Multinomial Distribution The binomial distribution applies only to cases involving two

Calculating Probabilities Based on a Saint Index survey, assume that when adults are asked to identify the most unpopular projects for their hometown, 54% include WalMart among their choices. Suppose we want to find the probability that when five adults are randomly selected, exactly two of them include WalMart. What is wrong with using the multiplication rule to find the probability of getting two adults who include WalMart followed by three people who do not include WalMart, as in this calculation: (0.54)(0.54)(0.46)(0.46)(0.46)?

Consistent Notation If we use the binomial probability formula (Formula 55) for finding the probability described in Exercise 1, what is wrong with letting p denote the probability of getting an adult who includes WalMart while x counts the number of adults who do not include WalMart?

Independent Events Based on a Saint Index survey, when 1000 adults were asked to identify the most unpopular projects for their hometown, 54% included WalMart among their choices. Consider the probability that among 30 different adults randomly selected from the 1000 who were surveyed, there are at least 18 who include WalMart. Given that the subjects surveyed were selected without replacement, are the 30 selections independent? Can they be treated as being independent? Can the probability be found by using the binomial probability formula? Explain.

Notation of 0 + Using the same survey from Exercise 3, the probability of randomly selecting 30 of the 1000 adults and getting exactly 28 who include WalMart is represented as 0 + . What does 0 + indicate? Does 0 + indicate that it is it impossible to get exactly 28 adults who include WalMart?

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Clinical Trial of YSORT The YSORT method of gender selection, developed by the Genetics & IVF Institute, is designed to increase the likelihood that a baby will be a boy. When 291 couples use the YSORT method and give birth to 291 babies, the weights of the babies are recorded.

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Clinical Trial of YSORT The YSORT method of gender selection, developed by the Genetics & IVF Institute, is designed to increase the likelihood that a baby will be a boy. When 291 couples use the YSORT method and give birth to 291 babies, the genders of the babies are recorded.

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Veggie Survey In an Idaho Potato Commission survey of 1000 adults, subjects are asked to select their favorite vegetables, and each response was recorded as potatoes or other.

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Veggie Survey In an Idaho Potato Commission survey of 1000 adults, subjects are asked to select their favorite vegetables, and responses of potatoes, corn, broccoli, tomatoes, or other were recorded.

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Surveying Senators The current Senate consists of 83 males and 17 females. Forty different senators are randomly selected without replacement, and the gender of each selected senator is recorded.

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied. Surveying Senators Ten different senators are randomly selected without replacement, and the numbers of terms that they have served are recorded

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Smartphone Survey In a RingCentral survey, 380 different smartphone users are randomly selected without replacement. Respondents were asked to identify the only thing that they can't live without. Responses consist of whether a smartphone was identified.

Identifying Binomial Distributions. In Exercises 512, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.Online Shopping In a Consumer Reports survey, 427 different women are randomly selected without replacement, and each woman is asked what she purchases online. Responses consist of whether clothing was identified.

Guessing Answers The math portion of the ACT test consists of 60 multiplechoice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions. a. Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. That is, find P(WWC), where C denotes a correct answer and W denotes a wrong answer. b. Beginning with WWC, make a complete list of the different possible arrangements of two wrong answers and one correct answer, then find the probability for each entry in the list. c. Based on the preceding results, what is the probability of getting exactly one correct answer when three guesses are made?

Win 4 Lottery In the New York State Win 4 lottery, you place a bet by selecting four digits. Repetition is allowed, and winning requires that your sequence of four digits matches the four digits that are later drawn. Assume that you place one bet with a sequence of four digits. a. Use the multiplication rule to find the probability that your first two digits match those drawn and your last two digits do not match those drawn. That is, find P(MMXX), where M denotes a match and X denotes a digit that does not match the winning number. b. Beginning with MMXX, make a complete list of the different possible arrangements of two matching digits and two digits that do not match, then find the probability for each entry in the list. c. Based on the preceding results, what is the probability of getting exactly two matching digits when you select four digits for the Win 4 lottery game?

