24. (Level 2) The following quote can be found inwikipedia (via “Screening for HIV: a review ofthe evidence for the U.S. Preventive Services TaskForce”. Ann. Intern. Med. 143 (1): 5573) regard-ing the accuracy of the enzyme immunoassay testfor HIV:

5
The
sensitivityof a test is the percentage of theresults that will be positive when HIV is present.Conversely, thespecificityof a test is the percentageof the results that will be negative when HIV isnot present.For the following problems assumethat the prevalence of HIV in the USA is 0.07%and for injection drug users the prevalence is 25%(Note: I made up these prevalence numbers for thisproblem).
(a) If you give a random American the enzyme
immunoassay test for HIV, what is the chancethe test will be positive.(b) Suppose you are a doctor and your patient testpositive for HIV using the enzyme immunoas-say test. What is the chance your patient hasHIV?
(c) Suppose you are a doctor and your patient testpositive for HIV using the enzyme immunoas-say test. Suppose, in addition, that your pa-tient is an injection drug user.What is thechance your patient has HIV?
26. (Level 2) The probability is 0.60 that a famousNigerian distance runner will enter the Bostonmarathon.If he does not enter, the probabilitythat last years winner will repeat is 0.66, but if heenters, the probability that last years winner willrepeat is only 0.18.What is the probability thatlast years winner will repeat?
27. (Level 2) With reference to the previous problem,suppose that we have been out of touch but hear onthe radio that last years winner won again. Whatis the probability that the Nigerian distance runnerdid not enter the race?
28. (Level 2) The probability that a certain movie will
get an award for good acting is 0.16, the probabilitythat it will get an award for good directing is 0.27,and the probability that it will get awards for bothis 0.11. What is the probability that the movie willget...(a) ...either or both awards?(b) ...only one of the two awards?(c) ...neither award?