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All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
In Problems 21.26, given a normal distribution with mean 25 and standard deviation 5, find the area under the normal curve from the mean to the indicated measurement 18.7In Problems 21.26, given a normal distribution with mean 25 and standard deviation 5, find the area under the normal curve from the mean to the indicated measurement 28.3In Problems 21.26, given a normal distribution with mean 25 and standard deviation 5, find the area under the normal curve from the mean to the indicated measurement 23.9In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. 48,60In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. 60,84Consider the normal distribution with mean 60 and standard deviation 12 . Find the area under the normal curve and above the given interval on the horizontal axis (57,63)In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. 54,66In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. ,54In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. 63,In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. 51,In Problems 27-34, consider the normal distribution with mean 60 and standard deviation 12. Find the area under the normal curve and above the given interval on the horizontal axis. ,78In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The mean of a normal distribution is a positive real numberIn Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The standard deviation of a normal distribution is a positive real numberIn Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If two normal distributions have the same mean and standard deviation, then they have the same shape.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If two normal distributions have the same mean, then they have the same standard deviationIn Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The area under a normal distribution and above the horizontal axis is equal to 1.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. In a normal distribution, the probability is 0 that a score lies more than 3 standard deviations away from the mean.In Problems 41-48, use the rule-of-thumb test to check whether anormal distribution (with the same mean and standard deviationas the binomial distribution) is a suitable approximation for thebinomial distribution with n=15,p=0.7In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=12,p=.6In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=15,p=.4In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=20,p=.6In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=100,p=.05In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=200,p=.03In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=500,p=.05In Problems 41-48, use the rule-of-thumb test to check whether a normal distribution (with the same mean and standard deviation as the binomial distribution) is a suitable approximation for the binomial distribution with n=400,p=.08The probability of success in a Bernoulli trial is p=.1. Explain how to determine the number of repeated trials necessary to obtain a binomial distribution that passes the rule-of-thumb test for using a normal distribution as a suitable approximation.For a binomial distribution with n=100, explain how to determine the smallest and largest values of p that pass the rule-of-thumb test for using a normal distribution as a suitable approximation.A binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 185220A binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 190205A binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 210220A binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 175185A binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 225 or moreA binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 212 or moreA binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 175 or moreA binomial experiment consists of 500 trials. The probability of success for each trial is .4. What is the probability of obtaining the number of successes indicated in Problems 51-58? Approximate these probabilities to two decimal places using a normal curve. (This binomial experiment easily passes the rule-of-thumb test, as you can check. When computing the probabilities, adjust the intervals as in Examples 3 and 4.) 188 or moreTo graph Problems 59-62, use a graphing calculator and refer to the normal probability distribution function with mean and standard deviation : fx=12ex2/22 Graph equation(1) with =5 and (A) =10 (B) =15 (C) =20 Graph all three in the same viewing window with Xmin=10,Xmax=40, Ymin=0 and Ymax=0.1To graph Problems 59-62, use a graphing calculator and refer to the normal probability distribution function with mean and standard deviation : fx=12ex2/22 Graph equation (1) with =5 and (A) =8 (B) =12 (C) =16 Graph all three in the same viewing window with Xmin=10,Xmax=30,Ymin=0,and Ymax=0.1.To graph Problems 59-62, use a graphing calculator and refer to the normal probability distribution function with mean and standard deviation : fx=12ex2/22 Graph equation (1) with =20 and (A) =2 (B) =4 Graph both in the same viewing window with Xmin=0,Xmax=40,Ymin=0,andYmax=0.2.To graph Problems 59-62, use a graphing calculator and refer to the normal probability distribution function with mean and standard deviation : fx=12ex2/22 Graph equation (1) with =18 and (A) =3 (B) =6 Graph both in the same viewing window with Xmin=0,Xmax=40,Ymin=0,andYmax=0.2(A) If 120 scores are chosen from a normal distribution with mean and standard deviation 8, how many scores x would be expected to satisfy 67x83 (B) Usea graphing calculator to generate 120 scores from the normal distribution with mean 75 and standard deviation 8. Determine the number of scores x such that 67x83, and compare your results with the answerto part (A).