Name Period Binomial/Geometric Distributions Homework A free-throw shooter has a 70% average for making free-throws. Out of 20 attempts, find the following probabilities: 1. P(10 makes) 2. P(at least 10 makes) 3. P(17 makes) 4. P(at most 17 makes) 5. P(20 makes) 6. P(5 makes) 7. P(16 or more makes) 8. P(11 makes) 9. P(at most 11 makes) 10. P(at least 11 makes) 11. P(between 12 to 17 makes) 12. P(from 12 to 17 makes inclusive) 13. How many free-throws do you expect this shooter to make?
14. If the probability that a light bulb is defective is.1, what is the probability that exactly 3 of 8 light bulbs are defective? At most 3 of 8 are defective? 15. Suppose that 30% of employees in a large factory are smokers. What is the probability that there will be exactly two smokers in a randomly chosen five-person work group? At least 2 smokers in the same group? 16. Joe DiMaggio had a career batting average of.325. What was the probability that he would get at least one hit in five official at-bats? 17. A manager notes that there is a.125 probability that any employee will arrive late for work. What is the probability that exactly one person in a six person department will arrive late? 18. A manufacturer has the following quality control check at the end of a production line: If at least 8 of 10 randomly selected articles meet all specifications, the whole shipment is approved. If, in reality, 85 % of a particular shipment meets all specifications, what is the probability that the shipment will make it through the control check? 19. A certain tennis player makes a successful serve 70% of the time. Assume that each serve is independent of the others. If she serves 6 times, what is the probability that she gets a) all 6 serves in? b) exactly 4 serves in? c) at least 4 serves in? d) no more than 4 serves in?
20. An orchard owner knows he ll have to use 6% of the apples he harvests for cider because they will have bruises or blemishes. He expects a tree to produce 300 apples. a) Describe an appropriate model for the number of cider apples that may come from a tree. Justify your model. b) Find the probability there will be no more than a dozen cider apples. c) Is it likely there will be more than 50 cider apples? Explain. 21. An Olympic archer is able to hit a bulls-eye 80% of the time. Assume each shot is independent of the others. If she shoots 15 arrows, what is the probability of each result described below? a) What is the expected number of bulls-eyes for the 15 attempts? What is the standard deviation? b) What is the probability that she never misses? c) She gets exactly 11 bulls-eyes? d) She gets between 9 and 13 bulls-eyes? e) She gets less than 8 bulls-eyes? f) She gets at least 8 bulls-eyes? g) She gets at most 10 bulls-eyes? h) She gets between 9 and 13 bulls-eyes inclusive? 22. An airline, believing that 5% of passengers fail to show up for flights, overbook the flights. Suppose a plane will hold 265 passengers, and the airline sells 275 tickets. What is the probability the airline will not have enough seats so someone gets bumped off the flight? 23. A lecture hall has 200 seats with folding arm tablets (writing surfaces), 30 of which are designed for left-handers. The average sizes of the classes that meet there is 188, and we can assume that 13% of the students are left-handed. What is the probability that a right-handed student in one of these classes is forced to use a left arm tablet?
For 24-29, solve using both binomial calculations AND the normal approximation to the binomial. Show work! 24. More than 200 billion grocery coupons are distributed each year for discounts exceeding $84 billion. However, according to a report in USA Today, only 3.2% of the coupons are redeemed. If a company distributes 5000 coupons, what is the probability that (a) more than 100 are redeemed? (b) fewer than 200 are redeemed? (c) between 50 and 120 are redeemed? 25. Suppose that 15% of the cars coming out of an assembly plant have some defect. In a delivery of 40 cars, what is the probability that exactly 5 cars have defects? 26. If 60% of the population supports massive federal budget cuts, what is the probability that in a survey of 250 people, at most 155 people support such cuts? 27. Assume that a baseball team has an average pitcher, that is, one whose probability of winning any decision is.5. If this pitcher has 30 decisions in a season, what is the probability that he will win at least 20 games? Use the normal approximation only. a).0505 b).2514 c).2743 d).3333 e).4300
28. In a test for ESP (extrasensory perception), a subject is told that cards contain a star, a circle, a wave or a square. As the researcher looks at each of 45 cards in turn, the subject names the shape on the card without benefit of looking at it. A subject who is guessing has a probability of 0.25 of guessing correctly on each card. If the subject does significantly better than guessing, it could be evidence of ESP. a) What are the mean and standard deviation of this binomial distribution? b) What is the probability the subject guesses between and including 6 to 13 correctly? c) What is the probability the subject guesses between 13 and 20 correctly? d) What is the probability the subject guesses exactly 12 correctly? e) What is the probability the subject guesses less than 15 correctly? f) What is the probability the subject guesses more than 17 correctly? g) What is the probability the subject guesses between 2 standard deviations of the mean correctly? 29. Dentists are increasingly concerned about the growing trend of local school districts to grant soft drink companies the exclusive rights to install soda machines in schools in return for money. According to the National Soft Drink Association, 62% of schools nationally have such contracts. Suppose that 30 schools around the country are randomly selected and asked if they have soft drink contracts. a) What are the mean and standard deviation of this binomial distribution? b) What is the probability that between and including 18 to 25 answer Yes? c) What is the probability that less than 20 answer Yes? d) What is the probability that exactly 17 answer Yes?
e) What is the probability that between 14 and 23 answer Yes? f) What is the probability that those within 2 standard deviations answer Yes? g) What is the probability that between 7 and 13 answer No? h) What is the probability that more than 25 answer Yes? 30. A potential buyer will sample videotapes from a large lot of new videotapes. If she finds at least one defective one, she ll reject the entire lot. If ten percent of the lot is defective, what is the probability that she ll find a defective tape by the 4 th videotape? 31. A certain golfer makes her putts 60% of the time. a. If she putts 10 times, what is the probability that she will make half or less? b. What is the probability that the 3 rd putt was the first one she made? c. If she putts 8 times, what is the probability that she will make 5 or more putts? d. What is the probability that the 4 th putt was the first one she misses? 32. A game is played with a spinner on a circle, like the minute hand on a clock. The circle is marked evenly from 1 to 100. The player spins the spinner and the resulting number is the number of seconds the player is given to solve a randomly selected mathematics problem. Suppose there are 30 students playing in the class. a. What is the probability that 10 of the students received over a minute to solve the problem? b. What is the probability that exactly half of the students received 30 or less seconds? c. What is the probability that the fourth student was the first to receive a minute or less? d. What is the probability that at least half of the students received 45 seconds or more? e. What is the probability that a minute or more was received by the 10 th student?