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Finite Mathematics and Its Applications 12th edition Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair - Solutions

The manager of a supermarket counts the number of customers waiting in the express checkout line at random times throughout the week. Her observations are found in the following frequency table:Construct the corresponding relative frequency table, and use it to estimate the probability that at most
A fair coin is tossed twice. Let X be the number of heads. (a) Determine the probability distribution of X. (b) Determine the probability distribution of 2X + 5.
An experiment consists of three binomial trials, each having probability 1/3 of success. (a) Determine the probability distribution table for the number of successes. (b) Use the table to compute the mean and the variance of the probability distribution.
1. An archer has probability .3 of hitting a certain target. What is the probability of hitting the target exactly two times in four attempts?2. A true-false exam consists of ten 10-point questions. The instructor informs the students that six of the answers are true and four are false. An
A pair of fair dice is rolled 12 times. Note: For each roll, the probability of getting a seven is 1/6. (a) What is Pr (the result is seven exactly twice)? (b) What is Pr (the result is seven at least twice)? (c) What is the expected number of times that the result is seven?
1. Table 2 gives the probability distribution of the random variable X. Compute the mean and the variance of the random variable.2. An urn contains four red balls and four white balls. An experiment consists of selecting at random a sample of four balls and recording the number of red balls in the
Determine whether or not the matrix is stochastic?1.2. 3. 4. 5. 6.
Draw a transition diagram corresponding to the stochastic matrix?1.2. 3. 4.
Referring to Example 5, consider a typical group of French women, of whom 47% currently work outside of the home. Assume that the same percentage of daughters follow in their mothers' footsteps as with the American women-that is, that given by the matrixUse A and A2 to determine the proportion of
Repeat Exercise 19 for the women of China, of whom 44% currently work outside of the home. (Round to the nearest whole percent.)Refer to Exercise 19,Referring to Example 5, consider a typical group of French women, of whom 47% currently work outside of the home. Assume that the same percentage of
A cell phone provider classifies its customers as Low users (less than 400 minutes per month) or High users (400 or more minutes per month). Studies have shown that 80% of the people who were Low users one month will be Low users the next month, and that 70% of the people who were High users one
A university faculty health plan offers an optional dental plan. During the open enrollment period each year, 90% of the people who currently have the dental plan reenroll for it and 10% opt out. Of the people who do not have the dental plan, 40% enroll for it and 60% stay out of the plan?(a) Draw
The Southwestern states were popular destinations in 2015. Suppose that Fig. 6 shows the percentages of people who moved in and out of the Southwest during 2015. At the beginning of 2015, about 12% of the U.S. population lived in the Southwest.Figure 6,(a) Set up the stochastic matrix that displays
For a certain group of states, it was observed that 70% of the Democratic governors were succeeded by Democrats and 30% by Republicans. Also, 40% of the Republican governors were succeeded by Democrats and 60% by Republicans. (a) Set up the 2 × 2 stochastic matrix with columns and rows labeled D
Each day, mice are put into a T-maze (a maze shaped like a "T"; Fig. 7). In this maze, they have the choice of turning to the left (are rewarded with cheese) or to the right (receive cheese along with mild shock). After the first day, their decision whether to turn left or right is influenced by
In 1913, Markov analyzed a long poem written by a Russian author. He found that vowels were followed by consonants 87.2% of the time (either in the same word or the next word) and that consonants were followed by vowels 66.3% of the time. (a) Set up the 2 × 2 stochastic matrix, with columns and
Refer to Example 7 (taxi zones). If, originally, 40% of the taxis start in zone I, 40% in zone II, and 20% in zone III, how will the taxis be distributed after each has taken one passenger? Refer to Example 7, Taxis pick up and deliver passengers in a city that is divided into three zones. Records
A group of physical fitness devotees works out in the gym every day. The workouts vary from strenuous to moderate to light. When their exercise routine was recorded, the following observation was made: Of the people who work out strenuously on a particular day, 40% will work out strenuously on the
According to the Higher Education Research Institute, 33% of students at baccalaureate-granting colleges who entered college in 2015 characterize their political views as Liberal, 45% as Middle-of-the-road, and 22% as Conservative. Suppose that, each year, these students changed their political