Using the Binomial Probability Table. In Exercises 1520, assume that random guesses are made for five multiplechoice questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. Use the Binomial Probability table(Table A1) to find the indicated probability for the number of correct answers. Find the probability that the number x of correct answers is exactly 3

Using the Binomial Probability Table. In Exercises 1520, assume that random guesses are made for five multiplechoice questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. Use the Binomial Probability table(Table A1) to find the indicated probability for the number of correct answers.Find the probability that the number x of correct answers is at least 3

Using the Binomial Probability Table. In Exercises 1520, assume that random guesses are made for five multiplechoice questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. Use the Binomial Probability table(Table A1) to find the indicated probability for the number of correct answers.Find the probability that the number x of correct answers is more than 2.

Using the Binomial Probability Table. In Exercises 1520, assume that random guesses are made for five multiplechoice questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. Use the Binomial Probability table(Table A1) to find the indicated probability for the number of correct answers.Find the probability that the number x of correct answers is fewer than 3.

Using the Binomial Probability Table. In Exercises 1520, assume that random guesses are made for five multiplechoice questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. Use the Binomial Probability table(Table A1) to find the indicated probability for the number of correct answers.Find the probability of no correct answers.

Using the Binomial Probability Table. In Exercises 1520, assume that random guesses are made for five multiplechoice questions on an ACT test, so that there are n = 5 trials, each with probability of success (correct) given by p = 0.20. Use the Binomial Probability table(Table A1) to find the indicated probability for the number of correct answers.Find the probability that all answers are correct

Using Technology or the Binomial Probability Formula. In Exercises 2124, assume that when blood donors are randomly selected, 45% of them have blood that is Group O (based on data from the Greater New York Blood Program).. If the number of blood donors is n = 8 , find the probability that the number with Group O blood is x = 3 .

Using Technology or the Binomial Probability Formula. In Exercises 2124, assume that when blood donors are randomly selected, 45% of them have blood that is Group O (based on data from the Greater New York Blood Program). If the number of blood donors is n = 16 , find the probability that the number with Group O blood is x = 6 .

Using Technology or the Binomial Probability Formula. In Exercises 2124, assume that when blood donors are randomly selected, 45% of them have blood that is Group O (based on data from the Greater New York Blood Program).. If the number of blood donors is n = 20 , find the probability that the number with Group O blood is x = 16 .

Using Technology or the Binomial Probability Formula. In Exercises 2124, assume that when blood donors are randomly selected, 45% of them have blood that is Group O (based on data from the Greater New York Blood Program).If the number of blood donors is n = 11 , find the probability that the number with Group O blood is x = 9

Using Computer Results. In Exercises 2528, refer to the accompanying Excel display. (In one of Mendel's hybridization experiments with peas, the probability of offspring peas having green pods is , or 0.75. ) The display lists the probabilities obtained by entering the values of n = 6 and p = 0.75. Those probabilities correspond to the numbers of peas with green pods in a group of six offspring peas. EXCEL Peas with Green Pods x P(x) 0 0.000 1 0.004 2 0.033 3 0.132 4 0.297 5 0.356 6 0.178Genetics Find the probability that at least two of the six offspring peas have green pods. If at least two offspring peas with green pods are required for further experimentation, is it reasonable to expect that at least two will be obtained?

Using Computer Results. In Exercises 2528, refer to the accompanying Excel display. (In one of Mendel's hybridization experiments with peas, the probability of offspring peas having green pods is , or 0.75. ) The display lists the probabilities obtained by entering the values of n = 6 and p = 0.75. Those probabilities correspond to the numbers of peas with green pods in a group of six offspring peas. EXCEL Peas with Green Pods x P(x) 0 0.000 1 0.004 2 0.033 3 0.132 4 0.297 5 0.356 6 0.179Genetics Find the probability that at most five of the six offspring peas have green pods.

Using Computer Results. In Exercises 2528, refer to the accompanying Excel display. (In one of Mendel's hybridization experiments with peas, the probability of offspring peas having green pods is , or 0.75. ) The display lists the probabilities obtained by entering the values of n = 6 and p = 0.75. Those probabilities correspond to the numbers of peas with green pods in a group of six offspring peas. EXCEL Peas with Green Pods x P(x) 0 0.000 1 0.004 2 0.033 3 0.132 4 0.297 5 0.356 6 0.180Genetics Find the probability that at most two of the six offspring peas have green pods. Is two an unusually low number of peas with green pods (among six)? Why or why not?

