Question

The risk of HIV:

The risk of HIV runs high in North America. In the at-risk population, about 1 in 30 people are HIV carriers, while in general population (people who are not at risk), 1 in 300 are. The at-risk population is 2% in total in North America. Doctors have developed a test for HIV and suppose that it correctly identifies carriers 95% of the time, while it correctly identifies the disease-free only 90%. As the test detects HIV only, you can assume that it is conditionally independent of being at risk, given carrier or not-carrier status.

a. If a random person is sampled, what is the probability that he/she is a carrier?

b. Given that a person has a positive test result and is not in the at-risk population, what is the probability that he/she is a carrier?

Answer 1

P[ test correctly identifies carriers ] = 95% = 0.95

P[ test correctly identifies the disease-free ] = 90% = 0.9

P[ at-risk population ] = 2% = 0.02

P[ person is an HIV carrier | at-risk population ] = 1/30

P[ not at-risk population ] = 1 - 0.02 = 0.98

P[ person is an HIV carrier | not at-risk population ] = 1/300

a)

P[ a random person selected is a carrier ] = P[ at-risk population ]*P[ person is an HIV carrier | at-risk population ] + P[ not at-risk population ]*P[ person is an HIV carrier | not at-risk population ]

P[ a random person selected is a carrier ] = 0.02*(1/30) + 0.98*(1/300)

P[ a random person selected is a carrier ] = 2/3000 + 9.8/3000

P[ a random person selected is a carrier ] = 11.8/3000

P[ a random person selected is a carrier ] = 0.0039

b) a person has a positive test result and is not in the at-risk population, what is the probability that he/she is a carrier.

P[ person is a carrier | a person has a positive test result and is not in the at-risk population ] = P[ person is a carrier and a person has a positive test result and is not in the at-risk population ] / P[ a person has a positive test result and is not in the at-risk population ]

P[ a person has a positive test result and is not in the at-risk population ] = P[ not at-risk population ]*P[ test correctly identifies carriers ] = 0.98*0.95 = 0.931

P[ person is a carrier and a person has a positive test result and is not in the at-risk population ] = P[ test correctly identifies carriers ]*P[ person is a carrier and is not in the at-risk population ]

P[ person is a carrier and is not in the at-risk population ] = P[ person is an HIV carrier | not at-risk population ]*P[ not at-risk population ] = 0.98*(1/300) = 9.8/3000

P[ person is a carrier and a person has a positive test result and is not in the at-risk population ] = 0.95*(9.8/3000) = 9.31/3000 = 0.931/300

P[ person is a carrier | a person has a positive test result and is not in the at-risk population ] = (0.931/300)/0.391

= 1/300 = 0.0033