8.1 Random Phenomena and Probability

1. Identify three random phenomena that occur in your life.

2. Random phenomena can’t be predicted for certain in the short term but exhibit regular patterns in the long term. Which of the data sets in parts (a) through (d) do not appear to be from the random phenomena of coin tossing? Explain.

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3. (a) Hold a penny upright on its edge under your forefinger on a hard surface, and then snap it with your other forefinger so that it spins for some time before falling. Based on 50 spins, estimate the probability of heads.

(b) Toss a thumbtack with a gently curved back (similar to Figure 8.4, page 344) on a hard surface 100 times. (To speed up the process, toss 10 at a time.) How many times did it land with the point up? What is the approximate probability of landing point up?

4. Suppose there is a forecast of a 70% chance of rain. Using the “long run” interpretation of the definition of probability, if it does not rain the next day, is it appropriate to say that the forecaster was “wrong”? Explain.

5. The table of random digits (Table 7.1, page 298) was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the first five lines in the table are 0s? This proportion is an estimate of the true probability, which in this case is known to be 0.1.

6. A student is randomly chosen from a large mathematics class.

7. A basketball player shoots four free throws.

8. A randomly chosen subject (participant) arrives for a study of exercise and fitness.

9. Consider flipping a dime, nickel, and penny.

10. At the start of class, an instructor gives a brief quiz on assigned reading. The quiz consists of two questions: a true-or-false question and a multiple-choice question with choices (a), (b), or (c). A student who hasn’t done the assignment guesses at both questions. Use a table to represent the sample space of possible answers for this two-question quiz, and then list the elements in the sample space.

8.2 Basic Rules of Probability

11. Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement about an event. (The probability is usually a much more exact measure of likelihood than is the verbal statement.)

12. Based on data from the American Community Survey, a high percentage of U.S. workers drive alone to work (see Tables 8.1 and 8.2, pages 348 and 349). A 2009 survey asked respondents about the number of vehicles they had access to within their household. The results appear below.

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13. In Spotlight 8.1 (page 345), we noted that a question posed to Blaise Pascal in 1654 by an amateur mathematician launched the formal study of probability. Here’s a simplified version of this “Problem of Points.” Suppose two players are playing a coin flip game where “heads” earns Player A one point and “tails” earns Player B one point. The winner is the first player to reach a total of four points. The game is interrupted with Player A ahead by a score of 3 to 2. Based on the sample space of possible ways that the game can be finished, what would be a fair division of the jackpot money between Players A and B?

14. Choose a young adult (aged 25 to 34 years) at random. The probability is 0.12 that the person chosen did not complete high school, 0.31 that the person has a high school diploma but no further education, and 0.29 that the person has at least a bachelor’s degree.

15. While it is a less common way (than probability) of expressing likelihood and does not follow the five basic Probability Rules, statements of “odds” are often encountered in gambling contexts. The odds against an event happening are the ratio . In a horse race, suppose the odds against a particular horse winning the race are 3:2 (or “3 to 2”); in other words, . What is the probability that the horse wins?

8.3 Rules of Probability: Independent and Dependent Events

16. The Punnett square is a diagram that biologists use to determine the probability of offspring having certain genetic makeup. Suppose that represents the gene for brown eyes and represents the gene for blue eyes. In genetics, capital letters refer to dominant traits, so a person receiving both and generally has brown eyes. This diagram shows the possibilities for the child of two parents. Each parent gives the child one of its two genes with equal probability. What is the probability that this child will receive the genetic makeup for brown eyes? Discuss how this relates to Example 2 (page 346).

17. Suppose two events and are mutually exclusive and both have positive probabilities of occurring. Can and also be independent events? Explain.

18. Given , , and , determine the following probabilities.

19. What is the probability that Laurie rolls a pair of dice and gets an even sum in each of her first three rolls in the game of Monopoly?

20. If you are in the championship round of a tournament and you are better than your opponent, are you better off having the championship determined by playing a single game or by playing a two-out-of-three- game series? You might try intuition, simulation, or a tree diagram.

21. Return to the probability model that you created for Workers and Vehicles in Exercise 12. Suppose a random sample of two U.S. workers is chosen.

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22. According to the 2013 March Supplement Survey, 22.3% of U.S. households are in the Midwest. In addition, 9.6% of households earn $75,000 or more per year and are located in the Midwest. Determine the probability that a randomly selected American household earns $75,000 or more per year, given that the house is located in the Midwest.

