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All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences

In a family with 2 children, excluding multiple births and assuming that a boy is as likely as a girl at each birth, what is the expected number of boys?A fair coin is flipped. If a head turns up, you win $1. If a tail turns up, you lose $1. What is the expected value of the game? Is the game fair?Repeat Problem 17, assuming an unfair coin with the probability of a head being .55 and a tail being .45.After paying $4 to play, a single fair die is rolled, and you are paid back the number of dollars corresponding to the number of dots facing up. For example, if a 5 turns up, $5 is returned to you for a net gain, or payoff, of $1 ; if a 1 turns up. $1 is returned for a net gain of $3 ; and so on. What is the expected value of the game? Is the game fair?Repeat Problem 19 with the same game costing $3.50 for each play.21E In Problem 21, for the game to be fair, how much should you lose if a head and a tail turn up?A friend offers the following game: She wins $1 from you if, on four rolls of a single die, a 6 turns up at least once; otherwise, you win $1 from her. What is the expected value of the game to you? To her?On three rolls of a single die, you will lose $10 if a 5 turns up at least once, and you will win $7 otherwise. What is the expected value of the game?A single die is rolled once. You win $5 if a 1 or 2 turns up and 10 if a 3, 4, or 5 turns up. How much should you lose if a 6 turns up in order for the game to be fair? Describe the steps you took to arrive at your answer.A single die is rolled once. You lose $12 if a number divisible by 3 turns up. How much should you win if a number not divisible by 3 turns up in order for the game to be fair? Describe the process and reasoning used to arrive at your answer.27E A coin is tossed three times. Suppose you lose $3 if 3 heads appear, lose $2 if 2 heads appear, and win $3 if 0 heads appear. How much should you win or lose if 1 head appears in order for the game to be fair?A card is drawn from a standard 52-card deck. If the card is a king, you win $10 ; otherwise, you lose $1. What is the expected value of the game?30E A 5-card hand is dealt from a standard 52-card deck. If the hand contains at least one king, you win $10 ; otherwise, you lose $1. What is the expected value of the game?A 5-card hand is dealt from a standard 52-card deck. If the hand contains at least one diamond, you win $10 ; otherwise, you lose $4. What is the expected value of the game?The payoff table for two courses of action, A1 or A2, is given below. Which of the two actions will produce the largest expected value? What is it?The payoff table for three possible courses of action is given below. Which of the three actions will produce the largest expected value? What is it?Roulette wheels in Nevada generally have 38 equally spaced slots numbered 00,0,1,2...,36. A player who bets $1 on any given number wins $35 (and gets the bet back) if the ball comes to rest on the chosen number; otherwise, the $1 bet is lost. What is the expected value of this game?In roulette (see Problem 35), the numbers from 1to36 are evenly divided between red and black. A player who bets $1 on black wins $1 (and gets the $1 bet back) if the ball comes to rest on black; otherwise (if the ball lands on red, 0, or 00), the $1 bet is lost. What is the expected value of the game?Five thousand tickets are sold at $1 each for a charity raffle. Tickets will be drawn at random and monetary prizes awarded as follows: 1 prize of $500 ; 3 prizes of 100, 5 prizes of $20, and 20 prizes of $5. What is the expected value of this raffle if you buy 1 ticket?Ten thousand raffle tickets are sold at $2 each for a local library benefit. Prizes are awarded as follows: 2 prizes of $1,000. 4 prizes of $500, and 10 prizes of $100. What is the expected value of this raffle if you purchase 1 ticket?41E 42E One thousand raffle tickets are sold at $1 each. Three tickets will be drawn at random (without replacement), and each will pay $200. Suppose you buy 5 tickets. (A) Create a payoff table for 0,1,2,and3 winning tickets among the 5 tickets you purchased. (If you do not have any winning tickets, you lose $5 ; if you have 1 winning ticket, you net 195 since your initial $5 will not be returned to you; and so on.) (B) What is the expected value of the raffle to you?Repeat Problem 43 with the purchase of 10 tickets.45E Use a graphing calculator to simulate the results of placing a $1 bet on black in each of 400 games of roulette (see Problems 36 and 45) and compare the simulated and expected gains or losses.A 3- card hand is dealt from a standard deck. You win $20 for each diamond in the hand. If the game is fair, how much should you lose if the hand contains no diamonds?48E Insurance. The annual premium for a $5,000 insurance policy against the theft of a painting is $150. If the (empirical) probability that the painting will be stolen during the year is .01, what is your expected return from the insurance company if you take out this insurance?Insurance. An insurance company charges an annual premium of $75 for a $200,000 insurance policy against a house burning down. If the (empirical) probability that a house bums down in a given year is .0003, what is the expected value of the policy to the insurance company?Decision analysis. After careful testing and analysis, an oil company is considering drilling in two different sites. It is estimated that site A will net $30 million if successful (probability .2 ) and lose $3 million if not (probability .8 ); site R will net $70 million if successful (probability .1 ) and lose $4 million if not (probability .9 ). Which site should the company choose according to the expected return for each site?Decision analysis. Repeat Problem 51, assuming that additional analysis caused the estimated probability of success in field B to be changed from .1to.11.Genetics. Suppose that at each birth, having a girl is not as likely as having a boy. The probability assignments for the number of boys in a 3 -child family are approximated empirically from past records and are given in the table. What is the expected number of boys in a 3 -child family?Genetics. A pink-flowering plant is of genotype RW. If two such plants are crossed, we obtain a red plant (RR) with probability .25, a pink plant (RW or WR) with probability .50, and a white plant (WW) with probability .25, as shown in the table. What is the expected