Using a Distribution to Find Probabilities In Exercise,

Problem 1E In Exercise, find the indicated probability using the geometric distribution. Find P(3) when p = 0.65.

Problem 2E In Exercise, find the indicated probability using the geometric distribution. Find P(1) when p = 0.45.

Problem 1UAE In Exercise, assume that the manufacturer’s claim is true—40% of toothpaste buyers prefer Brand A toothpaste. Use the graph of the binomial distribution and technology to answer the questions. Explain your reasoning. Interpreting the “Most Likely” Outcome In a random sample of 100, what is the most likely outcome? How likely is it?

Problem 32EC Poisson Distribution: Variance In Exercise, use the fact that the variance of a Poisson distribution is ?2 = µ. Bankruptcies The mean number of bankruptcies filed per hour by businesses in the United States in a recent year was about five. (a) Find the variance and the standard deviation. Interpret the results. (b) Find the probability that at most three businesses will file bankruptcy in any given hour.

Problem 3E In Exercise, find the indicated probability using the geometric distribution. Find P(5) when p = 0.09.

Problem 2UAE In Exercise, assume that the manufacturer’s claim is true—40% of toothpaste buyers prefer Brand A toothpaste. Use the graph of the binomial distribution and technology to answer the questions. Explain your reasoning. Interpreting the “Most Likely” Outcome In a random sample of 100, what is the probability that between 35 and 45 people, inclusive, prefer Brand A?

Problem 3UAE In Exercise, assume that the manufacturer’s claim is true—40% of toothpaste buyers prefer Brand A toothpaste. Use the graph of the binomial distribution and technology to answer the questions. Explain your reasoning. In a random sample of 100, you found 36 who prefer Brand A. Would the manufacturer’s claim be believable?

Problem 4E In Exercise, find the indicated probability using the geometric distribution. Find P(8) when p = 0.28.

Problem 4UAE In Exercise, assume that the manufacturer’s claim is true—40% of toothpaste buyers prefer Brand A toothpaste. Use the graph of the binomial distribution and technology to answer the questions. Explain your reasoning. In a random sample of 100, you found 25 who prefer Brand A. Would the manufacturer’s claim be believable?

Problem 5E In Exercise, find the indicated probability using the Poisson distribution. Find P(4) when µ = 5.

Problem 7E In Exercise, find the indicated probability using the Poisson distribution. Find P(2) when µ = 1.5.

Problem 8E In Exercise, find the indicated probability using the Poisson distribution. Find P(5) when µ = 9.8.

Problem 6E In Exercise, find the indicated probability using the Poisson distribution. Find P(3) when µ = 6.

Problem 9E In your own words, describe the difference between the value of x in a binomial distribution and in a geometric distribution.

Problem 10E In your own words, describe the difference between the value of x in a binomial distribution and in a Poisson distribution.

Problem 12E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Defective Parts An auto parts seller finds that 1 in every 100 parts sold is defective. Find the probability that (a) the first defective part is the tenth part sold, (b) the first defective part is the first, second, or third part sold, and (c) none of the first 10 parts sold are defective.

Problem 15E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Pass Completions Football player Tom Brady completes a pass 63.7% of the time. Find the probability that (a) the first pass he completes is the second pass, (b) the first pass he completes is the first or second pass, and (c) he does not complete his first two passes.

Problem 13E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Births The mean number of births per minute in the United States in a recent year was about eight. Find the probability that the number of births in any given minute is (a) exactly five, (b) at least five, and (c) more than five.

Problem 16E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Precipitation In Savannah, Georgia, the mean number of days in July with 0.01 inch or more of precipitation is 13. Find the probability that, next July, the number of days with 0.01 inch or more of precipitation in Savannah is (a) exactly 16 days, (b) at most 16 days, and (c) more than 16 days.

Problem 11E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Telephone Sales The probability that you will make a sale on any given telephone call is 0.19. Find the probability that you (a) make your first sale on the fifth call, (b) make your first sale on the first, second, or third call, and (c) do not make a sale on the first three calls.