Using Computer Results. In Exercises 2528, refer to the accompanying Excel display. (In one of Mendel's hybridization experiments with peas, the probability of offspring peas having green pods is , or 0.75. ) The display lists the probabilities obtained by entering the values of n = 6 and p = 0.75. Those probabilities correspond to the numbers of peas with green pods in a group of six offspring peas. EXCEL Peas with Green Pods x P(x) 0 0.000 1 0.004 2 0.033 3 0.132 4 0.297 5 0.356 6 0.181 Genetics Find the probability that five or more of the six offspring peas will have green pods. Is five an unusually high number of offspring peas with green pods? Why or why not?

In Exercises 2932, use either technology or the Binomial Probability table (Table A1)See You Later Based on a Harris Interactive poll, 20% of adults believe in reincarnation. Assume that six adults are randomly selected, and find the indicated probability. a. What is the probability that exactly five of the selected adults believe in reincarnation? b. What is the probability that all of the selected adults believe in reincarnation? c. What is the probability that at least five of the selected adults believe in reincarnation? d. If six adults are randomly selected, is five an unusually high number who believe in reincarnation?

In Exercises 2932, use either technology or the Binomial Probability table (Table A1) Live TV Based on a Comcast survey, there is a 0.8 probability that a randomly selected adult will watch primetime TV live, instead of online, on DVR, etc. Asssume that seven adults are randomly selected, and find the indicated probability. a. Find the probability that exactly two of the selected adults watch primetime TV live. b. Find the probability that exactly one of the selected adults watches primetime TV live. c. Find the probability that fewer than three of the selected adults watch primetime TV live. d. If we randomly select seven adults, is two an unusually low number for those who watch primetime TV live?

In Exercises 2932, use either technology or the Binomial Probability table (Table A1)Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability. a. Find the probability that none of the selected adults say that they were too young to get tattoos. b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos. c. Find the probability that the number of selected adults saying they were too young is 0 or 1. d. If we randomly select five adults, is one an unusually low number who say that they were too young to get tattoos?

In Exercises 2932, use either technology or the Binomial Probability table (Table A1)Tainted Currency Based on the American Chemical Society, there is a 0.9 probability that in the United States, a randomly selected dollar bill is tainted with traces of cocaine. Assume that eight dollar bills are randomly selected. a. Find the probability that all of them have traces of cocaine. b. Find the probability that exactly seven of them have traces of cocaine. c. Find the probability that the number of dollar billls with traces of cocaine is seven or more. d. If we randomly select eight dollar bills, is seven an unusually high number for those with traces of cocaine?

Death of Taxes Based on a Bellowes survey of adults, there is a 0.48 probability that a randomly selected adult uses a tax preparer to file taxes. Find the probability that among 20 randomly selected adults, exactly 12 use tax preparers to file their taxes. Among 20 random adults, is 12 an unusually high number for those who use tax preparers to file their taxes?

Career Choice Based on a Ridgid survey of high school students, 25% of high school students plan to choose a career in information technology. Find the probability that among 24 randomly selected high school students, exactly 6 plan to choose a career in information technology. Is it unlikely to find that among the 24 selected high school students, exactly 25% plan to choose a career in information technology?

OnTime Flights The U.S. Department of Transportation recently reported that 80.5% of U.S. airline flights arrived on time. Find the probability that among 12 randomly selected flights, exactly 10 arrive on time. Does that probability apply to the author, who must make 12 flights originating in New York?

Online Banking Based on data from a USA Today Snapshot, 72% of adults have security concerns about online banking. Find the probability that among 30 randomly selected adults, exactly 20 have security concerns. Among 30 random adults, is 20 an unusually low number for those who have security concerns?

Nielsen Rating CBS televised a recent Super Bowl football game between the New Orleans Saints and the Indianapolis Colts. That game received a rating of 45, indicating that among all U.S. households, 45% were tuned to the game (based on data from Nielsen Media Research). An advertiser wants to obtain a second opinion by conducting its own survey, and a pilot survey begins with 12 randomly selected households. a. Find the probability that none of the households is tuned to the Saints/Colts game. b. Find the probability that at least one household is tuned to the Saints/Colts game. c. Find the probability that at most one household is tuned to the Saints/Colts game. d. If at most one household is tuned to the Saints/Colts game, does it appear that the 45 share value is wrong? Why or why not?

Overbooking Flights When someone buys a ticket for an airline flight, there is a 0.0995 probability that the person will not show up for the flight (based on data from an IBM research paper by Lawrence, Hong, and Cherrier). The Beechcraft 1900C jet can seat 19 passengers. Is it wise to book 21 passengers for a flight on the Beechcraft 1900C? Explain.