number of W genes present in a crossing of this type?Lottery. A $2 Powerball lottery ticket has a 1/27.05 probability of winning $4, a 1/317.39 probability of winning $7, a 1/10,376.47 probability of winning $100, a 1/913,129.18 probability of winning $50,000, a 1/11,688,053.52 probability of winning $1,000,000 and a 1/292,201,338 probability of winning the Grand Prize. If the Grand Prize is currently $100,000,000, what is the expected value of a single Powerball lottery ticket?Lottery. Repeat Problem 55, assuming that the Grand Prize is currently $400,000,000.In a single deal of 5 cards from a standard 52-card deck, what is the probability of being dealt 5 clubs?Brittani and Ramon are members of a 15- person ski club. If the president and treasurer are selected by lottery, what is the probability that Brittani will be president and Ramon will be treasurer? (A person cannot hold more than one office.)Each of the first 10 letters of the alphabet is printed on a separate card. What is the probability of drawing 3 cards and getting the code word dig by drawing d on the first draw, i on the second draw, and g on the third draw? What is the probability of being dealt a 3- card hand containing the letters d,i, and g in any order?A drug has side effects for 50 out of 1,000 people in a test. What is the approximate empirical probability that a person using the drug will have side effects?A spinning device has 5 numbers, 1,2,3,4,and5, each as likely to turn up as the other. A person pays $3 and then receives back the dollar amount corresponding to the number turning up on a single spin. What is the expected value of the game? Is the game fair?6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE Answer Problems 18-25 using the following probability tree: PA Answer Problems 18-25 using the following probability tree: PBA Answer Problems 1825 using the following probability tree: PBA Answer Problems 18-25 using the following probability tree: PAB Answer Problems 18-25 using the following probability tree: PAB Answer Problems 18-25 using the following probability tree: PB Answer Problems 18-25 using the following probability tree: PAB 25RE (A) If 10 out of 32 students in a class were born in June, July, or August, what is the approximate empirical probability of any student being born in June, July, or August? (B) If one is as likely to be born in any of the 12 months of a year as any other, what is the theoretical probability of being born in either June, July, or August? (C) Discuss the discrepancy between the answers to parts (A) and (B).27RE 28RE 29RE 30RE In a single draw from a standard 52-card deck, what are the probability and odds for drawing (A) A jack or a queen? (B) A jack or a spade? (C) A card other than an ace?32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE From a standard deck of 52 cards, what is the probability of obtaining a 5-card hand (A) Of all diamonds? (B) Of 3 diamonds and 2 spades? Write answers in terms of CnrorPnr ; do not evaluate.47RE 48RE The command in Figure A was used on a graphing calculator to simulate 50 repetitions of rolling a pair of dice and recording the minimum of the two numbers. A statistical plot of the results is shown in Figure B. (A) Use Figure B to find the empirical probability that the minimum is 2. (B) What is the theoretical probability that the minimum is 2 ? (C) Using a graphing calculator to simulate 200 rolls of a pair of dice, determine the empirical probability that the minimum is 4 and compare with the theoretical probability.50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE Two fair (not weighted) dice are each numbered with a 3 on one side, a 2 on two sides, and a 1 on three sides. The dice are rolled, and the numbers on the two up faces are added. If X is the random variable associated with the sample space S=2,3,4,5,6: (A) Find the probability distribution of X. (B) Find the expected value of X.If you pay $3.50 to play the game in Problem 62 (the dice are rolled once) and you are returned the dollar amount corresponding to the sum on the faces, what is the expected value of the game? Is the game fair? If it is not fair, how much should you pay in order to make the game fair?64RE 65RE 66RE Market research. From a survey of 100 city residents, it was found that 40 read the daily newspaper, 70 watch the evening news, and 30 do both. What is the (empirical) probability that a resident selected at random (A) Reads the daily paper or watches the evening news? (B) Does neither? (C) Does one but not the other?Market research. A market research firm has determined that 40 of the people in a certain area have seen the advertising for a new product and that 85 of those who have seen the advertising have purchased the product. What is the probability that a person in this area has seen the advertising and purchased the product?Market analysis. A clothing company selected 1,000 persons at random and surveyed them to determine a relationship between the age of the purchaser and the annual purchases of jeans. The results are given in the table. Given the events A= person buys 2 pairs of jeans B= person is between 12 and 18 years old C= person does not buy more than 2 pairs of jeans D= person buys more than 2 pairs of jeans (A) Find PA,PB,PAB,PABandPBA. (B) Are events A and B independent? Explain. (C) Find PC,PD,PCDandPDC. (D) Are events C and D mutually exclusive? Independent? Explain.70RE 71RE 72RE 73RE 74RE Genetics. Six men in 100 and 1 woman in 100 are colorblind. A person is selected at random and is found to be color-blind. What is the probability that this person is a man? (Assume that the total population contains the same number of women as men.)76RE (A) Refer to the transition diagram in Figure 1. What is the probability that a person using brand A will switch to another brand when he or she runs out of toothpaste? (B) Refer to transition probability matrix P. What is the probability that a person who is not using brand A will not switch to brand A when he or she runs out of toothpaste? (C) In Figure 1, the sum of the probabilities on the arrows leaving each state is 1. Will this be true for any transition diagram? Explain your answer. (D) In transition probability matrix P, the sum of the probabilities in each row is 1. Will this be true for any transition probability matrix? Explain your answer.Refer to Example 4. States D and G are referred to as absorbing states because a student who enters either one of these states never leaves it. Absorbing