Problem 17E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Glass Manufacturer A glass manufacturer finds that 1 in every 500 glass items produced is warped. Find the probability that (a) the first warped glass item is the tenth item produced, (b) the first warped glass item is the first, second, or third item produced, and (c) none of the first 10 glass items produced are defective.

Problem 14E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Typographical Errors A newspaper finds that the mean number of typographical errors per page is four. Find the probability that the number of typographical errors found on any given page is (a) exactly three, (b) at most three, and (c) more than three.

Problem 18E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Winning a Prize A cereal maker places a game piece in each of its cereal boxes. The probability of winning a prize in the game is 1 in 4. Find the probability that you (a) win your first prize with your fourth purchase, (b) win your first prize with your first, second, or third purchase, and (c) do not win a prize with your first four purchases.

Problem 19E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Major Hurricanes A major hurricane is a hurricane with wind speeds of 111 miles per hour or greater. During the 20th century, the mean number of major hurricanes to strike the U.S. mainland per year was about 0.6. Find the probability that the number of major hurricanes striking the U.S. mainland in any given year is (a) exactly one, (b) at most one, and (c) more than one.

Problem 22E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Breaking Up Twenty-nine percent of Americans ages 16 to 21 years old say that they would break up with their boyfriend or girlfriend for $10,000. You randomly select seven 16- to 21-year-olds. Find the probability that the number of 16- to 21-year-olds who say that they would break up with their boyfriend or girlfriend for $10,000 is (a) exactly two, (b) more than three, and (c) between one and four, inclusive.

Problem 21E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Heart Transplants The mean number of heart transplants performed per day in the United States in a recent year was about six. Find the probability that the number of heart transplants performed on any given day is (a) exactly seven, (b) at least eight, and (c) no more than four.

Problem 23E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Education Fifty-four percent of parents would give up cable television to have their child’s education paid for. You randomly select five parents. Find the probability that the number of parents who would give up cable television to have their child’s education paid for is (a) exactly three, (b) less than four, and (c) at least three.

Problem 20E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Nuclear Energy Fifty-seven percent of U.S. adults favor using nuclear energy as a source of electricity in the United States. You randomly select eight U.S. adults. Find the probability the number of U.S. adults who favor using nuclear energy as a source of electricity in the United States is (a) exactly four, (b) less than five, and (c) at least three.

Problem 25E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Cheating Forty-two percent of adults say that they have cheated on a test or exam before. You randomly select six adults. Find the probability that the number of adults who say that they have cheated on a test or exam before is (a) exactly four, (b) more than two, and (c) at most five.

Problem 24E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Pilot Test The probability that a student passes the written test for a private pilot license is 0.75. Find the probability that the student (a) passes on the first attempt, (b) passes on the second attempt, and (c) does not pass on the first or second attempt.

Problem 26E Using a Distribution to Find Probabilities In Exercise, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities. Oil Tankers The mean number of oil tankers at a port city is eight per day. Find the probability that the number of oil tankers on any given day is (a) exactly eight, (b) at most three, and (c) more than eight.

Problem 27EC Comparing Binomial and Poisson Distributions An automobile manufacturer finds that 1 in every 2500 automobiles produced has a particular manufacturing defect. (a) Use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 6000 cars. (b) The Poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p. Repeat part (a) using a Poisson distribution and compare the results.

Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and \(N-k\) failures, the probability of selecting a sample of size n that has x successes and \(n-x\) failures is given by \(P(x)=\frac{\left({ }_{k} C_{x}\right)\left({ }_{N-k} C_{n-x}\right)}{{ }_{N} C_{n}}\). In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Find the probability that (a) all three microchips are not defective, (b) one microchip is defective and two are not defective, and (c) two microchips are defective and one is not defective. ________________ Equation Transcription: Text Transcription: N-k n-x P(x)=fraction{(_kC_x)(_{N-k}C_{n-x})}{_{N}C_{n}}

Problem 30EC Geometric Distribution: Mean and Variance In Exercise, use the fact that the mean of a geometric distribution is µ = 1/p and the variance is ?2 = q/p2. Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

Problem 29EC Geometric Distribution: Mean and Variance In Exercise, use the fact that the mean of a geometric distribution is µ = 1/p and the variance is ?2 = q/p2. Daily Lottery A daily number lottery chooses three balls numbered 0 to 9. The probability of winning the lottery is 1/1000. Let x be the number of times you play the lottery before winning the first time. (a) Find the mean, variance, and standard deviation. (b) How many times would you expect to have to play the lottery before winning? It costs $1 to play and winners are paid $500. Would you expect to make or lose money playing this lottery? Explain.