(A) If 250 scores are chosen from a normal distribution with mean 100 and standard deviation 10, how many scores x would be expected to be greater than 110 ? (B) Use a graphing calculator to generate 250 cores from the normal distribution with mean 100 and standard deviation 10. Determine the number of scores greater than 110, and compare your results with the answer to part (A).Sales Salespeople for a solar technology company have average annual sales of 200,000, with a standard deviation of 20,000 What percentage of the salespeople would be expected to make annual sales of 240,000 or more? Assume a normal distribution.Guarantees. The average lifetime for a car battery is 170 weeks, with a standard deviation of 10 weeks. If the company guarantees the battery for 3 years, what percentage of the batteries sold would be expected to be returned before the end of the warranty period? Assume a normal distribution.Quality control. A manufacturing process produces a critical part of average length 100 millimeters, with a standard deviation of 2 millimeters. All parts deviating by more than 5 millimeters from the mean must be rejected. What percentage of the parts must be rejected, on the average? Assume a normal distribution.Quality control. An automated manufacturing process produces a component with an average width of 7.55 centimeters, with a standard deviation of 0.02 centimeter. All components deviating by more than 0.05 centimeter from the mean must be rejected. What percentage of the parts must be rejected, on the average? Assume a normal distribution.Marketing claims. A company claims that 60 of the households in a given community use its product. A competitor surveys the community, using a random sample of 40 households, and finds only 15 households out of the 40 in the sample use the product. If the company’s claim is correct, what is the probability of 15 or fewer households using the product in a sample of 40 ? Conclusion? Approximate a binomial distribution with a normal distributionLabor relation A union representative 60 claims of the union membership will vote in favor of a particular settlement. A random sample of 100 members is polled, and out of these, 47 favor the settlement. What is the approximate probability of 47 or fewer in a sample of 100 favoring the settlement when 60 of all the membership favor the settlement? Conclusion? Approximate a binomial distribution with a normal distributionMedicine. The average healing time of a certain type of incision is 240 is hours, with standard deviation of 20 hours. What percentage of the people having this incision would heal in 8 days or less? Assume a normal distributionAgriculture. The average height of a hay crop is 38 inches, with a standard deviation of 1.5 inches. What percentage of the crop will be 40 inches or more? Assume a normal distributionGenetics. In a family with 2 children, the probability that both children are girls is approximately .25. In a random sample of 1,000 families with 2 children, what is the approximate probability that 220 or fewer will have 2 girls? Approximate a binomial distribution with a normal distribution.Genetics. In Problem 73, what is the approximate probability of the number of families with 2 girls in the sample being at least 225 and not more than 275 ? Approximate a binomialdistribution with a normal distribution.Testing. Scholastic Aptitude Tests (SATs) are scaled so that the mean score is 500 and the 100 standard deviation is. What percentage of students taking this test should score 700 or more? Assume a normal distribution.Politics. Candidate Harkins claims that she will receive 52 of the vote for governor. Her opponent, Mankey, finds that 470 out of a random sample of 1,000 registered voters favor Harkins. If Harkins's claim is correct, what is the probability that only 470 or fewer will favor her in a random sample of 1,000 ? Conclusion? Approximate a binomial distributionwith a normal distribution.Grading on a curve. An instructor grades on a curve by assuming that grades on a test are normally distributed. If the average grade is 70 and the standard deviation is 8, find the test scores for each grade interval if the instructor assigns grades as follows: 10As, 20Bs, 40Cs,20Ds, and 10Fs.Psychology. A test devised to measure aggressive-passive personalities was standardized on a large group of people. The scores were normally distributed with a mean of 50 and a standard deviation of 10. If we designate the highest 10 as aggressive, the next 20 as moderately aggressive, the middle 40 as average, the next 20 as moderately passive, and the lowest 10 as passive, what ranges of scores will be covered by these five designations?Use a bar graph and a broken-line graph to graph the data on voter turnout, as a percentage of the population eligible to vote, in U.S. presidential elections.Use a pie graph to graph the data on educational attainment in the U.S. population of adults 25 years of age or older.(A) Draw a histogram for the binomial distribution Px=3Cx,.4x.63x (B) What are the mean and standard deviation?For the set of sample measurements 1,1,2,2,2,3,3,4,4,5, find the (A)Mean (B)Median (C)Mode (D)Standard DeviationIf a normal distribution has a mean of 100 and a standard deviation of 10, then (A) How many standard deviations is 118 from the mean? (B) What is the area under the normal curve between the mean and 118 ?Given the sample of 25 quiz scores listed in the following table from a class of 500 students: (A) Construct a frequency table using a class interval of width 2, starting at 9.5. (B) Construct a histogram. (C) Construct a frequency polygon (D) Construct a cumulative frequency and relative cumulative frequency table. (E) Construct a cumulative frequency polygon.For the set of grouped sample data given in the table, (A) Find the mean. (B) Find the standard deviation. (C) Find the median(A) Construct a histogram for the binomial distribution Px=6Cx.5x.56x (B) What are the mean and standard deviation?What are the mean and standard deviation for a binomial distribution with p=.6 and n=1,000 ?In Problems 10 and 11, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. (A) If the data set x1,x2,xn has mean x, then the data set x1+5,x2+5,,xm+5 has mean x+5 (B) If the data set x1,x2,,xn has standard deviation s, then the data set x1+5,x2+5,,xn+5 has standard deviation s+5.In Problems 10 