63. For customers purchasing a refrigerator at a certain appli- an...

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63. For customers purchasing a refrigerator at a certain appli- ance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refriger- ator had an icemaker, and C be the event that the customer purchased an extended warranty. Relevant probabilities are P(A) = .75 P(B|A) = .9 P(B|A') = .8 P(C I A n B) = .8 = .6 P(C|A' n B) = .7 a. Construct a tree diagram consisting of first-, second-, and third-generation branches, and place an event label and appropriate probability next to each branch. b. Compute c. Compute P(B n C). d. Compute P(C). e. Compute P(A)BN , C), the probability of a U.S. pur- chase given that an icemaker and extended warranty are also purchased.78. A boiler has five identical relief valves. The probability that any particular valve will open on demand is .95. Assuming independent operation of the valves, calculate P(at least one valve opens) and P(at least one valve fails to open).80. Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work. If components work independently of 1 2 3 4 one another and P(component works) = .9, calculate P(system works).84. Seventy percent of all vehicles examined at a certain emis- sions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the following probabilities: a. P(all of the next three vehicles inspected pass) b. P(at least one of the next three inspected fails) c. P(exactly one of the next three inspected passes) d. P(at most one of the next three vehicles inspected passes) e. Given that at least one of the next three vehicles passes inspection, what is the probability that all three pass (a conditional probability)?

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