Question

Before solving. Kindly put this in a table so that all bottom and right marginals all add up to 1.

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 0.97 the test will detect it. If a bag does not contain large amounts of liquid, there is a 0.06 probability the test will conclude that it does (a false positive).
Suppose that in reality only 1 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.

Answer 1

Let L be the event that there is liquid in the bag

Let + be the event of a positive test.

So,

P(+ | L) = 0.97

P(+ | L') = 0.06

P(L) = 1/100 = 0.01

P(L') = 1-P(L)

= 0.99

Data:

	Liquid	No Liquid	Total
+	0.0097	0.0594	0.0691
-	0.0003	0.9306	0.9309
Total	0.01	0.99

A.

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

= 0.97 * 0.01 + 0.06 * 0.99

= 0.0691

B.

P( L | + ) = P( L ∩ + ) / P(+)

= P( + | L ) P(L) / P(+)

= 0.97 * 0.01/ 0.0691

= 0.1404

C.

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

= 1 - 0.1404

= 0.8596

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