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All Textbook Solutions for Finite Mathematics and Applied Calculus (MindTap Course List)
58E59E60E61E62E63E64E65EIf higher and higher powers of P approach a fixed matrix Q, explain why the rows of Q must be steady-state distributions vectors.1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14REA die is constructed in such a way that rolling a 6 is twice as likely as rolling each other number. That die is rolled four times. Let X be the number of times a 6 is rolled. Evaluate the probabilities in Exercises 1320. The probability that 6 comes up at most twice16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46REMac vs. Windows On average, 5% of all hits by Mac OS users and 10% of all hits by Windows users result in orders for books at OHaganBooks.com. Due to online promotional efforts, the site traffic is approximately 10 hits per hour by Mac OS users, and 20 hits per hour by Windows users. Compute the probabilities in Exercises 4348. (Round all answers to three decimal places.) How many orders for books can OHaganBooks.com expect in the next hour from Mac OS users?48RE49RE50RE51RE52RE53RE54RE55RE56RE1CS2CS3CS4CS5CS6CS7CS8CSIn Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] Roll two dice; X= the sum of the numbers facing up.2EIn Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] Select a stock at random; X= your profit, to the nearest dollar, if you purchase one share and sell it one year later.In Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] Select an electric utility company at random; X= the exact amount of electricity, in gigawatt hours, it supplies in a year.5EIn Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] Watch a soccer game; X= 5 the total number of goals scored.7EIn Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] Your class is given a mathematics exam worth 100 points; X is the average score, rounded to the nearest whole number.In Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] According to quantum mechanics, the energy of an electron in a hydrogen atom can assume only the values k/1,k/4,k/9,k/16, for a certain constant value k. X= the energy of an electron in a hydrogen atom.In Exercises 110, classify the random variable X as finite, discrete infinite, or continuous, and indicate the values that X can take. [hinT: See Quick Examples 510.] According to classical mechanics, the energy of an electron in a hydrogen atom can assume any positive value. X= the energy of an electron in a hydrogen atom.11E12E13E14EIn Exercises 1118, (a) say what an appropriate sample space is; (b) complete the following sentence: X is the rule that assigns to each ; and (c) list the values of X for all the outcomes. [hinT: See Example 1.] X is the number of red marbles that Tonya has in her hand after she selects four marbles from a bag containing four red marbles and two green ones and then notes how many there are of each color.16EIn Exercises 1118, (a) say what an appropriate sample space is; (b) complete the following sentence: X is the rule that assigns to each ; and (c) list the values of X for all the outcomes. [hinT: See Example 1.] The mathematics final exam scores for the students in your study group are 89%, 85%, 95%, 63%, 92%, and 80%.18E19E20E21E22EIn Exercises 2128, give the probability distribution for the indicated random variable, draw the corresponding histogram, and calculate the indicated probability. [hinT: See Example 3.] Three fair coins are tossed, and X is the square of the number of heads showing. Calculate P(1X9).24E25E26E27E28E2010 Income Distribution up to $100,000 The following table shows the distribution of household incomes in 2010 for a sample of 1,000 households in the United States with incomes up to $100,000:2 Income Bracket ($) 0 19,999 20,000 39,999 40,000 59,999 60,000 79,999 80,000 99,999 Households 240 290 180 170 120 a. Let X be the (rounded) midpoint of a bracket in which a household falls. Find the relative frequency distribution of X, and graph its histogram. [hinT: See Example 4.] b. Shade the area of your histogram corresponding to the probability that a randomly selected U.S. household in the sample has a value of X above 50,000. What is this probability?2003 Income Distribution up to $100,000 Repeat Exercise 29, using the following data from a sample of 1,000 households in the United States in 2003:3 Income Bracket ($) 0 19,999 20,000 39,999 40,000 59,999 60,000 79,999 80,000 99,999 Households 270 280 200 150 10031E32E33EHousing Prices Going into the Real Estate Bubble The following table shows the average percentage increase in the price of a house from 1980 to 2001 in nine regions of the United States:7 Region Percent Increase New England 300 Pacific 225 Middle Atlantic 225 South Atlantic 150 Mountain 150 West North