Problem 31EC Poisson Distribution: Variance In Exercise, use the fact that the variance of a Poisson distribution is ?2 = µ. Golf In a recent year, the mean number of strokes per hole for golfer Phil Mickelson was about 3.9. (a) Find the variance and standard deviation. Interpret the results. (b) Find the probability that he would play an 18-hole round and have more than 72 strokes?

In Exercises 1 4, find the indicated probability using the geometric distribution Find P132 when p = 0.65.

In Exercises 1 4, find the indicated probability using the geometric distribution . Find P112 when p = 0.45.

In Exercises 1 4, find the indicated probability using the geometric distribution . Find P152 when p = 0.09.

In Exercises 1 4, find the indicated probability using the geometric distribution Find P182 when p = 0.28.

In Exercises 58, find the indicated probability using the Poisson distribution Find P142 when m = 5.

In Exercises 58, find the indicated probability using the Poisson distribution Find P132 when m = 6.

In Exercises 58, find the indicated probability using the Poisson distribution . Find P122 when m = 1.5.

In Exercises 58, find the indicated probability using the Poisson distribution Find P152 when m = 9.8.

In your own words, describe the difference between the value of x in a binomial distribution and in a geometric distribution.

In your own words, describe the difference between the value of x in a binomial distribution and in a Poisson distribution

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Telephone Sales The probability that you will make a sale on any given telephone call is 0.19. Find the probability that you (a) make your first sale on the fifth call, (b) make your first sale on the first, second, or third call, and (c) do not make a sale on the first three calls.

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Defective Parts An auto parts seller finds that 1 in every 100 parts sold is defective. Find the probability that (a) the first defective part is the tenth part sold, (b) the first defective part is the first, second, or third part sold, and (c) none of the first 10 parts sold are defective.

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Births The mean number of births per minute in the United States in a recent year was about eight. Find the probability that the number of births in any given minute is (a) exactly five, (b) at least five, and (c) more than five. (Source: Centers for Disease Control and Prevention)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Typographical Errors A newspaper finds that the mean number of typographical errors per page is four. Find the probability that the number of typographical errors found on any given page is (a) exactly three, (b) at most three, and (c) more than three.

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Pass Completions Football player Tom Brady completes a pass 63.7% of the time. Find the probability that (a) the first pass he completes is the second pass, (b) the first pass he completes is the first or second pass, and (c) he does not complete his first two passes. (Source: National Football League)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Precipitation In Savannah, Georgia, the mean number of days in July with 0.01 inch or more of precipitation is 13. Find the probability that, next July, the number of days with 0.01 inch or more of precipitation in Savannah is (a) exactly 16 days, (b) at most 16 days, and (c) more than 16 days. (Source: National Climatic Data Center)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Glass Manufacturer A glass manufacturer finds that 1 in every 500 glass items produced is warped. Find the probability that (a) the first warped glass item is the tenth item produced, (b) the first warped glass item is the first, second, or third item produced, and (c) none of the first 10 glass items produced are defective.

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Winning a Prize A cereal maker places a game piece in each of its cereal boxes. The probability of winning a prize in the game is 1 in 4. Find the probability that you (a) win your first prize with your fourth purchase, (b) win your first prize with your first, second, or third purchase, and (c) do not win a prize with your first four purchases.