23. According to the 2013 March Supplement Survey, 32.0% of U.S. households are in the South. In addition, 7.9% of households earn $100,000 or more per year and are located in the South. Determine the probability that a randomly selected Southern household earns $100,000 or more per year.

24. According to the 2013 March Supplement Survey, 18.8% of U.S. households are in the Northeast. In addition, 21.2% of Northeastern households earn $75,000 or more per year. What percentage of households earn $75,000 or more per year and are located in the Northeast?

25. Each year, the study Monitoring the Future: A Continuing Study of American Youth surveys twelfth-grade students on a wide range of topics related to behaviors, attitudes, and values. One of the survey questions asks students to identify their sex. Another question asks students to rate their intelligence compared with others their age. Assume that the sample used in this survey is representative of twelfth-grade students. Here are some of the results from the more than 13,000 students who participated in the survey: 49.8% were male; 50.2% were female; 60.3% of the females rated their intelligence as above average; 68.6% of the males rated their intelligence as above average. Since the number of participants in the survey was large, the survey percentages should be close to the probabilities for the population of twelfth-grade students.

26. The concern over drug use among teens has prompted some schools to consider mandatory drug testing of its students. However, schools should be warned that drug tests, while generally reliable, are not perfect. Sometimes a person who does use drugs gets a negative result (a false negative). Sometimes a person who does not use drugs tests positive (a false positive). Companies that produce tests for drugs usually provide information on two characteristics of their test: , the probability that their test correctly identifies a drug user, and , the probability that it correctly reports the absence of drugs.

Suppose that a large high school is considering a mandatory drug testing program. The particular test under consideration correctly identifies 95% of the users and correctly reports the absence of drugs in 90% of the nonusers .

8.4 Discrete Probability Models

27.Table 8.10 gives a probability model for total household income.

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28. Role-playing games like Dungeons & Dragons use many different types of dice. One type of die has a tetrahedral (pyramidal) shape with four triangular faces (see Figure 8.27). Each triangular face has a number (1, 2, 3, or 4) next to each of its edges. Because the top of this die is not a face but a point, the way to read it is by the number at the top of the face that is visible when the die comes to rest. Suppose that the intelligence of a character is determined by rolling this four-sided die twice and adding 1 to the sum of the results.

29. North Carolina State University posts the grade distributions for its courses online. Students in Statistics 101 in a recent semester earned 21% As, 43% Bs, 30% Cs, 5% Ds, and 1% Fs. Here is the probability model for the grade of a randomly chosen Statistics 101 student.

30. How do rented housing units differ from units occupied by their owners? Here are probability models for the number of rooms for owner-occupied units and renter-occupied units, according to the Census Bureau:

Make probability histograms of these two models, using the same scale. What are the most important differences between the models for owner-occupied and rented housing units?

31. In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate—that is, satisfies the rules of probability. If not, give specific reasons for your answer.

32. What is the probability that a housing unit has five or more rooms? Use the models in Exercise 30 to answer this question for both owner-occupied and rented units.

33. Balanced six-sided dice with altered labels can produce interesting distributions of outcomes. Construct the probability model (sample space and assignment of probabilities for each sum) for rolling the dice that is featured in Joseph Gallian’s article “Weird Dice” in the February 1995 issue of Math Horizons. Instead of using the regular values {1, 2, 3, 4, 5, 6}, one die has the labels 1, 2, 2, 3, 3, 4, and the other die has the labels 1, 3, 4, 5, 6, 8. How does this model compare with the model for regular dice?

8.5 Equally Likely Outcomes

34. If you play the lottery, there are two possibilities— you could either win or not win. Explain whether or not this means that you have a 1 out of 2 chance (i.e., a 50% probability) of winning.

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35. A party host gives a door prize to one guest chosen at random. There are 42 men and 48 women at the party. What is the probability that the prize goes to a woman?

36. At a party a cooler is filled with cans of drinks: 12 Cokes, 6 Diet Cokes, 4 lime seltzers, and 8 lemon seltzers. You grab a can from the cooler at random.

37. Suppose you have 10 books in a stack but you only have space for five books on your bookshelf.

38. Abby, Boaz, Carmen, Dani, and Eduardo work in a firm’s public relations office. Their employer must choose two of them to attend a conference in Paris. To avoid unfairness, the choice will be made by drawing two names from a hat. (This is an SRS of size 2.)

39. You toss a balanced coin 10 times and write down the resulting sequence of heads and tails, such as HTTTHHTHHH.

40. In the Texas Hold ‘Em style of poker, play begins with each player being dealt two cards face down. From a standard 52-card deck, how many possible 2-card hands could be dealt to you?