and 11, discuss the validity of each statement. If thestatement is always true, explain why. If not, give a counterexample. (A) If X represents a binomial random variable with mean , then PX=.5 (B) If X represents a normal random variable with mean then PX=.5. (C) The area of a histogram of a binomial distribution is equal to the area above the x axis and below a normal curve.If the probability of success in a single trial of a binomial experiment with 1,000 trials is 6, what is the probability of obtaining at least 550 and no more than 650 successes in 1,000 trials?Given a normal distribution with mean 50 and standard deviation 6, find the area under the normal curve (A) Between 41 and 62 (B) From 59 onA data set is formed by recording the sums of 100 rolls of a pair of dice. A second data set is formed by again rolling a pair of dice 100 times but recording the product, not the sum, of the two numbers. (A) Which of the two data sets would you expect to have the smaller standard deviation? Explain. (B) To obtain evidence for your answer to part (A), use a graphing calculator to simulate both experiments, and compute the standard deviations of each of the two data sets.For the sample quiz scores in Problem 6, find the mean and standard deviation using the data (A) Without grouping (B) Grouped, with class interval of width 2, starting at 9.5A fair die is rolled five times. What is the probability of rolling (A) Exactly three 6s ? (B) At least three 6s ?Two dice are rolled three times. What is the probability of getting a sum of 7 at least once?Ten students take an exam worth 100 points. (A) Construct a hypothetical set of exam scores for the ten students in which both the median and the mode are 30 points higher than the mean. (B) Could the median and mode both be 50 points higher than the mean? Explain.In the last presidential election, 39 of a city’s registered voters actually cast ballots. (A)In a random sample of 20 registered voters from that city, what is the probability that exactly 8 voted in the last presidential election? (B) Verify by the rule-of-thumb test that the normal distribution with mean 7.8 and standard deviation 2.18 is a good approximation of the binomial distribution with n=20 and p=.39. (C) For the normal distribution of part (B), Px=8=0. Explain the discrepancy between this result and your answer from part (A)A random variable represents the number of wins in a 12 - game season for a football team that has a probability of .9 of winning any of its games. (A) Find the mean and standard deviation of the random variable. (B) Find the probability that the team wins each of its 12 games. (C) Use a graphing calculator to simulate 100 repetitions of the binomial experiment associated with the random variable, and compare the empirical probability of a perfect season with the answer to part (B).Retail sales. The daily number of bad checks received by a large department store in a random sample of 10 days out of the past year was 15,12,17,5,5,8,13,5,16, and 4. Find the (A) Mean (B) Median (C) Mode (D) Standard deviationPreference survey. Find the mean, median, and/or mode, whichever are applicable, for the following employee cafeteria service survey:Plant safety. The weekly record of reported accidents in a large auto assembly plant in a random sample of 35 weeks from the past 10 years is listed below (A) Construct a frequency and relative frequency table using class intervals of width 2, starting at 29.5 (B) Construct a histogram and frequency polygon (C) Find the mean and standard deviation for the grouped dataPersonnel screening.The scores on a screening test for new technicians are normally distributed with mean 100 and standard deviation 10. Find the approximate percentage of applicants taking the test who score (A) Between 92 and 108 (B) 115 or higherMarket research A newspaper publisher claims that 70 of the people in a community read their newspaper. Doubting the assertion, a competitor randomly surveys 200 people in the community. Based on the publisher's claim (and assuming a binomial distribution) (A) Compute the mean and standard deviation. (B) Determine whether the rule-of-thumb test warrants the use of a normal distribution to approximate this binomial distribution. (C) Calculate the approximate probability of finding at least 130 and no more than 155 readers in the sample (D) Determine the approximate probability of finding 125 or fewer readers in the sample. (E) Use a graphing calculator to graph the relevant normal distribution.Health care. A small town has three doctors on call for emergency service. The probability that any one doctor will be available when called is .90. What is the probability that at least one doctor will be available for an emergency call?Suppose that a and k are both saddle values of the matrix A=abcdefghijkl (A) Show that a must equal k after explaining why each of the following must be true ac,ck,ai and ik (B) Show that c and i must equal k as well.Repeat Example 1 for the HDTV game matrix discussed at the beginning of this section: C499549R49954955704066Determine which of the matrix games below are non-strictly determined: A=13022410B=2143056108In Problems 1-8, is the matrix game strictly determined? 2131In Problems 1-8, is the matrix game strictly determined? 5312In Problems 1-8, is the matrix game strictly determined? 723501In Problems 1-8, is the matrix game strictly determined? 232252In Problems 1-8, is the matrix game strictly determined? 014321112In Problems 1-8, is the matrix game strictly determined? 113271041In Problems 1-8, is the matrix game strictly determined? 425310123In Problems 1-8, is the matrix game strictly determined? 153424687In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 1021In Problems 9-16 , the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 1243In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 531548213In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 241023104In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 020321150In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 302010221In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 273130543In Problems 9-16, the matrix for a strictly determined game is given. Find the value of the game. Is the game fair? 42110142117E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32EFor the matrix game of Problem 31, would you rather be player R or player C ? Explain.For the matrix game of Problem 32, would you rather be player R or player C ? Explain.