states are discussed in detail in Section 9.3. (A) How can absorbing states be recognized from a transition diagram? Draw a transition diagram with two states, one that is absorbing and one that is not, to illustrate. (B) How can absorbing states be recognized from a transition matrix? Write the transition matrix for the diagram you drew in part (A) to illustrate.An insurance company classifies drivers as low-risk if they are accident-free for one year. Past records indicate that 98 of the drivers in the low-risk category L one year will remain in that category the next year, and 78 of the drivers who are not in the low-risk category L one year will be in the low-risk category the next year. (A) Draw a transition diagram. (B) Find the transition matrix P. (C) If 90 of the drivers in the community are in the low-risk category this year, what is the probability that a driver chosen at random from the community will be in the low-risk category next year? Year after next?Find P4 and use it to find S4 for AAP=AA.8.2.3.7 and AAS0=.8.2 If a graphing calculator or a computer is available for computing matrix products and powers of a matrix, finding state matrices for any number of trials becomes a routine calculation.Use P8 and a graphing calculator to find S8 for P and S0 as given in Matched Problem 2. Round values in S8 to six decimal places.Refer to Example 4. At the end of each year the faculty examines the progress that each advanced-level student has made on the required thesis. Past records indicate that 30 of advanced-level students A complete the thesis requirement C and 10 are dropped from the program for insufficient progress D, never to return. The remaining students continue to work on their theses. (A) Draw a transition diagram. (B) Find the transition matrix P. (C) What is the probability that an advanced-level student completes the thesis requirement within 4 years? Is dropped from the program for insufficient progress within 4 years?In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 254132 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 456937 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 322541 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 693745 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 322541 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 456937 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 254132 In Problems 1-8, find the matrix product, if it is defined. (If necessary, review Section 4.4.) 693745 In Problems 9-14, use the transition matrix ABP=AB.7.3.1.9 to find S1 and S2 for the indicated initial state matrix S0. S0=01 In Problems 9-14, use the transition matrix ABP=AB.7.3.1.9 to find S1 and S2 for the indicated initial state matrix S0. S0=10 In Problems 9-14, use the transition matrix ABP=AB.7.3.1.9 to find S1 and S2 for the indicated initial state matrix S0. S0=.6.4 In Problems 9-14, use the transition matrix ABP=AB.7.3.1.9 to find S1 and S2 for the indicated initial state matrix S0. S0=.2.8 In Problems 9-14, use the transition matrix ABP=AB.7.3.1.9 to find S1 and S2 for the indicated initial state matrix S0. S0=.25.75 In Problems 9-14, use the transition matrix ABP=AB.7.3.1.9 to find S1 and S2 for the indicated initial state matrix S0. S0=.75.25 In Problems 15-20, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=10 In Problems 15-20, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=01 In Problems 15-20, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.3.7 In Problems 15-20, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.9.1 In Problems 15-20, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.5.5 In Problems 15-20, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.2.8 In Problems 21-26, use the transition matrix ABCP=ABC.2.4.4.7.2.1.5.3.2 to find S1 and S2 for the indicated initial state matrix S0. S0=010 In Problems 21-26, use the transition matrix ABCP=ABC.2.4.4.7.2.1.5.3.2 to find S1 and S2 for the indicated initial state matrix S0. S0=001 In Problems 21-26, use the transition matrix ABCP=ABC.2.4.4.7.2.1.5.3.2 to find S1 and S2 for the indicated initial state matrix S0. S0=.50.5 In Problems 21-26, use the transition matrix ABCP=ABC.2.4.4.7.2.1.5.3.2 to find S1 and S2 for the indicated initial state matrix S0. S0=.5.50 In Problems 21-26, use the transition matrix ABCP=ABC.2.4.4.7.2.1.5.3.2 to find S1 and S2 for the indicated initial state matrix S0. S0=.1.3.6 In Problems 21-26, use the transition matrix ABCP=ABC.2.4.4.7.2.1.5.3.2 to find S1 and S2 for the indicated initial state matrix S0. S0=.4.3.3 In Problems 27-32, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=100 In Problems 27-32, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=010 In Problems 27-32, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=0.4.6 In Problems 27-32, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.80.2 In Problems 27-32, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.5.2.3 In Problems 27-32, use the transition diagram to find S1 and S2 for the indicated initial state matrix S0. S0=.2.7.1 Draw the transition diagram that corresponds to the transition matrix of Problem 9.Find the transition matrix that corresponds to the transition diagram of Problem 15.Draw the transition matrix that corresponds to the transition diagram of Problem 27.Find the transition diagram that corresponds to the transition matrix of Problem 21.In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? .3.710 In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? .9.1.4.8 In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? .5.5.7.3 In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? 