44 Applicants who wish to be admitted to a certain professio

44. Applicants who wish to be admitted to a certain professional school in a large university are required to take a screening test devised by an educational testing service. From past results, the testing services has estimated that 70% of all applicants are eligible for admission and that 92% of those who are eligible for admission pass the exam, whereas 12% of those who are ineligible for admission pass the exam. Using these results, what is the probability that an applicant for admission:

a. passed the exam?

b. Passed the exam but was actually ineligible?

Solution

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Applicants who wish to be admitted to a certain professional school in a large university are required to take a screening test devised by an educational testing service. From past results, the testing service has established that 70% of all applicants are eligible for admission and that 87% of those who are eligible for admission pass the exam, whereas 12% of those who are ineligible for admission pass the exam. (Round your answers to three decimal places.)
(a) What is the probability that an applicant for admission passed the exam?

(b) What is the probability that an applicant for admission passed the exam but was actually ineligible?

Answer

(a) This is the sum of the probabilities that somebody is eligible and passed, and that somebody is ineligible and passed. The probability that someone is eligible and passed is (70%)(87%) = 60.9%. The probability that someone is ineligible and passed is (30%)(12%) = 3.6%. Therefore, the probability that somebody passed the exam is 60.9% + 3.6% = 64.5%.

(b) The probability that an applicant passed the exam but was ineligible was shown to be 3.6% in part (a). The probability that an applicant passes the exam is 64.5%. Therefore, the probability that an applicant who passed was ineligible is 3.6%/64.5% = 5.6%.