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 0.09 probability the test will conclude that it does (a false positive).
Suppose that in reality only 4 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. 1
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. 2
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

Let L be the event that there is liquid in the bag
Let + the be event of a positive test.

we are given:

P(+ | L) = 0.94
P(+ | L') = 0.09
P(L) = 0.04

Find P(+), use the Law of Total Probability to do this:

P(+) = P(+|L)P(L) + P(+ | L')P(L')
= 0.94 * 0.04 + 0. 09* (1- 0.04)
= 0.124

Find P( L | + ) = P( L + ) / P(+) . . . definition of conditional probability
= P( + | L ) P(L) / P(+) . . . again using definition of conditional probability
= 0.94 * 0.04/ 0.124
= 0.3032


P( L' | + ) = 1 - P( L | + ) = 1 - 0.3032= 0.6968

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