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. There exists a payoff matrix that has exactly two saddle values.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. There exists a payoff matrix having a saddle value that appears exactly twice.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The smallest entry in any payoff matrix is a saddle value.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The largest entry in any payoff matrix is a saddle value.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample If a payoff matrix has a row consisting of all 0s and a column consisting of all 0s, then the game is fair.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example If a strictly determined matrix game is fair, then at least one of the payoffs is 0.Is there a value of m such that the following is not a strictly determined matrix game? Explain. 3m0142EPrice war a small town on a major highway has only two gas stations: station R, a major brand station, and station C an independent. A market research firm provided the following payoff matrix, where each entry indicates the percentage of customers who go to station R for the indicated prices per gallon of unleaded gasoline. Find saddle values and optimal strategies for each company. StationC1.351.40StationR1.401.4550704050Investment Suppose that you want to invest $10,000 for a period of 5 years. After getting financial advice, you come up with the following game matrix, where you R are playing against the economy C. Each entry in the matrix is the expected payoff (in dollars) after 5 years for an investment of $10,000 in the corresponding row designation, with the state of the economy in the corresponding column designation. (The economy is regarded as a rational player who can make decisions against the investor in any case, the investor would like to do the best possible, irrespective of what happens to the economy.) Find saddle values and optima strategies for each player. EconomyCFallNochangeRiseInvestorR5-YearCDBlue-chipstockSpeculativestock5,8705,8705,8702,0004,0007,0005,0002,00010,000Store location two competitive pet shops want to open stores at Lake Tahoe, where there are currently no pet shops. The following figure shows the percentages of the total Tahoe population serviced by each of the three main business centers. If both shops open in the same business center, then they split all the business equally; if they open in two different centers, then they each get all the business in the center in which they open plus half the business in the third center. Where should the two pet shops open? Set up a game matrix and solve.Store location Two competing auto parts companies ( R and C ) are trying to decide among three small towns ( E, F, and G ) for new store locations. All three towns have the same business potential. If both companies operate in the same town, they split the business evenly (payoff is 0 to both). If, however, they operate in different towns, the store that is closer to the third town will get all of that town’s business. For example, if R operates in E and C in E, the payoff is 1 ( R E has gained one town). If. on the other hand. operates in F and C in, the payoff is 1 ( R has lost one town to C ). Write the payoff matrix, find all saddle values, and indicate optimal strategies for both stores.Let M=abcd (A) Show that if the row minima belong to the same column, at least one of them is a saddle value. (B) Show that if the column maxima belong to the same row, at least one of them is a saddle value. (C) Show that if a+db+c=0 then M has a saddle value (that is, M is strictly determined). (D) Explain why part (C) implies that the denominator D in Theorem 4 will never be 0(A) Using Theorem 4, give conditions on a,b,c, and d to guarantee that the Non strictly determined matrix game is fair. M=abcd (B) Construct the matrix of payoffs for a two-finger Morra game that is Non strictly determined and fair. (C) How many such matrices are there? Explain.Solve the following version of the two-finger Morra game (which is equivalent to the penny-matching game in Section 11.1): M=1111Solve the matrix game: M=112324113In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) 10231401In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) 01231410In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) 01231401In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) 10231410In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) .5.52314.5.5In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) .4.62314.5.5In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) .4.62314.7.3In Problems 1-8, calculate the matrix product. (If necessary, review Section 4.4) .5.52314.7.3In Problems 9-18, which rows and columns of the game matrix are recessive? 2135In Problems 9-18, which rows and columns of the game matrix are recessive? 1320In Problems 9-18, which rows and columns of the game matrix are recessive? 353101In Problems 9-18, which rows and columns of the game matrix are recessive? 243125In Problems 9-18, which rows and columns of the game matrix are recessive? 350421In Problems 9-18, which rows and columns of the game matrix are recessive? 250413In Problems 9-18, which rows and columns of the game matrix are recessive? 201051321In Problems 9-18, which rows and columns of the game matrix are recessive? 223411301In Problems 9-18, which rows and columns of the game matrix are recessive? 010001100In Problems 9-18, which rows and columns of the game matrix are recessive? 