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Major Hurricanes A major hurricane is a hurricane with wind speeds of 111 miles per hour or greater. During the 20th century, the mean number of major hurricanes to strike the U.S. mainland per year was about 0.6. Find the probability that the number of major hurricanes striking the U.S. mainland in any given year is (a) exactly one, (b) at most one, and (c) more than one. (Source: National Hurricane Center)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Nuclear Energy Fifty-seven percent of U.S. adults favor using nuclear energy as a source of electricity in the United States. You randomly select eight U.S. adults. Find the probability the number of U.S. adults who favor using nuclear energy as a source of electricity in the United States is (a) exactly four, (b) less than five, and (c) at least three. (Source: Gallup Poll)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Heart Transplants The mean number of heart transplants performed per day in the United States in a recent year was about six. Find the probability that the number of heart transplants performed on any given day is (a) exactly seven, (b) at least eight, and (c) no more than four. (Source: U.S. Department of Health and Human Services)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Breaking Up Twenty-nine percent of Americans ages 16 to 21 years old say that they would break up with their boyfriend or girlfriend for $10,000. You randomly select seven 16- to 21-year-olds. Find the probability that the number of 16- to 21-year-olds who say that they would break up with their boyfriend or girlfriend for $10,000 is (a) exactly two, (b) more than three, and (c) between one and four, inclusive. (Source: Bank of America Student Banking & Seventeen)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Education Fifty-four percent of parents would give up cable television to have their childs education paid for. You randomly select five parents. Find the probability that the number of parents who would give up cable television to have their childs education paid for is (a) exactly three, (b) less than four, and (c) at least three. (Source: Gerber Life College Plan Survey)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Pilot Test The probability that a student passes the written test for a private pilot license is 0.75. Find the probability that the student (a) passes on the first attempt, (b) passes on the second attempt, and (c) does not pass on the first or second attempt.

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Cheating Forty-two percent of adults say that they have cheated on a test or exam before. You randomly select six adults. Find the probability that the number of adults who say that they have cheated on a test or exam before is (a) exactly four, (b) more than two, and (c) at most five. (Source: Rasmussen Reports)

Using a Distribution to Find Probabilities In Exercises 1126, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.Oil Tankers The mean number of oil tankers at a port city is eight per day. Find the probability that the number of oil tankers on any given day is (a) exactly eight, (b) at most three, and (c) more than eight.

Comparing Binomial and Poisson Distributions An automobile manufacturer finds that 1 in every 2500 automobiles produced has a particular manufacturing defect. (a) Use a binomial distribution to find the probability of finding 4 cars with the defect in a random sample of 6000 cars. (b) The Poisson distribution can be used to approximate the binomial distribution for large values of n and small values of p. Repeat part (a) using a Poisson distribution and compare the results

Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and N - k failures, the probability of selecting a sample of size n that has x successes and n - x failures is given by P1x2 = 1kCx21N - kCn - x2 NCn . In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Find the probability that (a) all three microchips are not defective, (b) one microchip is defective and two are not defective, and (c) two microchips are defective and one is not defective

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is m = 1p and the variance is s2 = qp2 .Daily Lottery A daily number lottery chooses three balls numbered 0 to 9. The probability of winning the lottery is 1/1000. Let x be the number of times you play the lottery before winning the first time. (a) Find the mean, variance, and standard deviation. (b) How many times would you expect to have to play the lottery before winning? It costs $1 to play and winners are paid $500. Would you expect to make or lose money playing this lottery? Explain.

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is m = 1p and the variance is s2 = qp2 .Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

Poisson Distribution: Variance In Exercises 31 and 32, use the fact that the variance of a Poisson distribution is s2 = m.. Golf In a recent year, the mean number of strokes per hole for golfer Phil Mickelson was about 3.9. (a) Find the variance and standard deviation. Interpret the results. (b) Find the probability that he would play an 18-hole round and have more than 72 strokes? (Source: PGATour.com)

Poisson Distribution: Variance In Exercises 31 and 32, use the fact that the variance of a Poisson distribution is s2 = m.Bankruptcies The mean number of bankruptcies filed per hour by businesses in the United States in a recent year was about five. (a) Find the variance and the standard deviation. Interpret the results. (b) Find the probability that at most three businesses will file bankruptcy in any given hour. (Source: Administrative Office of the U.S. Courts)