For a non-MI patient who has both tests:

1. What is the probability that both tests are negative?
2. What is the probability that both tests are positive?

Hints: (1) Answer is not 0.12 -- tests are not to be assumed to be independent.

           (2) Using 2-by-2 table to structure your calculations can help.

3. What is the probability that at least one test is positive?

Answer 1

P[ Negative finding of CPK test for patient who does not have a MI ] = 0.6

P[ Negative finding of SGOT test for patient who does not have a MI ] = 0.7

P[ Negative finding of at least one test for patient who does not have a MI ] = 0.9

a) What is the probability that both tests are negative?

P[ Negative finding of both tests for patient who does not have a MI ] = P[ Negative finding of SGOT test for patient who does not have a MI ] + P[ Negative finding of CPK test for patient who does not have a MI ] - P[ Negative finding of at least one test for patient who does not have a MI ]

P(A^B|C) = P(A|C) + P(B|C) - P(AUB|C)

P[ Negative finding of both tests for patient who does not have a MI ] = 0.6 + 0.7 - 0.9

P[ Negative finding of both tests for patient who does not have a MI ] = 0.4

b) What is the probability that both tests are positive?

P[ Positive finding of both tests for patient who does not have a MI ] = 1 - P[ Negative finding of at least one test for patient who does not have a MI ]

P(A'^B'|C) = P((AUB)' | C ) = 1 - P(AUB|C)

P[ Positive finding of both tests for patient who does not have a MI ] = 1 - 0.9

P[ Positive finding of both tests for patient who does not have a MI ] = 0.1

c) What is the probability that at least one test is positive?

P[ Positive finding of at least one test for patient who does not have a MI ] = 1 - P[ Negative finding of both tests for patient who does not have a MI ]

P(A'UB'|C) = P((A^B)' | C ) = 1 - P(A^B|C)

P[ Positive finding of at least one test for patient who does not have a MI ] = 1 - 0.4

P[ Positive finding of at least one test for patient who does not have a MI ] = 0.6