0110 In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? .1.3.6.2.4.4 42E In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? .5.1.40.5.5.2.1.7 In Problems 37-44, could the given matrix be the transition matrix of a Markov chain? .3.3.4.7.2.2.1.8.1 In Problems 45-50, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.In Problems 45-50, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.In Problems 45-50, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.In Problems 45-50, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.In Problems 45-50, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.In Problems 45-50, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.In Problems 51-56, are there unique values of a,b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why. ABCP=ABC0.5ab0.4.2c.1 In Problems 51-56, are there unique values of a,b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why. ABCP=ABCa0.9.2.3b.6c0 In Problems 51-56, are there unique values of a,b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why. ABCP=ABC0a.30b0c.80 54E In Problems 51-56, are there unique values of a,b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why. ABCP=ABC.2.17a.4c.5b.4 56E In Problems 57-60, use the given information to draw the transition diagram and find the transition matrix. A Markov chain has two stales, A and B. The probability of going from state A to state B in one trial is .7, and the probability of going from state B to state A in one trial is .9.A Markov chain has two states, A and B. The probability of going from state A to state A in one trial is .6, and the prob-ability of going from state B to state B in one trial is .2.59E 60E Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP3=ABC.35.348.302.262.336.402.212.298.49 Find the probability of going from state A to state B in two trials.Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP2=ABC.35.348.302.262.336.402.212.298.49 Find the probability of going from state B to state C in two trials.Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP3=ABC.35.348.302.262.336.402.212.298.49 Find the probability of going from state C to state A in three trials.Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP3=ABC.35.348.302.262.336.402.212.298.49 Find the probability of going from state B to state B in three trials.65E Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP3=ABC.35.348.302.262.336.402.212.298.49 Find S2 for S0=010 and explain what it represents.67E 68E Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP3=ABC.35.348.302.262.336.402.212.298.49 Using a graphing calculator to compute powers of P, find the smallest positive integer n such that the corresponding entries in Pn and Pn+1 are equal when rounded to two decimal places.Problems 61-70 refer to the following transition matrix P and its powers ABCP=ABC.6.3.1.2.5.3.1.2.7 ABCP2=ABC.43.35.22.25.37.38.17.27.56 ABCP3=ABC.35.348.302.262.336.402.212.298.49 Using a graphing calculator to compute powers of P, find the smallest positive integer n such that the corresponding entries in Pn and Pn+1 are equal when rounded to three decimal places.In Problems 71-74, given the transition matrix P and initial-state matrix S0, find P4 and use P4 to find S4. ABP=AB.1.9.6.4;S0=.8.2 In Problems 71-74, given the transition matrix P and initial-state matrix S0, find P4 and use P4 to find S4. ABP=AB.8.2.3.7;S0=.4.6 In Problems 71-74, given the transition matrix P and initial-state matrix S0, find P4 and use P4 to find S4. ABCP=ABC0.4.6001100;S0=.2.3.5 In Problems 71-74, given the transition matrix P and initial-state matrix S0, find P4 and use P4 to find S4. ABCP=ABC010.80.2100;S0=.4.2.4 A Markov chain with two states has transition matrix P. If the initial-state matrix is S0=10, discuss the relationship between the entries in the kth -state matrix and the entries in the kth power of P.Repeat Problem 75 if the initial-state matrix is S0=01.Given the transition matrix ABCDP=ABCD.2.2.3.30100.2.2.1.50001 (A) Find P4. (B) Find the probability of going from state A to state D in four trials. (C) Find the probability of going from state C to state B in four trials. (D) Find the probability of going from state B to state A in four trials.78E Show that if P=a1a1bb is probability matrix, then P2 is a probability matrix.Show that if P=a1a1bbandS=c1c are probability matrices, then SP is a probability matrix.Use a graphing calculator and the formula Sk = S0Pk (Theorem 1) to compute the required state matrices in Problems 81-84. The transition matrix for a Markov chain is P=.4.6.2.8 (A) If S0=01, find S2,S4,S8, Can you identify a state matrix S that the matrices Sk seem to be approaching? (B) Repeat part (A) for S0 = 10. (C) Repeat part (A) for S0 = .5.5. (D) Find SP for any matrix S you identified in part (A)-(C). (E) Write a brief verbal description of the long-term behavior of the state matrices of this Markov chain based on your observations in parts (A)-(D).Use a graphing calculator and the formula Sk=S0Pk (Theorem 1) to compute the required state matrices in Problems 81-84. Repeat Problem 81forP=.9.4.1.6.Use a graphing calculator and the formula Sk=S0Pk (Theorem 1) to compute the required state matrices in Problems 81-84. Refer to Problem 81.FindPkfork=2,4,8,.... Can you identify a matrix Q that the matrices Pk are approaching? If so, how is Q related to the results you discovered in Problem 81?Use a graphing calculator and the formula Sk=S0Pk (Theorem 1) to compute the required state matrices in Problems 81-84. Refer to Problem 82.FindPkfork=2,4,8,..... Can you identify a matrix Q that the matrices Pk are approaching? If so, how is Q related to the results you discovered in Problem 82?Scheduling. An outdoor restaurant in a summer resort closes only on rainy days. From past records, it is found that from May through September, when it rains one day, the probability of rain for the next day is .4; when it does not rain one day, the probability of rain for the next day is .06. (A) Draw a transition diagram. (B) Write the transition matrix. (C) If it rains on Thursday, what is the probability that the restaurant will be closed on Saturday? On Sunday?Scheduling. Repeat Problem 85 if the probability of rain following a rainy day is .6 and the probability of rain following a nonrainy day is .1.Advertising. A television advertising campaign is conducted during the football season to promote a well-known brand X shaving cream. For each of several weeks, a survey is made, and it is found that each week, 80 of those using brand X continue to use it and 20 switch to another brand. It is also found that of those not using brand X,20 switch to brand X while the other 80 continue using another brand. (A) Draw a transition diagram. (B) Write the transition matrix. (C) If 20 of the people are using brand X at the start of the advertising campaign, what percentage will be using it 1 week later? 