200010003Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 2110Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 2351Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 1301Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 1203Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 1224Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 315210Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 43241542Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 12305503Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 32345132Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 03312212Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 534120703Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 431302593Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 301213632Solve the matrix games in Problems 19-32, indicating optimal strategies P* and Q* for R and C, respectively, and the value v of the game. (Both strictly and non strictly determined games are included, so check for this first.) 123201152In Problems 33-38, discuss the validity of each statement, if them statement is always true, explain why. If not, give a counter example. If a matrix game is strictly determined, then both players have optimal strategies that are pure.In Problems 33-38, discuss the validity of each statement, if them statement is always true, explain why. If not, give a counterexample . If both players of a matrix game have optimal strategies that are mixed, then the game is non strictly determined.In Problems 33-38, discuss the validity of each statement, if them statement is always true, explain why. If not, give a counter example If a payoff matrix has a row consisting of all 0s, then that row is recessiveIn Problems 33-38, discuss the validity of each statement, if the statement is always true, explain why. If not, give a counterexample. Every payoff matrix either has a recessive row or a recessive column.In Problems 33-38, discuss the validity of each statement, if the statement is always true, explain why. If not, give a counterexample If the first-column entries of a 22 payoff matrix are equal, then the game is strictly determined.In Problems 33-38, discuss the validity of each statement, if the statement is always true, explain why. If not, give a counterexample . If a matrix game is fair, then both players have optimal strategies that are pure.You R and a friend C are playing the following matrix game, where the entries indicate your winnings from C in dollars .In order to encourage your friend to play, since you cannot lose as the matrix is written, you pay her $3 before each game. CR0210435402610103 (A) If selection of a particular row or a particular column is made at random, what is your expected value? (B) If you make whatever row choice you wish and your opponent continues to select a column at random, what is your expected value, assuming that you optimize your choice? (C) If you both make your own choices, what is your expected value, assuming that you both optimize your choices?You R and a friend C are playing the following matrix game, where the entries indicate your winnings from C in dollars . To encourage your friend to play, you pay her $4 before each game. The jack, queen, king, and ace from the hearts, spades, and diamonds are taken from a standard deck of cards. The game is based on a random draw of a single card from these 12 cards. The play is indicated at the top and side of the matrix. CHSDRJQKA106221447126 (A) If you select a row by drawing a single card and your friend selects a column by drawing a single card (after replacement), what is your expected value of the game? (B) If your opponent chooses an optimal strategy (ignoring the cards) and you make your row choice by drawing a card, what is your expected value? (C) If you both disregard the cards and make your own choices, what is your expected value, assuming that you both choose optimal strategies?For M=abcdP=p1p2Q=q1q2 Show that PMQ=EP,QUsing the fundamental theorem of game theory, prove that P*MQ*=vShow non strictly that the determined solution formulas matrix game (Theorem meet the 4 ) conditions for a 22 for a solution stated in Theorem 2.Show that if a 22 matrix game has a saddle value, then either one row is recessive or one column is recessive.Explain how to construct a 22 matrix game M for which the optimal strategies are P*=.9.1andQ*=.3.7Explain how to construct a 22 matrix game M for which the optimal strategies are P*=.6.4andQ*=.8.2In Problems 47 and 48, derive the formulas of Theorem 4 for the solution of any 22 non strictly determined matrix game by rewriting and analyzing EP,Q=ap1q2+bp1q2+cp2q1+dp2q2 (4) (See the solution of the two-finger Morra game on pages G9-G11.) (A) Let p2=1p1 and q2=1q1 and simplify (4) to show that EP,Q=Dp1dcq1+bdp1+d where D=a+db+c. (B) Show that if p1 is chosen so that Dp1dc=0, then v=adbcD, regardless of the value of p1.In Problems 47 and 48, derive the formulas of Theorem 4 for the solution of any 22 non strictly determined matrix game by rewriting and analyzing EP,Q=ap1q2+bp1q2+cp2q1+dp2q2 (4) (See the solution of the two-finger Morra game on pages G9-G11.) (A) Let p2=1p1 and q2=1q1 and simplify (4) to show that EP,Q=Dq1dbp1+cdq1+d where Dq1db=0. (B) Show that if q1 is chosen so that Dq1db=0 then v=adbcD, regardless of the value of p1.Bank promotion A town has only two banks, bank R band bank C, and both compete equally for the town’s business. Every week, each bank decides on the use of one, and only one, of the following means of promotion: TV, radio, newspaper. and mail. A market research firm provided the following payoff matrix, which indicates the percentage of market gain or loss for each choice of action by R and by C (we assume that any gain by R is a loss by C, and vice versa): CTVRadioPaperMailRTVRadioPaperMail0101121111010110 (A) Find optimal strategies for bank R and bank C. What is the value of the game? (B) What is the expected value of the game for R if bank R always chooses TV and bank C uses its optimal strategy? (C) What is the expected value of the game for R if bank C always chooses radio and bank R uses its optimal strategy? (D) What is the expected value of the game for R if both banks always use the newspaper?Viewer ratings A city has two competitive television stations, station R and station C. Every month, each