1. 2. 3. 4. 5. Consider randomly selecting a single individual and having that person test...

 

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1. 2. 3. 4. 5. Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A₁, A2, and A3 by A1-likes vehicle #1, A2 =likes vehicle # 2, A3 likes vehicle #3. Suppose that P(4₁)=0.65, P(A₂)=0.55, P(43)=0.70, P(A₁A₂)=0.80, P(A₂43)=0.40, P(A₁A₂A3) = 0.88. a. What is the probability that the individual likes both vehicle #1 and vehicle # 3 but not #2? b. What is the probability that the individual likes only one of the vehicles, i.e., (only #1 or only #2 or only #3)? c. Are A₁, A₂ and A3 independent events? Answer in two different ways. d. If you learn that the individual did not like vehicle #2, what now is the probability that he/she liked at least one of the other two vehicles? Four customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker and C be the event that the customer purchased an extended warranty. Relevant probabilities are |P(A) = 0.75, P(B|A) = 0.9, P(B|A') = 0.8, P(C|An B) = 0.8, |PC|An B') = 0.6, P(C|A' n B) = 0.7, P(C|A'n B) = 0.3 a. Construct a tree diagram consisting of first, second, and third generation branches and place and event labeled and appropriate probabilities next to each branch. b. Compute P(AUB), P (AUC), P(B UC), P(A UBUC). c. Are A, B and Cindependent? For any events A and B with P(B) > 0, prove that a. P(A/B) + P(A|B) = 1 b. P(A/B) > P(A) c. P(A¹B) < P(A') Starting at a fixed time, each car entering an intersection is observed to see whether it turns left (L), straight ahead (S), or goes right (R). The experiment terminates as soon as a car is observed to turn left. Construct a tree diagram consisting of L, S, and R branches. Let X = the number of cars observed. What are possible X values? List five outcomes of the sample space and their associated X values. List the elements of the event Ax = {s EN: X(s) = x} for each x EX and write the properties of the events. An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X = number of months between successive payments. The cdf of X is as follows: 0 x < 1 0.30 1<x<3 0.40 3<x< 4 0.45 4≤ x < 6 0.60 6≤x≤ 12 12 ≤ x. 1 F(x) = a. What is the pmf of X ? b. Compute the probability that the number of payments is in between 3 and 6, inclusive. c. Given that the number of payments is at most 6, what is the probability that the number of payments is at most 2. 1. 2. 3. 4. 5. Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A₁, A2, and A3 by A1-likes vehicle #1, A2 =likes vehicle # 2, A3 likes vehicle #3. Suppose that P(4₁)=0.65, P(A₂)=0.55, P(43)=0.70, P(A₁A₂)=0.80, P(A₂43)=0.40, P(A₁A₂A3) = 0.88. a. What is the probability that the individual likes both vehicle #1 and vehicle # 3 but not #2? b. What is the probability that the individual likes only one of the vehicles, i.e., (only #1 or only #2 or only #3)? c. Are A₁, A₂ and A3 independent events? Answer in two different ways. d. If you learn that the individual did not like vehicle #2, what now is the probability that he/she liked at least one of the other two vehicles? Four customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker and C be the event that the customer purchased an extended warranty. Relevant probabilities are |P(A) = 0.75, P(B|A) = 0.9, P(B|A') = 0.8, P(C|An B) = 0.8, |PC|An B') = 0.6, P(C|A' n B) = 0.7, P(C|A'n B) = 0.3 a. Construct a tree diagram consisting of first, second, and third generation branches and place and event labeled and appropriate probabilities next to each branch. b. Compute P(AUB), P (AUC), P(B UC), P(A UBUC). c. Are A, B and Cindependent? For any events A and B with P(B) > 0, prove that a. P(A/B) + P(A|B) = 1 b. P(A/B) > P(A) c. P(A¹B) < P(A') Starting at a fixed time, each car entering an intersection is observed to see whether it turns left (L), straight ahead (S), or goes right (R). The experiment terminates as soon as a car is observed to turn left. Construct a tree diagram consisting of L, S, and R branches. Let X = the number of cars observed. What are possible X values? List five outcomes of the sample space and their associated X values. List the elements of the event Ax = {s EN: X(s) = x} for each x EX and write the properties of the events. An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X = number of months between successive payments. The cdf of X is as follows: 0 x < 1 0.30 1<x<3 0.40 3<x< 4 0.45 4≤ x < 6 0.60 6≤x≤ 12 12 ≤ x. 1 F(x) = a. What is the pmf of X ? b. Compute the probability that the number of payments is in between 3 and 6, inclusive. c. Given that the number of payments is at most 6, what is the probability that the number of payments is at most 2.

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ANSWER The concept of intersection and union of the events is us... View the full answer