2 weeks later?Car rental. A car rental agency has facilities at both JFK and LaGuardia airports. Assume that a car rented at either airport must be returned to one or the other airport. If a car is rented at LaGuardia, the probability that it will be returned there is .8 ; if a car is rented at JFK. the probability that it will be returned there is .7. Assume that the company rents all its 100 cars each day and that each car is rented (and returned) only once a day. If we start with 50 cars at each airport, then (A) What is the expected distribution on the next day? (B) What is the expected distribution 2 days later?Homeowner's insurance. In a given city, the market for homeowner’s insurance is dominated by two companies: National Property and United Family. Currently, National Property insures 50 of homes in the city. United Family insures 30, and the remainder are insured by a collection of smaller companies. United Family decides to offer rebates to increase its market share. This has the following effects on insurance purchases for the next several years: each year 25 of National Property’s customers switch to United Family and 10 switch to other companies; 10 of United Family’s customers switch to National Property and 5 switch to other companies: 15 of the customers of other companies switch to National Property and 35 switch to United Family. (A) Draw a transition diagram. (B) Write the transition matrix. (C) What percentage of homes will be insured by National Property next year? The year after next? (D) What percentage of homes will be insured by United Family next year? The year after next?Service contracts. A small community has two heating services that offer annual service contracts for home heating: Alpine Heating and Badger Furnaces. Currently, 25 of homeowners have service contracts with Alpine. 30 have service contracts with Badger, and the remainder do not have service contracts. Both companies launch aggressive advertising campaigns to attract new customers, with the following effects on service contract purchases for the next several years: each year 35 of homeowners with no current service contract decide to purchase a contract from Alpine and 40 decide to purchase one from Badger. In addition, 10 of the previous customers at each company decide to switch to the other company, and 5 decide they do not want a service contract. (A) Draw a transition diagram. (B) Write the transition matrix. (C) What percentage of homes will have service contracts with Alpine next year? The year after next? (D) What percentage of homes will have service contracts with Badger next year? The year after next?Travel agent training. A chain of travel agencies maintains a training program for new travel agents. Initially, all new employees are classified as beginning agents requiring extensive supervision. Every 6 months, the performance of each agent is reviewed. Past records indicate that after each semiannual review, 40 of the beginning agents are promoted to intermediate agents requiring only minimal supervision. 10 are terminated for unsatisfactory performance, and the remainder continue as beginning agents. Furthermore, 30 of the intermediate agents are promoted to qualified travel agents requiring no supervision. 10 are terminated for unsatisfactory performance, and the remainder continue as intermediate agents. (A) Draw a transition diagram. (B) Write the transition matrix. (C) What is the probability that a beginning agent is promoted to qualified agent within 1 year? Within 2 years?Welder training. All welders in a factory begin as apprentices. Every year the performance of each apprentice is reviewed. Past records indicate that after each review, 10 of the apprentices are promoted to professional welder, 20 are terminated for unsatisfactory performance, and the remainder continue as apprentices. (A) Draw a transition diagram. (B) Write the transition matrix. (C) What is the probability that an apprentice is promoted to professional welder within 2 years? Within 4 years?Health plans. A midwestern university offers its employees three choices for health care: a clinic-based health maintenance organization (HMO), a preferred provider organization (PPO). and a traditional fee-for-service program (FFS). Each year, the university designates an open enrollment period during which employees may change from one health plan to another. Prior to the last open enrollment period, 20 of employees were enrolled in the HMO, 25 in the PPO, and the remainder in the FFS. During the open enrollment period, 15 of employees in the HMO switched to the PPO and 5 switched to the FFS, 20 of the employees in the PPO switched to the HMO and 10 to the FFS, and 25 of the employees in the FFS switched to the HMO and 30 switched to the PPO. (A) Write the transition matrix. (B) What percentage of employees were enrolled in each health plan after the last open enrollment period? (C) If this trend continues, what percentage of employees will be enrolled in each plan after the next open enrollment period?Dental insurance. Refer to Problem 93. During the open enrollment period, university employees can switch between two available dental care programs: the low-option plan (LOP) and the high-option plan (HOP). Prior to the last open enrollment period, 40 of employees were enrolled in the LOP and 60 in the HOP. During the open enrollment program, 30 of employees in the LOP switched to the HOP and 10 of employees in the HOP switched to the LOP. (A) Write the transition matrix. (B) What percentage of employees were enrolled in each dental plan after the last open enrollment period? (C) If this trend continues, what percentage of employees will be enrolled in each dental plan after the next open enrollment period?Housing trends. The 2000 census reported that 41.9 of the households in the District of Columbia were homeowners and the remainder were renters. During the next decade, 15.3 of homeowners became renters, and the rest continued to be homeowners. Similarly. 17.4 of renters became homeowners, and the rest continued to rent. (A) Write the appropriate transition matrix. (B) According to this transition matrix, what percentage of households were homeowners in 2010 ? (C) If the transition matrix remains the same, what percentage of households will be homeowners in 2030 ?Housing trends. The 2000 census reported that 66.4 of the households in Alaska were homeowners, and the remainder were renters. During the next decade, 37.2 of the homeowners became renters, and the rest continued to be homeowners. Similarly, 71.5 of the renters became homeowners, and the rest continued to rent. (A) Write the appropriate transition matrix. (B) According to this transition matrix, what percentage of households were homeowners in 2010 ? (C) If the transition matrix remains the same, what percentage of households will be homeowners in 2030 ?