48) A chain of fast-food restaurants offers an instant-winner promotion. Drink cups and packages of French fries come with peel-off coupons. According to the promotion, 1 in 20 packages is a winner.
(a) A group of friends buys four drinks and three orders of fries. What’s the probability that they win, assuming the prizes are attached to the cups and fry containers independently?
(b) Assuming independence, how many drinks and fries would the group have to order to have a 50% chance of finding a winner?
(c) Use Boole’s inequality to estimate the probability in part (a). Do you expect this approach to work well in this application?
50) A friend of yours took a multiple-choice quiz and got all six questions right, but now claims to have guessed on every question. If each question offers four possible answers, do you believe her? Explain why you either believe her or not.
52) Intel has long dominated the market for CPU chips, which are the brains in modern desktop and laptop computers. That domination was shaken in 2005 when rival AMD made inroads with a new line of processors. Intel captured 70% of the market for laptop computers in the fourth quarter of 2004, but only 42% in the same period of 2005. In some brands, the use of Intel chips fell more. For example, 70% of HP laptops in 2004 used Intel chips compared to 31% in 2005.
(a) What is the probability that someone who bought a laptop in 2004 and again in 2005 purchased both systems with Intel chips? Identify any assumptions that you make and offer some justification.
(b) Four customers in 2005 all buy HP laptop. If the customers decide independently, what is the probability that at least three of the laptops have Intel chips?
(c) Compare these changes from 2004 to 2005: the change in probability that at least one of four customers picks an Intel-based HP laptop and the change in the probability of one customer picking an Intel-based HP laptop.
54) A study reported in The New England Journal of Medicine revealed surprisingly large differences in rates of lung cancer among smokers of different races. For each group, the study reported a rate among smokers per 100,000.
Race |
Male |
Female |
Black |
264 |
161 |
White |
158 |
134 |
Japanese-American |
121 |
50 |
Latino |
79 |
47 |
(a) What is the probability that a black male smoker develops lung cancer?
(b) What is the probability that at least one of four Japanese-American women who smoke develops cancer? Do you need any assumptions for this calculation?
(c) If the four women were from the same family, would you question any assumptions used in answering the previous item?
56) A fast-food chain randomly attaches coupons for prizes to the packages used to serve French fries. Most of the coupons say “Play again”, but a few are winners. Seventy-five percent of the coupons pay nothing with the rest evenly divided between “Win a free order of fries” and Win a free sundae.”
(a) If each member of a family of three orders fries with her or his meal, what is the probability that someone in the family is a winner?
(b) What is the probability that one member of the family gets a free order of fries and another gets the sundae? The third wins nothing.
(c) The fries normally cost $1 and the sundae $2. What are the chances of the family winning $5 or more in prize money?
Subject | Business |
Due By (Pacific Time) | 4/25/2013 |
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