Central 125 West South Central 75 East North Central 150 East South Central 125 Let X be the percentage increase in the price of a house in a randomly selected region. a. What are the values of X? b. Compute the frequency and probability distribution of X. [hinT: See Example 5.] c. What is the probability that, in a randomly selected region, the percentage increase in the cost of a house exceeded 200%?35E36E37E38ECar Purchases To persuade his parents to contribute to his new car fund, Carmine has spent the last week surveying the ages of 2,000 cars on campus. His findings are reflected in the following frequency table: Age of Car (years) 0 1 2 3 4 5 6 7 8 9 10 Number of Cars 140 350 450 650 200 120 50 10 5 15 10 Carmines jalopy is 6 years old. He would like to make the following claim to his parents: x percent of students have cars newer than mine. Use a relative frequency distribution to find x.40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64EIn Exercises 110, you are performing five independent Bernoulli trials with p=.1 and q=.9. Calculate the probability of the stated outcome. Check your answer using technology. [hinT: See Quick Example 8.] Two successes2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23EAlien Retirement The probability that a randomly chosen citizen-entity of Cygnus is of pension age13 is approximately .8. What is the probability that, in a randomly selected sample of four citizen-entities, all of them are of pension age? [hinT: See Example 2.]25E26E27E28EManufacturing Your manufacturing plant produces air bags, and it is known that 10% of them are defective. Five air bags are tested. a. Find the probability that three of them are defective. b. Find the probability that at least two of them are defective.Manufacturing Compute the probability distribution of the binomial variable described in Exercise 29, and use it to compute the probability that if five air bags are tested, at least one will be defective and at least one will not.31EOther Teenage Pastimes According to the study cited in Exercise 31, the probability that a randomly selected teen-ager studied at least once during a week was only .52. What is the probability that less than half of the students in your study group of 10 have studied in the last week?33E34ETriple Redundancy To ensure reliable performance of vital computer systems, aerospace engineers sometimes employ the technique of triple redundancy, in which three identical computers are installed in a space vehicle. If one of the three computers gives results different from the other two, it is assumed to be malfunctioning and is ignored. This technique will work as long as no more than one computer malfunctions. Assuming that an onboard computer is 99% reliable (that is, the probability of its failing is .01), what is the probability that at least two of the three computers will malfunction?IQ Scores Mensa is a club for people who have high IQ scores. To qualify, your IQ must be at least 132, putting you in the top 2% of the general population. If a group of 10 people are chosen at random, what is the probability that at least 2 of them qualify for Mensa?Standardized Tests Assume that on a standardized test of 100 questions, a person has a probability of 80% of answering any particular question correctly. Find the probability of correctly answering between 75 and 85 questions, inclusive. (Assume independence, and round your answer to four decimal places.)38EProduct Testing It is known that 43% of all the Zero Fat hamburger patties produced by your factory actually contain more than 10 grams of fat. Compute the probability distribution for n=50 Bernoulli trials. a. What is the most likely value for the number of burgers in a sample of 50 that contain more than 10 grams of fat? b. Complete the following sentence: There is an approximately 71% chance that a batch of 50 Zero Fat patties contains ____ or more patties with more than 10 grams of fat. c. Compare the graphs of the distributions for n=50 trials and n=20 trials. What do you notice?Product Testing It is known that 65% of all the ZeroCal hamburger patties produced by your factory actually contain more than 1,000 calories. Compute the probability distribution for n=50 Bernoulli trials. a. What is the most likely value for the number of burgers in a sample of 50 that contain more than 1,000 calories? b. Complete the following sentence: There is an approximately 73% chance that a batch of 50 ZeroCal patties contains ____ or more patties with more than 1,000 calories. c. Compare the graphs of the distributions for n=50 trials and n=20 trials. What do you notice?Quality Control A manufacturer of light bulbs chooses bulbs at random from its assembly line for testing. If the probability of a bulbs being bad is .01, how many bulbs does the manufacturer need