(A) Suppose that the toothpaste company started with only 5 of the market instead of 10. Write the initial-state matrix and find the next six state matrices. Discuss the behavior of these state matrices as you proceed to higher states. (B) Repeat part (A) if the company started with 90 of the toothpaste market.Which of the following matrices are regular? (A) P=.3.710 (B) P=1010 (C) P=010.50.5.50.5 The transition matrix for a Markov chain is P=.6.4.1.9 Find the stationary matrix S and the limiting matrix P.Refer to Matched Problem 1 in Section 9.1, where we found the following transition matrix for an insurance company: LLP=LL.98.02.78.22L=low-riskL=notlow-risk A mail-order company classifies its customers as preferred, standard, or infrequent depending on the number of orders placed in a year. Past records indicate that each year, 5 of preferred customers are reclassified as standard and 12 as infrequent, 5 of standard customers are reclassified as preferred and 5 as infrequent, and 9 of infrequent customers are reclassified as preferred and 10 as standard. Assuming that these percentages remain valid, what percentage of customers are expected to be in each category in the long run?Repeat Example 5 for P=.3.6.1.2.3.5.1.2.7 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 1000 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 0010 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 1001 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 0110 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 100000001 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 100010001 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 001000000 In Problems 1-8. without using a calculator, find P100. (If necessary, review Section 4.4.) [Hint: First find P2.] 011001000 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .6.4.4.6 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .3.7.2.6 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .1.9.5.4 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .5.5.8.2 13E 14E 15E 16E In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .6.4.1.9.3.7 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .2.5.3.6.3.1 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? 010001.5.50 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .20.8001.70.3 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? .1.3.6.8.1.1001 In Problems 9-22, could the given matrix be the transition matrix of a regular Markov chain? 001.90.1010 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.1.9.6.4 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.8.2.3.7 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.5.5.3.7 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.9.1.7.3 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.5.1.4.3.700.6.4 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.4.1.5.2.800.5.5 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.8.20.5.1.40.6.4 For each transition matrix P in Problems 23-30, solve the equation SP=S to find the stationary matrix S and line limiting matrix P. P=.2.80.6.1.30.9.1 Problems 31-34 refer to the regular Markov chain with transition matrix P=.5.5.2.8 For S=.2.5, calculate SP. Is S a stationary matrix? Explain.Problems 31-34 refer to the regular Markov chain with transition matrix P=.5.5.2.8 For S=.61.5, calculate SP. Is S a stationary matrix? Explain.Problems 31-34 refer to the regular Markov chain with transition matrix P=.5.5.2.8 For S=00, calculate SP. Is S a stationary matrix? Explain.Problems 31-34 refer to the regular Markov chain with transition matrix P=.5.5.2.8 For S=2757, calculate SP. Is S a stationary matrix? Explain.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The nn identity matrix is the transition matrix for a regular Markov chain.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. The nn matrix in which each entry equals 1n is the transition matrix for a regular Markov chain.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If the 22 matrix P is the transition matric for a regular Markov chain, then, at most, one of the entries of P are equal to 0.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If the 33 matrix P is the transition matric for a regular Markov chain, then, at most, two of the entries of P are equal to 0.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 transition matrix P for a Markov chain has a stationary matrix S, then P is regular.In Problems 35-40, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If P is the transition matrix for a Markov chain, then P has a unique stationary matrix.In Problems 41-44, approximate the stationary matrix S for each transition matrix P by computing powers of the transition matrix P. Round matrix entries to four decimal places. P=.51.49.27.73 In Problems 41-44, approximate the stationary matrix S for each transition matrix P by computing powers of the transition matrix P. Round matrix entries to four decimal places. P=.68.32.19.81 In Problems 41-44, approximate the stationary matrix S for each transition matrix P by computing powers of the transition matrix P. Round matrix entries to four decimal places. P=.5.500.5.5.8.1.1 In Problems 41-44, approximate the stationary' matrix S for each transition matrix P by computing powers of the transition matrix P. Round matrix entries to four decimal places. P=.2.2.6.50.5.50.5 A red urn contains 2 red marbles and 3 blue marbles, and a blue urn contains 1 red marble and 4 blue marbles. A marble is selected from an urn, the color is noted, and the marble is returned to the urn from which it was drawn. The next marble is drawn from the urn whose color is the same as the marble just drawn. This is a Markov process with two states: draw from the red urn or draw from the blue urn. (A) Draw a transition diagram for this process. (B) Write the transition matrix. (C) Find the stationary matrix and describe the long-run behavior of this process.Repeat Problem 45 if the red urn contains 5 red and 3 blue marbles, and the blue urn contains 1 red and 3 blue marbles.Given the transition matrix P=0110 A) Discuss the behavior of the state matrices S1,S2,S3,.... for the initial-state matrix S0=.2.8. (B) Repeat part (A) for S0=.5.54. (C) Discuss the behavior of Pk,k=2,3,4,.... (D) Which of the conclusions of Theorem 1 are not valid for this matrix? Why is this not a contradiction?Given the transition matrix P=001100010 (A) Discuss the behavior of the state matrices S1,S2,S3,.... for the initial-state matrix S0=.2.3.5. (b) Repeat part (A) for S0=131313. (C) Discuss the behavior of Pk,k=2,3,4,..... (D) Which of the conclusions of Theorem 1 are not valid for this matrix? Why is this not a contradiction?The transition matrix for a Markov chain is P=100.2.2.6001 (A) Show that R=100 and S=001 are both stationary matrices for P. Explain why this does not contradict Theorem 1A. (B) Find another stationary matrix for P. [Hint: Consider T=aR+1aS,where0a1.] (C) How many different stationary matrices does P have?The transition matrix for a Markov chain is P=.70.3010.20.8 (A) Show that R=.40.6 and S=010 are both stationary matrices for P. Explain why this does not contradict Theorem 1A. (B) Find another stationary matrix for P. [Hint: Consider T=aR+1aS,where0a1.] (C) How many different stationary matrices does P have?Problems 51 and 52 require the use of a graphing calculator. 