Question

Part 1 The three most popular options on a certain type of new car are a...

Part 1

The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 48% of all purchasers request A, 59% request B, 74% request C, 68% request A or B, 85% request A or C, 83% request B or C, and 90% request A or B or C, determine the probabilities of the following events. [Hint: "A or B" is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.]

(a) The next purchaser will request at least one of the three options.

(b) The next purchaser will select none of the three options.

(c) The next purchaser will request only an automatic transmission and not either of the other two options.

(d) The next purchaser will select exactly one of these three options.

Part 2

A certain system can experience three different types of defects. Let Ai (i = 1,2,3) denote the event that the system has a defect of type i. Suppose that the following probabilities are true.

P(A1) = 0.14      P(A2) = 0.10     P(A3) = 0.07
P(A1A2) = 0.16      P(A1A3) = 0.17
P(A2A3) = 0.14      P(A1A2A3) = 0.02

(a) What is the probability that the system does not have a type 1 defect?

(b) What is the probability that the system has both type 1 and type 2 defects?

(c) What is the probability that the system has both type 1 and type 2 defects but not a type 3 defect?

(d) What is the probability that the system has at most two of these defects?

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