to test before the probability of finding at least one bad one rises to more than .5? (You may have to use trial and error to solve this.)42E43E44E45E46E47E48E49E50E51E52E53E54E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23EIn Exercises 1728, calculate the expected value of the given random variable X. [Exercises 23, 24, 27, and 28 assume familiarity with counting arguments and probability (see Section 8.4).] [hinT: See Quick Example 6.] X is the number of green marbles that Suzan has in her hand after she selects four marbles from a bag containing three red marbles and two green ones.25E26E27E28E29E30E31E32E33E34ESchool Enrollment The following table shows the approximate numbers of school goers in the United States (residents who attended some educational institution) in 1998, broken down by age group:27 Age 36.9 712.9 1316.9 1722.9 2326.9 2742.9 Population (millions) 12 24 15 14 2 5 Use the rounded midpoints of the given measurement classes to compute the probability distribution of the age X of a school goer. (Round probabilities to two decimal places.) Hence, compute the expected value of X. What information does the expected value give about residents enrolled in schools?36E37E38E39E2003 Income Distribution up to $100,000 Repeat Exercise 39, using the following data32 from a sample of 1,000 households in the United States in 2003: Income Bracket ($) 019,999 20,00039,999 40,00059,999 60,00079,999 80,00099,999 Households 270 280 200 150 100Highway Safety Exercises 4144 are based on the following table, which shows crashworthiness ratings for several categories of motor vehicles.33 In all of these exercises, take X as the crash-test rating of a small car, Y as the crash-test rating for a small SUV, and so on as shown in the table. Overall Frontal Crash Test Rating Number Tested 3(Good) 2(Acceptable) 1(Marginal) 0(Poor) Small Cars, X 16 1 11 2 2 Small SUVs, Y 10 1 4 4 1 Medium SUVs, Z 15 3 5 3 4 Passenger Vans, U 13 3 0 3 7 Midsize Cars, V 15 3 5 0 7 Large Cars, W 19 9 5 3 2 Compute the probability distributions and expected values of X and Y. On the basis of the results, which of the two types of vehicles performed better in frontal crashes?42EHighway Safety Exercises 4144 are based on the following table, which shows crashworthiness ratings for several categories of motor vehicles.33 In all of these exercises, take X as the crash-test rating of a small car, Y as the crash-test rating for a small SUV, and so on as shown in the table. Overall Frontal Crash Test Rating Number Tested 3(Good) 2(Acceptable) 1(Marginal) 0(Poor) Small Cars, X 16 1 11 2 2 Small SUVs, Y 10 1 4 4 1 Medium SUVs, Z 15 3 5 3 4 Passenger Vans, U 13 3 0 3 7 Midsize Cars, V 15 3 5 0 7 Large Cars, W 19 9 5 3 2 On the basis of expected values, which of the following categories performed best in crash tests: small cars, midsize cars, or large cars?Highway Safety Exercises 4144 are based on the following table, which shows crashworthiness ratings for several categories of motor vehicles.33 In all of these exercises, take X as the crash-test rating of a small car, Y as the crash-test rating for a small SUV, and so on as shown in the table. Overall Frontal Crash Test Rating Number Tested 3(Good) 2(Acceptable) 1(Marginal) 0(Poor) Small Cars, X 16 1 11 2 2 Small SUVs, Y 10 1 4 4 1 Medium SUVs, Z 15 3 5 3 4 Passenger Vans, U 13 3 0 3 7 Midsize Cars, V 15 3 5 0 7 Large Cars, W 19 9 5 3 2 On the basis of expected values, which of the following categories performed best in crash tests: small SUVs, medium SUVs, or passenger vans?45ERoulette A roulette wheel has the numbers 1 through 36, 0, and 00. A bet on two numbers pays 17 to 1 (that is, if you bet $1 and one of the two numbers you bet comes up, you get back your $1 plus another $17). How much do you expect to win with a $1 bet on two numbers? [hinT: See Example 4.]47E48E49E50EExercises 51 and 52 assume familiarity with counting arguments and probability (see Section 8.4). Camping Kents Tents has four red tents and three green tents in stock. Karin selects four of them at random. Let X be the number of red tents she selects. Give the probability distribution of X and find the expected number of red tents selected.Exercises 51 and 52 assume familiarity with counting arguments and probability (see Section 8.4). Camping Kents Tents has five green knapsacks and four yellow ones in stock. Curt selects four of them at random. Let X be the number of green knapsacks he selects. Give the probability distribution of X, and find the expected number of green knapsacks selected.Elimination Tournaments In an elimination tournament the teams are arranged in opponent pairs for the first round, and the winner of each round goes on to the next round until the champion emerges. The following diagram illustrates a 16-team tournament bracket, in which