51. Refer to the transition matrix P in Problem 49. What matrix P do the powers of P appear to be approaching? Are the rows of P stationary matrices for P ?Problems 51 and 52 require the use of a graphing calculator. 51. Refer to the transition matrix P in Problem 50. What matrix P do the powers of P appear to be approaching? Are the rows of P stationary matrices for P ?The transition matric for a Markov chain is P=.1.5.4.3.2.5.7.1.2 Let Mk denote the maximum entry in the second column of Pk. Note that M1=.5. (A) Find M2,M3,M4, and M5 to three decimal places. (B) Explain why MkMk+1 for all positive integers k.The transition matric for a Markov chain is P=0.2.8.3.3.4.6.1.3 Let mk denote the maximum entry in the third column of Pk. Note that m1=.3. (A) Find m2,m3,m4, and m5 to three decimal places. (B) Explain why mkmk+1 for all positive integers k.Transportation. Most railroad cars are owned by individual railroad companies. When a car leaves its home railroad’s tracks, it becomes part of a national pool of cars and can be used by other railroads. The rules governing the use of these pooled cars are designed to eventually return the car to the home tracks. A particular railroad found that each month, 11 of its boxcars on the home tracks left to join the national pool, and 29 of its boxcars in the national pool were returned to the home tracks. If these percentages remain valid for a long period of time, what percentage of its boxcars can this railroad expect to have on its home tracks in the long run?Transportation. The railroad in Problem 55 also has a fleet of tank cars. If 14 of the tank cars on the home tracks enter the national pool each month, and 26 of the tank cars in the national pool are returned to the home tracks each month, what percentage of its tank cars can the railroad expect to have on its home tracks in the long run?Labor force. Table 1 gives the percentage of the U.S. female population who were members of the civilian labor force in the indicated years. The following transition matrix P is proposed as a model for the data, where L represents females who are in the labor force and L represents females who are not in the labor force: NextdecadeLLCurrentdecadeLL.92.08.2.8=P (A) Let S0=.433.567, and find S1,S2,S3, and S4. Compute the matrices exactly and then round entries to three decimal places. (B) Construct a new table comparing the results from part (A) with the data in Table 1. (C) According to this transition matrix, what percentage of the U.S. female population will be in the labor force in the long run?Home ownership. The U.S. Census Bureau published the home ownership rates given in Table 2. The following transition matrix P is proposed as a model for the data, where H represents the households that own their home. FouryearslaterHHCurrentyearHH.95.05.15.85=P (A) Let S0=.654.346, and find S1,S2, and S3. Compute both matrices exactly and then round entries to three decimal places. (B) Construct a new table comparing the results from part (A) with the data in Table 2. (C) According to this transition matrix, what percentage of households will own their home in the long run?Market share. Consumers can choose between three long distance telephone services: GTT, NCJ, and Dash. Aggressive marketing by all three companies results in a continual shift of customers among the three services. Each year. GTT loses 5 of its customers to NCJ and 20 to Dash. NCJ loses 15 of its customers to GTT and 10 to Dash, and Dash loses 5 of its customers to GTT and 10 to NCJ. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?Market share. Consumers in a certain area can choose between three package delivery services: APS, GX, and WWP. Each week, APS loses 10 of its customers to GX and 20 to WWP, GX loses 15#37; of its customers to APS and 10 to WWP, and WWP loses 5 of its customers to APS and 5 to GX. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?Insurance. An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 40 of those in the poor category are moved to satisfactory, and 20 of those in the satisfactory category are moved to preferred. Also, 20 in the preferred category are moved to the satisfactory category, and 20 in the satisfactory category are moved to the poor category. Customers are never moved from poor to preferred, or conversely, in a single year. Assuming that these percentages remain valid over a long period of time, how many customers are expected in each category in the long run?Insurance. Repeat Problems 61 if 40 of preferred customers are moved to the satisfactory category each year, and all other information remains the same.Problems 63 and 64 require the use of a graphing calculator Market share. Acme Soap Company markets one brand of soap, called Standard Acme SA, and Best Soap Company markets two brands. Standard Best SB and Deluxe Best DB. Currently, Acme has 40 of the market, and the remainder is divided equally between the two Best brands. Acme is considering the introduction of a second brand to get a larger share of the market. A proposed new brand, called brand X, was test-marketed in several large cities, producing the following transition matrix for the consumers' weekly buying habits: SBDBSAXP=SBDBSAX.4.1.3.2.3.2.2.3.1.2.2.5.3.3.1.3 Assuming that P represents the consumers' buying habits over a long period of time, use this transition matrix and the initial-state matrix S0=.3.3.40 to compute successive state matrices in order to approximate the elements in the stationary matrix correct to two decimal places. If Acme decides to market this new soap, what is the long-run expected total market share for their two soaps?Problems 63 and 64 require the use of a graphing calculator Market share. Refer to Problem 63. The