station makes exactly one choice for the Thursday 89P.M. time slot from the program categories shown in the following matrix. Each matrix entry is an average viewer index rating gain (or loss) established by a rating firm using data collected over the past 5 years. (Any R gain for station is a loss for station C, and vice versa.) CNaturefilmsTalksShowsSportsEventsMoviesRTravelNewsSitcomsSoaps0121203211100210 (A) Find the optimal strategies for station R and station C. What is the value of the game? (B) What is the expected value of the game for R if station R always chooses travel and station C uses its optimal strategy? (C) What is the expected value of the game for R if station C always chooses movies and station R uses its optimal strategy? (D) What is the expected value of the game for R if station R always chooses sitcoms and station C always chooses sports events?Investment You have inherited $10,000 just prior to a presidential election and wish to invest it in solar energy and oil stocks. An investment advisor provides you with a payoff matrix that indicates your probable 4 -year gains, depending on which party comes into office. How should you invest your money so that you would have the largest expected gain irrespective of how the election turns out? PlayerC(fate)RepublicanDemocratPlayerR(you)SolarenergyOil1,0005,0004,0003,000 Note: For a one-time play (investment), you would split your investment proportional to the entries in your optimal strategy matrix. Assume that fate is a very clever player. Then if fate deviates from its optimal strategy, you know you will not do any worse than the value of the game, and you may do better.Corporate farming For a one-time play (investment), you would split your investment proportional to the entries in your optimal strategy matrix. Assume that fate is a very clever player. Then if fate deviates from its optimal strategy, you know you will not do any worse than the value of the game, and you may do better. Weather(fate)WetNormalDryCorporatefarmWheatRice278323 The managers would like to determine the best strategy against the weather's “best strategy" to destroy them. Then, no matter what the weather does, the farm will do no worse than the value of the game and may do a lot better. This information could be very useful to the company when applying for loans. Note: For each year that the payoff matrix holds, the farm can split the planting between wheat and rice proportional to the size of entries in its optimal strategy matrix. (A) Find the optimal strategies for the farm and the weather, and the value of the game. (B) What is the expected value of the game for the farm if the weather (fate) chooses to play the pure strategy “wet" for many years, and the farm continues to play its optimal strategy? (C) Answer part (B), replacing “wet" with “normal." (D) Answer part (B), replacing “wet" with “dry.”Show that M=1132 is a strictly determined matrix game. Nevertheless, apply the geometric procedure given for non strictly determined matrix games to M . Does something go wrong? Do you obtain the correct optimal strategies? Explain.Solve the following matrix game using geometric linear programming methods: M=2413In problem 1-6, find the smallest integer k0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 3526In problem 1-6, find the smallest integer k0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 4112In problem 1-6, find the smallest integer k0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 0120In problem 1-6, find the smallest integer k0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 6851In problem 1-6, find the smallest integer k0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 1247In problem 1-6, find the smallest integer k0 such that adding k to each entry of the given matrix produces a matrix with all positive payoffs. 3213In problem 7-12, solve the matrix game using a geometric linear programming approach 2312In problem 7-12, solve the matrix game using a geometric linear programming approach 2121In problem 7-12, solve the matrix game using a geometric linear programming approach 1326In problem 7-12, solve the matrix game using a geometric linear programming approach 4623In problem 7-12, solve the matrix game using a geometric linear programming approach 2156In problem 7-12, solve the matrix game using a geometric linear programming approach 6211Is there a better way to solve the matrix game in Problem 11 than the geometric linear programming approach? Explain.Is there a better way to solve the matrix game in Problem 12 than the geometric linear programming approach? Explain.Explain why the value of a matrix game is positive if all of the payoffs are positive.Explain why the value of a matrix game is negative if all of the payoffs are negative.In Problem 17-20, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If all payoffs of a matrix game are zero then the game is fair.In Problem 17-20, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If all payoffs of a matrix game is positive then all payoffs are positive.In Problem 17-20, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If half of the payoffs of a game matrix are positive and half are negative then the game is fair.In Problem 17-20, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If a matrix game is fair then some payoffs are positive and some are negative.In Problems 21-24 remove recessive rows and columns; then solve using geometric linear programming techniques. 1204153210In Problems 21-24 remove recessive rows and columns; then solve using geometric linear programming techniques. 193102310In Problems 21-24 remove recessive rows and columns; then solve using geometric linear programming techniques. 13326482In Problems 21-24 remove recessive rows and columns; then solve using geometric linear programming techniques. 