XSORT Method of Gender Selection When testing a method of gender selection, we assume that the rate of female births is 50%, and we reject that assumption if we get results that are unusual in the sense that they are very unlikely to occur with the 50% rate. In a preliminary test of the XSORT method of gender selection, 14 births included 13 girls. a. Assuming a 50% rate of female births, find the probability that in 14 births, the number of girls is 13. b. Assuming a 50% rate of female births, find the probability that in 14 births, the number of girls is 14. c. Assuming a 50% rate of female births, find the probability that in 14 births, the number of girls is 13 or more. d. Do these preliminary results suggest that the XSORT method is effective in increasing the likelihood of a baby being a girl? Explain.

Challenged Calls in Tennis In a recent U.S. Open tennis tournament, among 20 of the calls challenged by players, 8 were overturned after a review using the HawkEye electronic system. Assume that when players challenge calls, they are successful in having them overturned 50% of the time. a. Find the probability that among 20 challenges, exactly 8 are successfully overturned. b. The probability that among 20 challenges, 8 or fewer are overturned is 0.252. Does this result suggest that the success rate is less than 50%? Why or why not?

Composite Sampling. Exercises 41 and 42 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease. HIV The probability of a randomly selected adult in the United States being infected with the human immunodeficiency virus (HIV) is 0.006 (based on data from the Kaiser Family Foundation). In tests for HIV, blood samples from 24 people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?

Composite Sampling. Exercises 41 and 42 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease.. STD Based on data from the Centers for Disease Control, the probability of a randomly selected person having gonorrhea is 0.00114. In tests for gonorrhea, blood samples from

Acceptance Sampling. Exercises 43 and 44 involve the method of acceptance sampling, whereby a shipment of a large number of items is accepted if tests of a sample of those items result in only one or none that are defective.Aspirin The Medassist Pharmaceutical Company receives large shipments of aspirin tablets and uses this acceptance sampling plan: Randomly select and test 40 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 5000 aspirin tablets actually has a 3% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Acceptance Sampling. Exercises 43 and 44 involve the method of acceptance sampling, whereby a shipment of a large number of items is accepted if tests of a sample of those items result in only one or none that are defective.Chocolate Chip Cookies The Killington Market chain uses this acceptance sampling plan for large shipments of packages of its generic chocolate chip cookies: Randomly select 30 packages and determine whether each is within specifications (not too many broken cookies, acceptable taste, distribution of chocolate chips, and so on). The entire shipment is accepted if at most 2 packages do not meet specifications. A shipment contains 1200 packages of chocolate chips, and 2% of those packages do not meet specifications. What is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

Geometric Distribution If a procedure meets all the conditions of a binomial distribution except that the number of trials is not fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by P ( x ) = p ( 1 p ) x 1 , where p is the probability of success on any one trial. Subjects are randomly selected for the National Health and Nutrition Examination Survey conducted by the National Center for Health Statistics, Centers for Disease Control. Find the probability that the first subject to be a universal blood donor (with group O and type Rh blood) is the fifth person selected. The probability that someone is a universal donor is 0.06.

Multinomial Distribution The binomial distribution applies only to cases involving two types of outcomes, whereas the multinomial distribution involves more than two categories. Suppose we have three types of mutually exclusive outcomes denoted by A, B, and C. Let P ( A ) = p 1 , P ( B ) = p 2 , and P ( C ) = p 3 . In n independent trials, the probability of x 1 outcomes of type A, x 2 outcomes of type B, and x 3 outcomes of type C is given by n ! ( x 1 ) ! ( x 2 ) ! ( x 3 ) ! p 1 x 1 p 2 x 2 p 3 x 3 A roulette wheel in the Hard Rock casino in Las Vegas has 18 red slots, 18 black slots, and 2 green slots. If roulette is played 12 times, find the probability of getting 5 red outcomes, 4 black outcomes, and 3 green outcomes.

Hypergeometric Distribution If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers matching the ones you selected), while the remaining B objects are of the other type (such as lottery numbers different from the ones you selected), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and n x objects of type B is P ( x ) = A ! ( A x ) ! x ! B ! ( B n + x ) ! ( n x ) ! ( A + B ) ! ( A + B n ) ! n !