41. A computer assigns three-character log-in IDs that may contain the digits 0 to 9 as well as the letters to , with repeats allowed.

42. Consider a typical combination lock on a locker or briefcase.

43. You may have heard that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time could eventually type a particular chosen text, such as the complete works of Shakespeare. Let’s focus on a monkey who just types the letters , and in random order in three-letter sequences.

44. Mozart composed a 16-bar Viennese minuet (“Musical Dice Game”) in which bars 1 through 7 each have 11 choices, bar 8 has 2, bars 9 through 15 each have 11, and bar 16 has 1. How many possible versions of this minuet are there?

45. In poker, a royal flush is a five-card hand containing (in any order) an ace, king, queen, jack, and 10, all of the same suit.

46. The King James Version of the Old Testament has its 39 books canonized in a different order than the Hebrew Bible does. What mathematical expression would yield the number of possible orders of these 39 books? Is this number larger than you expected?

47. Use technology (see Spotlight 8.3, page 369) for this exercise. A university IT department receives a shipment of 30 printers—20 are inkjet printers and 10 are laser printers. A particular technician randomly chooses 7 of the printers to process (check that it works, tag the printer, etc.). What is the probability that exactly 4 of the printers are inkjets and 3 are laser printers (we’ll call this event )?

8.6 Continuous Probability Models

48. Books on reserve at a university library can be checked out for at most 2 hours. The density curve for the amount of time the book is checked out is the shaded triangle shown in Figure 8.28.

49. Generate two random real numbers between 0 and 1 and take their sum. The sum can take any value between 0 and 2. The density curve is the shaded triangle shown in Figure 8.29.

50. Suppose two data values are each rounded to the nearest whole number. Make a density curve for the sum of the two roundoff errors, assuming each error has a continuous uniform distribution. (Exercise 49 might help.)

51. On the TV show The Price Is Right, the “Range Game” involves a contestant being told that the suggested retail price of a prize lies between two numbers that are $600 apart. The contestant has one chance to position a red window with a span of $150 that will contain the price. On one episode, the price of a piano is between $8900 and $9500. If we assume a uniform continuous distribution (i.e., that all prices within the $600 interval are equally likely), what is the probability that the contestant will be successful?

8.7 The Mean and Standard Deviation of a Probability Model

52. You have a campus errand that will take only 15 minutes. The only parking space anywhere nearby is a faculty-only space, which is checked by campus police about once every hour. If you’re caught, the fine is $25.

53. Exercise 29 gives a probability model for the grade of a randomly chosen student in Statistics 101 at North Carolina State University, using the 4-point scale. What is the mean grade in this course? What is the standard deviation of the grades?

54. In Exercise 28, you gave a probability model for the intelligence of a character in a role-playing game. What is the mean intelligence for these characters?

55. Exercise 30 gives probability models for the number of rooms in owner-occupied and rented housing units. Find the mean number of rooms for each type of housing. Make probability histograms for the two models and mark the mean on each histogram. You see that the means describe an important difference between the two models: Owner-occupied units tend to have more rooms.

56. Typographical and spelling errors can be either “nonword errors” or “word errors.” A nonword error is not a real word, as when “the” is typed as “teh.” A word error is a real word, but not the right word, as when “lose” is typed as “loose.” When undergraduates write a 250-word essay (without checking spelling), the number of nonword errors has the following probability model:

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The number of word errors has the following model:

57. Find the mean for the following probability models. Explain how you determined your answers.

58. The idea of insurance is that we all face risks that are unlikely but carry a high cost. Think of a fire destroying your home. Insurance spreads the risk: We all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn down. An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year is per person. (The great majority of us have no loss, but a few lose their homes. The $250 is the average loss.) The company plans to sell fire insurance for $250 plus enough to cover its costs and profit. Explain clearly why it would be unwise to sell only 12 policies. Then explain why selling thousands of such policies is a safe business.

59. A company is considering offering an extended warranty on new washing machines. The company looks at past repair records and determines that there are two outcomes: an 85% probability of needing no repairs, and a 15% probability of needing a $200 repair during the warranty period. To help the company set a price for an extended warranty on new washing machines, determine the mean outcome for this model. This would be the break-even price for a company selling the extended warranties. (The company, of course, will charge more than this in order to make a profit.)