55131526(A) Let P and Q be strategies for the 22 matrix game M. Let k be a constant, and let J be the matrix with all 1s as entries. Show that the matrix product PkJQ equals the 11 matrix k. (B) Generalize part (A) to the situation where M and J are mn matrix.Use properties of matrix addition and multiplication to deduce from Problem 25 that if P* and Q* are optimal strategies for the game M with value v, then they are also optimal strategies for the game M+KJ with value v+k.Solve the matrix games in problems 27-30 by using geometric linear programming methods. Bank promotion Problem 49A, Exercise 11.2Solve the matrix games in problems 27-30 by using geometric linear programming methods. Viewer ratings Problem 50A, Exercise 11.2Solve the matrix games in problems 27-30 by using geometric linear programming methods. Investment. Problem 51, Exercise 11.2Solve the matrix games in problems 27-30 by using geometric linear programming methods. Corporate farming. Problem 52A, Exercise 11.2Outline a procedure for solving the 45 matrix game M=22161364175360442735 without actually solving the game.Suppose that the investor in Example 1 wishes to invest 10,000 in long- and short-term bonds, as well as in gold, and he is concerned about inflation. After some analysis, he estimates that the return (in thousands of dollars) at the end of a year will be as indicated in the following payoff matrix: Inflation(fate)Up3Down3%Goldlong-termbondsShort-termbonds333211 Again, assume that fate is a very good player that will attempt to reduce the investor's return as much as possible. Find the optimal strategies for both the investor and for fate. What is the value of the game?In Problems 1-4, solve each matrix game 140012In Problems 1-4, solve each matrix game. 112201In Problems 1-4, solve each matrix game. 012103230In Problems 1-4, solve each matrix game. 120012201In Problems 5-8, outline a procedure for solving the matrix game, then solve it. 422101132132In Problems 5-8, outline a procedure for solving the matrix game, then solve it. 567453234267In Problems 5-8, outline a procedure for solving the matrix game, then solve it. 2131124011110112In Problems 5-8, outline a procedure for solving the matrix game, then solve it. 2531012210132320Scissors, paper ,stone game This game is well known in many parts of the world. Two players simultaneously present a hand in one of three positions: an open hand (paper), a closed fist (stone), or two open fingers (scissors). The payoff is 1 unit according to the rule “Paper covers stone, stone breaks scissors, and scissors cut paper.” If both players present the same form, the payoff is 0. (A) Set up the payoff matrix for the game. (B) Solve the game using the simplex method discussed in this section. (Remove any recessive rows and columns, if present, before you start the simplex method.Player R has a $2, a $5,and a $10 bill. Player C has a $1, a $5, and a $10 bill. Each player selects and shows (simultaneously) one of his or her three bills. If the total value of the two bills shown is even, R wins C ’s bill; if the value is odd, C wins R ’ s bill. (Which player would you rather be?) (A) Set up the payoff matrix for the game. (B) Solve the game using the simplex method discussed in this section. (Remove any recessive rows and columns, if present, before you start the simplex method.)Headphone sales. A department store chain is about to order deluxe, standard, and discount headphones for next year’s inventory. The state of the nation’s economy (fate) during the year will be an important factor on sales for that year. Records over the past 5 years indicate that if the economy is up, the company will net 3, 1, and 0 million dollars, respectively, on sales of deluxe, standard, and discount models. If the economy is down, the company will net 1,1, and 3 million dollars, respectively, on sales of deluxe, standard, and discount models. (A) Set up a payoff matrix for this problem. (B) Find optimal strategies for both the company and fate (the economy). What is the value of the game? (C) How should the company’s budget be allocated to each grade of headphone to maximize the company’s return respective of what the economy does the following year?Tour agency A tour agency organizes standard and luxury tours for the following year. Once the agency has committed to these tours, the schedule cannot be changed. The state of the economy during the following year has a direct effect on tour sales. From past records the agency has established the following payoff matrix (in millions of dollars): Economy(fate)DownNochangeUpstandardLuxury102103 (A) Find optimal strategies for both the agency and fate (the economy). What is the value of the game? (B) What proportion of each type of tour should be arranged for in advance in order for the agency to maximize its return irrespective of what the economy does the following year? (C) What is the expected value of the game to the agency if they organize only luxury tours and fate plays the strategy “down”? If the agency plays its optimal strategy and fate plays the strategy “no change”? Discuss these and other possible scenarios.In Problems 1 and 2, is the matrix game strictly determined? 5342In Problems 1 and 2, is the matrix game strictly determined? 1421In Problems 3-8, determine the value V of the matrix game. Is the game fair? 5230In Problems 3-8, determine the value V of the matrix game. Is the game fair? 7391In Problems 3-8, determine the value V of the matrix game. Is the game fair? 9352In Problems 3-8, determine the value V of the matrix game. Is the game fair? 61035In Problems 3-8, determine the value V of the matrix game. Is the game fair? 8754In Problems 3-8, determine the value V of the matrix game. Is the game fair? 34529RE10RE11RE12REDelete as many recessive rows and columns as possible, then write the reduced matrix game: 235130011Problems 14-17 refer to the matrix game: M=2101 Solve M using formulas from Section 11.2Problems 14-17 refer to the matrix game: M=2101 Write the two linear programming problems corresponding to M after adding 3 to each payoff.Problems 14-17 refer to the matrix game: M=2101 Solve the matrix game M using linear programming and a geometric approach.Problems 14-17 refer to the matrix game: M=2101 Solve the matrix game M using linear programming and the simplex method.In Problems 18-21, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If a matrix game is fair, then it is strictly determined.In Problems 18-21, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If a game matrix has a saddle value equal to 0, then the game is fair.In Problems 18-21, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. A game matrix can have at most one recessive row.In Problems 18-21, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If all payoffs of a matrix game are negative, then the value of the game is negative.In Problems 22-26, solve each matrix game (first check for saddle values, recessive rows, and recessive columns). 