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 0.98 the test will detect it. If a bag does not contain large amounts of liquid, there is a 0.07 probability the test will conclude that it does (a false positive).Suppose that in reality only 4 in 100 bags actuallycontain large amounts of liquid.
1. What is the probability a randomly selected bag will have apositive test? Give your answer to four decimal places.
2. Given a randomly selected bag has a positive test, what is theprobability it actually contains a large amount of liquid? Giveyour answer to four decimal places.
3. Given a randomly selected bag has a positive test, what is theprobability it does not contain a large amount of liquid? Give youranswer to four decimal places.

Answer 1

Answer:

1. 0.1445

2. 0.3426

3. 0.6574

Step-by-step explanation:

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) = 0.99

P(+ | L') = 0.10

P(L) = 0.05

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

P(+) = P(+|L)P(L) + P(+ | L')P(L')

= 0.99 * 0.05 + 0.1 * (1- 0.05)

= 0.1445

2. Find P( L | + ) = P( L ∩ + ) / P(+) . . . definition of conditional probability

= P( + | L ) P(L) / P(+) . . . again using definition of conditional probability

= 0.99 * 0.05 / 0.1445

= 0.3426

3. P( L' | + ) = 1 - P( L | + ) = 1 - 0.3426 = 0.6574