the 16 participating teams are arranged on the left under Round 1 and the winners of each round are added as the tournament progresses. The top team in each game is considered the home team, so the top-to-bottom order matters. To seed a tournment means to select which teams to play each other in the first round according to their preliminary ranking. For instance, in professional tennis and NCAA basketball the seeding is set up in the following order based on the preliminary rankings: 1 versus 16, 8 versus 9, 5 versus 12, 4 versus 13, 6 versus 11, 3 versus 14, 7 versus 10, and 2 versus 15.35 Exercises 5356 are based on various types of elimination tournaments. Someone offers you the following bet: If a randomly chosen seeding of a 16-team tournament results in the top-ranked team playing the bottom-ranked team, the second-ranked team playing the second-lowest ranked team, and so on, you win $1 million; otherwise, you lose $1. What are your expected winnings on this bet? [hinT: See Exercise 37 in Section 8.4.]54E55E56E57E58E59EInsurance The Blue Sky Flight Insurance Company insures passengers against air disasters, charging a prospective passenger $20 for coverage on a single plane ride. In the event of a fatal air disaster, it pays out $100,000 to the named beneficiary. In the event of a nonfatal disaster, it pays out an average of $25,000 for hospital expenses. Given that the probability of a planes crashing on a single trip37 is .00000087, and assuming that a passenger involved in a plane crash has a .9 chance of being killed, determine the profit (or loss) per passenger that the insurance company expects to make on each trip. [hinT: Use a tree to compute the probabilities of the various outcomes.]61E62E63E64E65E66E67E68E69E70E71E72E73E74EIn Exercises 18, compute the (sample) variance and standard deviation of the given data sample. (You calculated the means in the Section 9.3 exercises. Round all answers to two decimal places.) [hinT: See Quick Examples 1 and 2.] 1, 5, 5, 7, 142E3E4E5E6E7E8EIn Exercises 914, calculate the standard deviation of X for each probability distribution. (You calculated the expected values in the Section 9.3 exercises. Round all answers to two decimal places.) [hinT: See Quick Example 5.] x 0 1 2 3 P(X=x) .5 .2 .2 .110E11E12E13E14E15E16E17E18E19EIn Exercises 1524, calculate the expected value, the variance, and the standard deviation of the given random variable X. (You calculated the expected values in the Section 9.3 exercises. Round all answers to two decimal places.) X is the lower number when two dice are rolled.21EIn Exercises 1524, calculate the expected value, the variance, and the standard deviation of the given random variable X. (You calculated the expected values in the Section 9.3 exercises. Round all answers to two decimal places.) X is the number of green marbles that Suzan has in her hand after she selects four marbles from a bag containing three red marbles and two green ones.In Exercises 1524, calculate the expected value, the variance, and the standard deviation of the given random variable X. (You calculated the expected values in the Section 9.3 exercises. Round all answers to two decimal places.) Twenty darts are thrown at a dartboard. The probability of hitting a bulls-eye is .1. Let X be the number of bulls-eyes hit.In Exercises 1524, calculate the expected value, the variance, and the standard deviation of the given random variable X. (You calculated the expected values in the Section 9.3 exercises. Round all answers to two decimal places.) Thirty darts are thrown at a dartboard. The probability of hitting a bulls-eye is 15. Let X be the number of bulls-eyes hit.25E26EUnemployment Following is a sample of unemployment rates (in percentage points) in the United States sampled from the period 19902004:40 4.2, 4.7, 5.4, 5.8, 4.9. a. Compute the mean and standard deviation of the given sample. (Round your answers to one decimal place.) b. Assuming that the distribution of unemployment rates in the population is symmetric and bell shaped, 95% of the time, the unemployment rate is between ____ and ____ percent.28E29E30E31E32E33E34E2010 Income Distribution up to $100,000 The following table shows the distribution of household incomes in 201046 for a sample of 1,000 households in the United States with incomes up to $100,000: Income ($1,000) 10 30 50 70 90 Households 240 290 180 170 120 Compute the expected value m and the standard deviation s of the associated random variable X. If we define a lower-income family as one whose income is more than one standard deviation below the mean and a higher-income family as one whose income is at least one standard deviation above the mean, what is the income gap between higherand lower-income families in the United States? (Round your answers to the nearest $1,000.)