128024013In Problems 22-26, solve each matrix game (first check for saddle values, recessive rows, and recessive columns). 153743223021In Problems 22-26, solve each matrix game (first check for saddle values, recessive rows, and recessive columns). 031121In Problems 22-26, solve each matrix game (first check for saddle values, recessive rows, and recessive columns). 264747353398In Problems 22-26, solve each matrix game (first check for saddle values, recessive rows, and recessive columns). 112022321Does every strictly determined 22 matrix game have a recessive row or column? Explain.Does every strictly determined 33 matrix game have a recessive row or column? Explain.Finger game Consider the following finger game between Ron (rows) and Cathy (columns): Each points either 1 or 2 fingers at the other. If they match, Ron pays Cathy $2. If Ron points 1 finger and Cathy points 2, Cathy pays Ron $3. If Ron points 2 fingers and Cathy points 1, Cathy pays Ron $1. (A) Set up a payoff matrix for this game. (B) Use formulas from Section 11.2 to find the optimal strategies for Ron and for Cathy.Refer to Problem 29. Use linear programming and a geometric approach to find the expected value of the game for Ron. What is the expected value for Cathy?Agriculture A farmer decides each spring whether to plant com or soybeans. Com is the better crop under wet conditions, soybeans under dry conditions. The following payoff matrix has been determined, where the entries are in tens of thousands of dollars. Weather(fate)WetDryFarmerCornSoyabeans84210 Use linear programming and the simplex method to find optimal strategies for the farmer and the weather.Agriculture Refer to Problem 31. Use formulas from Section 11.2 to find the expected value of the game to the farmer. What is the expected value of the game to the farmer if the weather plays the strategy “ dry” for many years and the farmer always plants soybeans?Advertising A small town has two competing grocery stores, store R and store C. Every week each store decides to advertise its specials using either a newspaper ad or a mailing. The following payoff matrix indicates the percentage of market gain or loss for each choice of action by store R and store C. CPaperMailRPaperMail1654 Use linear programming and a geometric approach to find optimal strategies for store R and store C.Advertising Refer to Problem 33. Use linear programming and the simplex method to find the expected value of the game for store R. If store R plays its optimal strategy and store C always places a newspaper ad, what is the expected value of the game for store C ?State the real number property that justifies the indicated statement. (a) 8+3+y=8+3+y (b) x+y+z=z+x+y (c) a+bx+y=ax+y+bx+y (d) 5xy+0=5xy (e) If xy=1,x0, then y=1/xRound 693 to the nearest integer. Fractions with denominator 100 are called Percentages. They are used so often that they have their own notation: 3100=37100=7.5110100=110 So 3 is equivalent to 0.003, 7.5 is equivalent to 0.075, and so on.You intend to give a 20 tip, rounded to the nearest dollar, on a restaurant bill of $78.47. How much is the tip?In Problems 1-6, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. Commutative property : uv=?In Problems 1-6, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. Commutative property + : x+7=?In Problems 1-6, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. Associative property +: 3+7+y=?In Problems 1-6, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. Associative property : xyz=?In Problems 1-6, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. Identity property : 1u+v=?In Problems 1-6, replace each question mark with an appropriate expression that will illustrate the use of the indicated real number property. Commutative property + : 0+9m=?In Problems 7-26, indicate true (T) or false (F) 58m=58mIn Problems 7-26, indicate true (T) or false (F) a+cb=a+bcIn Problems 7-26, indicate true (T) or false (F) 5x+7x=5+7xIn Problems 726, indicate true (T) or false (F) uvw+x=uvw+uvxIn Problems 7-26, indicate true (T) or false (F) 2a2xy=2a4x+yIn Problems 7-26, indicate true (T) or false (F) 85=815In Problems 7-26, indicate true (T) or false (F) x+3+2x=2x+x+3In Problems 7-26, indicate true (T) or false (F) x3y5yx=15y2x2In Problems 7-26, indicate true (T) or false (F) 2xx+3=2xx+3In Problems 7-26, indicate true (T) or false (F) 2xx3=2xx3In Problems 7-26, indicate true (T) or false (F) 313=1In Problems 7-26, indicate true (T) or false (F) 0.5+0.5=0In Problems 7-26, indicate true (T) or false (F) x2y2=1x2y2In Problems 7-26, indicate true (T) or false (F) x+2x=x+2xIn Problems 7-26, indicate true (T) or false (F) ab+cd=a+cb+dIn Problems 7-26, indicate true (T) or false (F) kk+b=11+bIn Problems 7-26, indicate true (T) or false (F) x+8x+6=x+8x+x+86In Problems 7-26, indicate true (T) or false (F) uu2v+vu2v=u+vu2vIn Problems 7-26, indicate true (T) or false (F) If x22x+3=0, then either x2=0 or 2x+3=0.In Problems 7-26, indicate true (T) or false (F) If either x2=0 or 2x+3=0, then x22x+3=0If uv=1, does either u or v have to be 1 ? ExplainIf uv=0, does either u or v have to be 0 ? Explain.Indicate whether the following are true (T) or false (F): (A) All integers are natural numbers. (B) All rational number are rational numbers. (C) All natural numbers are real numbers.Indicate whether the following are true (T) or false (F): (a) All natural numbers are integers. (b) All real numbers are irrational. (c) All rational numbers are real numbers.Given an example of a real number that is not a rational number.