chemists at Acme Soap Company have developed a second new soap, called brand Y. Test-marketing this soap against the three established brands produces the following transition matrix: SBDBSAXP=SBDBSAX.3.2.2.3.2.2.2.4.2.2.4.2.1.2.3.4 Proceed as in Problem 63 to approximate the elements in the stationary matrix correct to two decimal places. If Acme decides to market brand Y, what is the long-run expected total market share for Standard Acme and brand Y ? Should Acme market brand X or brand Y ?Genetics. A given plant species has red, pink, or white flowers according to the genotypes RR, RW. and WW, respectively. If each of these genotypes is crossed with a pink flowering plant (genotype RW), then the transition matrix is NextgenerationRedPinkWhiteThisgenerationRedPinkWhite.5.50.25.5.250.5.5 Assuming that the plants of each generation are crossed only with pink plants to produce the next generation, show that regardless of the makeup of the first generation, the genotype composition will eventually stabilize at 25 red, 50 pink, and 25 white. (Find the stationary matrix.)Gene mutation. Suppose that a gene in a chromosome is of type A or type B. Assume that the probability that a gene of type A will mutate to type B in one generation is 104 and that a gene of type B will mutate to type A is 106. (A) What is the transition matrix? (B) After many generations, what is the probability that the gene will be of type A ? Of type B ? (Find the stationary matrix.)Rapid transit. A new rapid transit system has just started operating. In the first month of operation, it is found that 25 of commuters are using the system, while 75 still travel by car. The following transition matrix was determined from records of other rapid transit systems: NextmonthRapidtransitCarCurrentmonthRapidtransitCar.8.2.3.7 (A) What is the initial-state matrix? (B) What percentage of commuters will be using the new system after 1 month? After 2 months? (C) Find the percentage of commuters using each type of transportation after the new system has been in service for a long time.Politics: filibuster. The Senate is in the middle of a floor debate, and a filibuster is threatened. Senator Hanks, who is still vacillating, has a probability of .1 of changing his mind during the next 5 minutes. If this pattern continues for each 5 minutes that the debate continues and if a 24 -hour filibuster takes place before a vote is taken, what is the probability that Senator Hanks will cast a yes vote? A no vote? (A) Complete the following transition matrix: Next5minutesYesNocurrent5minutesYesNo.9.1 (B) Find the stationary matrix and answer the two questions. (C) What is the stationary matrix if the probability of Senator Hanks changing his mind ( .1 ) is replaced with an arbitrary probability p ?The population center of the 48 contiguous states of the United States is the point where a flat, rigid map of the contiguous states would balance if the location of each person was represented on the map by a weight of equal measure. In 1790, the population center was 23 miles east of Baltimore, Maryland. By 1990, the center had shifted about 800 miles west and 100 miles south to a point in southeast Missouri. To study this shifting population, the U.S. Census Bureau divides the states in to four regions as shown in the figure. Problems 69 and 70 deal with population shifts among these regions. Population shifts. Table 3 gives the percentage of the U.S. population living in the south region during the indicated years. The following transition matrix P is proposed as a model for the data, where S represents the population that lives in the south region: NextdecadeSSCurrentdecadeSS.61.39.21.79=P (A) Let S0=.309.691 and find S1,S2,S3, and S4. Compute the matrices exactly and then round entries to three decimal places. (B) Construct a new table comparing the results from part (A) with the data in Table 3. (C) According to this transition matrix, what percentage of the population will live in the south region in the long run?The population center of the 48 contiguous states of the United States is the point where a flat, rigid map of the contiguous states would balance if the location of each person was represented on the map by a weight of equal measure. In 1790, the population center was 23 miles east of Baltimore, Maryland. By 1990, the center had shifted about 800 miles west and 100 miles south to a point in southeast Missouri. To study this shifting population, the U.S. Census Bureau divides the states into four regions as shown in the figure. Problems 69 and 70 deal with population shifts among these regions. Population shifts. Table 3 gives the percentage of the U.S. population living in the south region during the indicated years. The following transition matrix P is proposed as a model for the data, where N represents the population that lives in the south region: NextdecadeNNCurrentdecadeNN.61.39.09.91=P (A) Let S0=.241.759 and find S1,S2,S3, and S4. Compute the matrices exactly and then round entries to three decimal places. (B) Construct a new table comparing the results from part (A) with the data in Table 4. (C) According to this transition matrix, what percentage of the population will live in the northeast region in the long run?(A) For the initial-state matrix S0=abc, find the first four state matrices, S1,S2,S3, and S4, in the Markov chain with transition matrix P=001010100 (B) Do the state matrices appear to be approaching a stationary matrix? Discuss.Determine whether each statement is true or false. Use examples and verbal arguments to support your conclusions. (A) A Markov chain with two states, one nonabsorbing and one absorbing, is always an absorbing chain. (B) A Markov chain with two states, both of which are absorbing, is always an absorbing chain. (C) A Markov chain with three states, one nonabsorbing and two absorbing, is always an absorbing chain.Identify any absorbing states for the following transition matrices: AABCP=ABC.50.50100.5.5 BABCP=ABC010100001 Use a transition diagram to determine whether P is the transition matrix for an absorbing Markov chain. AABCP=ABC.50.50100.5.5 BABCP=ABC010100001 Repeat Example 3 if 10 of farmers sell to company A each year, 40 sell to company B. and the remainder continue farming.Repeat Example 4 for the standard form P found in Matched Problem 3.Repeat Example 5 for the following transition diagram: E=entry-levelstudentsA=advanced-levelstudentsD=dropoutsG=graduates

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