1. 2. 3. 4. 5. Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A, A2, and A3 by A1-likes vehicle #1, A2 =likes vehicle # 2, A3 likes vehicle #3. Suppose that P(4)=0.65, P(A)=0.55, P(43)=0.70, P(AA)=0.80, P(A43)=0.40, P(AAA3) = 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 A, A 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(AB) < 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