The Transportation Safety Authority (TSA) has developed a new test to detect lar

Question

The Transportation Safety Authority (TSA) has developed a new test to detect large amounts of liquid in luggage bags. Based on many test runs, the TSA determines that if a bag does contain large amounts of liquid, there is a probability of 0.94 the test will detect it. If a bag does not contain large amounts of liquid, there is a 0.08 probability the test will conclude that it does (a false positive). Suppose that in reality only 5 in 100 bags actually contain large amounts of liquid. a. What is the probability a randomly selected bag will have a positive test? Give your answer to four decimal places. b. Given a randomly selected bag has a positive test, what is the probability it actually contains a large amount of liquid? Give your answer to four decimal places. c. Given a randomly selected bag has a positive test, what is the probability it does not contain a large amount of liquid? Give your answer to four decimal places.

Explanation / Answer

a)P(positive test) =P(large amount of liquid and tested positive+ not large amount of liquid and tested positive)

=(5/100)*0.94+(1-5/100)*0.08 =0.123

b)

P(large amount of liquid|positive test) =P(large amount of liquid and tested positive)/P(positive test)

=(5/100)*0.94/0.123=0.3821

c)P(not large amount of liquid|positive test)=P(not large amount of liquid and tested positive)/P(positive test)

=(1-5/100)*0.08/0.123 =0.6179

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