60. An American roulette wheel has 38 slots numbered 0, 00, and 1 to 36. The ball is equally likely to come to rest in any of these slots when the wheel is spun. The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains multiples of 3—that is, 3, 6, 9,…, 36. Joe places a $1 “column bet” that pays out $3 (so he gains $2) if any of these numbers comes up.

61. This table shows the prizes and respective probabilities for a lottery:

On average, how much money from a $1 ticket comes back to you in prizes?

62. A friend cuts your cake into two pieces: one is of the cake and the other is .

63. On five-choice questions on the SAT, you get 1 point for a correct answer and lose point for a wrong answer.

64. In August 2006, El Paso had a storm that was called a “500-year flood.”

8.8 The Central Limit Theorem

65. Newly manufactured automobile radiators may have small leaks. Most have no leaks, but some have one, two, or more. The number of leaks in radiators made by one supplier has mean 0.15 and standard deviation 0.4. The distribution of the number of leaks cannot be normal because only whole-number counts are possible. The supplier ships 400 radiators per day to an auto assembly plant. Take to be the mean number of leaks in these 400 radiators. Over several years of daily shipments, what interval of values will contain the middle 95% of the many values?

66. The scores of eighth-grade students on the National Assessment of Educational Progress (NAEP) mathematics test in 2007 have a distribution that is approximately normal, with mean and standard deviation .

67. Antonio measures the alcohol content of whiskey for his Chemistry 101 lab. He actually measures the mass of 5 milliliters (ml) of whiskey—a chemical calculation—and then finds the percentage of alcohol from the mass. The standard deviation of students’ measurements of mass is milligrams (mg). Antonio repeats the measurement three times and records the mean of his three measurements.

68. In Exercise 60, you found the mean and standard deviation of the outcome of a column bet in roulette. The central limit theorem says that the average outcome of a large number of bets has a distribution that is close to normal.

69. Averages of several measurements are less variable than individual measurements. The true mass of the whiskey sample in Exercise 67 is 4.6 grams, or 4600 mg. Antonio’s measurements have the normal distribution with mean 4600 mg and standard deviation 10 mg. In this case, the mean of his three measurements also has a normal distribution.

70. Exercise 28 gives the probability model for the intelligence assigned by chance to a character in a role-playing game. You found the mean intelligence of such characters in Exercise 54. Jermaine plays this character often. What interval covers (approximately) the middle 68% of average intelligence scores for 100 of Jermaine’s games?

71. The scores of high school seniors on the ACT college entrance examination in 2010 were roughly normal with mean and standard deviation .

72. Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person. For example, 70% of vehicles on the roads in the Minneapolis-St. Paul metropolitan area are occupied by just the driver. You choose 84 vehicles at random.

73. Among high-performing (and low-performing) schools, there is an unrepresentatively ________ proportion of smaller schools. Explain whether you would complete this sentence with the word “high” or “low,” in light of the formula for the standard deviation of the sampling distribution of the mean.

Chapter Review

74. Give either an intuitive or an algebraic argument to explain why .

75. License plates in Florida have the form A12BCD— that is, a letter followed by two digits followed by three more letters.

76. After you tell Jerry the probability that you calculated in Exercise 75, he realizes that he’s unlikely to get a plate ending in AAA. So he asks you, “What’s the probability I will get a plate in which all four letters are from my name?” These letters are J, E, R, and Y.

77. Choose at random a person aged 19 to 25 years and ask, “In the past four days, how many days did you do physical exercise or work out?” Based on a large sample survey, here is a discrete probability model for the answer you will get:

78. Using the information in Exercise 77, what is the mean number of days that randomly chosen 19- to 25-year-olds worked out in the past four days? If you interview many people in this age group, what does the law of large numbers say about the average number of days that these people work out?

79. Use the information in Exercise 77 and your result from Exercise 78 to answer these questions.

80. In Example 22 (page 378), we saw that a $1 bet on red has a mean outcome of . It turns out not all $1 bets in American roulette have the same mean outcome. The “five-number bet” pays an additional $6 if one of those five numbers comes up—otherwise, the player loses his $1.

81. Suppose you select 10 people at random. Find the probability of each event below:

82. In the 1970s, a group of children in Lyme, Connecticut, developed rheumatoid arthritis. However, it took until the 1980s for researchers to determine the cause of their disease, now known as Lyme disease—deer ticks infected with Borrelia burgdorferi. ELISA (enzyme-linked immunosorbent assay) is a commonly administered first test for Lyme disease. It correctly identifies patients with Lyme disease (the test is positive) 93.7% of the time and gives false positive results (the test is positive for patients who do not have Lyme disease) 6% of the time.