1. Assume that P(A) = 0.4, P(B) = 0.3, and P(C) = 0.1. (21p) (a) If the events A, B, and C are disjoint, find the probability that the union of these events occurs. (b) Draw a Venn diagram to illustrate your answer to part a) (c) Find the probability of the complement of the union of A, B, and C. 2. Conditional probabilities. Suppose that P(A) = 0.5, P(B) = 0.3, and P (B | A) = 0.2. (21p) (a) Find the probability that both A and B occur. (b) Use a Venn diagram to explain your calculation. (c) What is the probability of the event that B occurs, and A does not? 3. Draw a U.S. household at random and record the primary source of energy to generate heat for warmth of the household using space-heating equipment. "At random" means that we give every household the same chance to be chosen. That is, we choose an SRS of size 1. Here is the distribution of primary sources for U.S. households: (14p) Primary source Probability Natural gas 0.50 Electricity 0.35 Distillate fuel oil 0.06 Liquefied petroleum gases 0.05 Wood 0.02 Other 0.02 (a) Show that this is a legitimate probability model. (b) What is the probability that a randomly chosen U.S. household uses natural gas or electricity as its primary source of energy for space heating?4. Select a first-year college student at random and ask what his or her academic rank was in high school. Here are the probabilities, based on proportions from a large sample survey of first-year students: (18p) Rank Top 20% Second 20% Third 20% Fourth 20% Lowest 20% Probability 0.41 0.23 0.29 0.06 0.01 (a) Choose two first-year college students at random. Why is it reasonable to assume that their high school ranks are independent? (b) What is the probability that both were in the top 20% of their high school classes? (c) What is the probability that the first was in the top 20% and the second was in the lowest 20%? 5. For more than two decades, Jude Werra, president of an executive recruiting firm, has tracked executive resumes to determine the rate of misrepresenting education credentials and/or employment information. On a biannual basis, Werra reports a now nationally recognized statistic known as the "Liars Index." In 2013, Werra reported that 18.4% of executive job applicants lied on their resumes. Suppose an executive job hunter randomly selects five resumes from an executive job applicant pool. Let X be the number of misleading resumes found in the sample. (16p) (a) What are the possible values of X? (b) Use the binomial formula to find the P (X = 2). 6. Suppose the average number of emails received by a particular employee at your company is five emails per hour. Suppose the count of emails received can be adequately modeled as a Poisson random variable. (10p) (a) What is the probability of this employee receiving exactly five emails in any given hour