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Analytics (BUS 221) Assignment 2: Due Feb. 16, by 5 p.m. You can work in groups of up to 4 people. Put the name of each group member on the first page, in alphabetical order, by last name. Assignments must be turned in hard copies, not emails. Be neat, please. 1. Fun with Bayes' Theorem: A Morgan Stanley Consumer Research Survey sampled men and women ans asked each whether they preferred to drink plain bottled water or a sports drink such a Gatorade or Propel Fitness water. Suppose 200 men and 200 women participated in the study; and 280 of those surveyed preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women. Let M = the event the consumer is a man; W = the event the consumer is a woman; B = the event the consumer preferred plain bottled water; S = the event the consumer preferred sports drink. a. b. What the conditional probabilities c. What are the joint probabilities d. Given the consumer is a man, what is the probability he will prefer a sports drink to bottled water? e. Given the consumer is a woman, what is the probability she will prefer a sports drink to bottled water? f. 2. What is the probability that a person in the study preferred plain bottled water? Is the preference for a sports drink independent of whether the consumer is a man or a woman? Explain using probability information. P M | S P M S and and P W | S P W S ? ? Fun with Binomial Probabilities: You own 5 bonds. All five bonds have a 10% probability of defaulting within one year. Their default probabilities are statistically independent from one bond to the other. a. b. What is the probability that exactly one bond will default? c. 3. What is the mean and standard deviation for the number of bonds that will default within 1 year? What is the probability that more than one bond will default? Fun with Poisson Probabilities: The number of defaults per month in a large bank loan portfolio follows a Poisson distribution. On average, there are two defaults per month. The number of defaults is independent from one month to the next. a. What is the probability that there are 5 defaults over the span of the one month? b. What is the probability that there are more than 2 defaults over the span of the one month? c. What is the probability that there are exactly 10 defaults over the span of the next 6 months? 4. Fun with Geometric Probabilities: Suppose that to get out of jail in the board game of MONOPOLY you must either roll doubles or pay $50. a. b. Calculate the probability for getting out of jail in the third roll of the dice. c. Calculate the probability for getting out of jail in 3 rolls or less. a. 5. With two dice, what is the probability of rolling doubles? Estimate the expected number of rolls it takes to roll a double. Fun with Hypergeometric Probabilities: You have a well-shuffled deck of 52 playing cards (13 hearts 13 , diamonds , 13 clubs , and 13 spades ). 5 cards are dealt out to you. Out of these 5 cards, what are the probabilities of getting a. zero spades. b. 2 or fewer spades. c. exactly 5 spades. d. at least 1 spade