According to the Higher Education Research Institute, 80% of students at baccalaureate-granting colleges who entered college in 2015 lived in College residence halls, 15% lived with Family, and 5% lived in Other types of housing. Suppose that each year, these students changed their residences as
A sociologist studying living patterns in a certain region determines that, each year, the population shifts between urban, suburban, and rural areas as shown in Fig. 8.(a) Set up a stochastic matrix that displays these transitions.(b) What percentage of people who live in urban areas in 2017 will
A retailer stocks three brands of breakfast cereal. A survey is taken of 5000 people who purchase cereal weekly from this retailer. Each week, Crispy Flakes loses 12% of its customers to Crunchy Nuggets and 19% to Toasty Cinnamon Twists. Crunchy Nuggets loses 16% of its customers to Crispy Flakes
Birth weights can be classified as Low (less than 6 pounds), Average (between 6 and 8 pounds), and High (greater than 8 pounds). Birth weights of female babies tend to be influenced by the birth weights of their mothers. Suppose that this tendency is given by the following matrix:(a) If the initial
A college cafeteria offers three choices of dessert for lunch: ice cream (I), cake (C), and pie (P). A study of the students who order dessert every day showed that, of those students who order I on a certain day, the next day 60% of these students will order I and the others will be equally likely
The Ehrenfest urn model was originally proposed as a model for diffusion of gases, but has since come to be applied in a wide variety of fields. This problem is a simplified version of the model. Consider two urns, Urn A and Urn B, containing a total of four balls. At each time interval, a ball is
A study of occupational mobility in a certain community classified occupations as P (professional), WC (white collar), SM (skilled manual), and UM (unskilled manual). Currently, 4% of the adults are P, 20% are WC, 40% are SM, and 36% are UM. Studies predict that, of the children of P parents, 40%
Find the third and fourth distribution matrices for the given stochastic matrix and initial distribution. (Round entries to two decimal places.)1.2.
Compute the first five powers of each matrix. (Round to two decimal places.)1.2. 3.
Stochastic matrices for which some power contains no zero entries are called regular matrices. In Exercises 1 and 2, conjecture whether or not the given matrix is regular, by looking at the first few powers of the matrix.1.2.
Let A be the stochastic matrixand let the initial distribution be (a) Generate the next four distribution matrices. (b) Calculate A4B, and confirm that it is the same as the fourth distribution matrix?
Repeat Exercise 47 for the matricesAnd Refer to Exercise 47, Let A be the stochastic matrix and let the initial distribution be (a) Generate the next four distribution matrices. (b) Calculate A4B, and confirm that it is the same as the fourth distribution matrix?
1. Consider the matrices of Exercise 47. Beginning with the initial distribution matrix, generate 10 more distributions. Continue to generate 10 more. The matrices will get closer and closer to a certain 2 Ã— 1 matrix. What is that matrix?Refer to Exercise 47,Let A be the stochastic
Write a stochastic matrix corresponding to the transition diagram?1.2. 3. 4.
1. What is a Markov process? 2. What is a transition matrix? A stochastic matrix? A distribution matrix?
1. Explain how to find the stable matrix of an absorbing stochastic matrix? 2. What is the fundamental matrix of an absorbing stochastic matrix, and how is it used?
1. What is An? Give an interpretation of the entries of An. 2. How is the nth distribution matrix calculated from the initial distribution matrix?
1. Define regular stochastic matrix. 2. Define the stable matrix and the stable distribution of a regular stochastic matrix?
1. Explain how to find the stable distribution of a regular stochastic matrix? 2. What is meant by an absorbing state of a stochastic matrix?
What is an absorbing stochastic matrix?
Determine whether or not the given matrix is stochastic. If so, determine if it is regular, absorbing, or neither.1.2. 3. 4.
In a certain factory, some machines are properly adjusted and some need adjusting. Technicians randomly inspect machines and make adjustments. Suppose that, of the machines that are properly adjusted on a particular day, 80% will also be properly adjusted the following day and 20% will need
Find the stable matrix for the absorbing stochastic matrix?
Figure 1 gives the layout of a house with four rooms connected by doors. Room I contains a mousetrap, and room II contains cheese. A mouse, after being placed in one of the rooms, will search for cheese; if unsuccessful after one minute, it will exit to another room by selecting one of the doors at
Which of the following is the stable distribution for the regular stochastic matrix(a) (b) (c)
A city has two competing news stations. From a survey of regular listeners, it was determined that, of those who listen to station A on a particular day, 90% listen to station A the next day and 10% listen to station B. Of those who listen to station B on a particular day, 20% listen to station A
Workday traffic conditions from 9:00 am to 10:00 am on the Baltimore Beltway can be characterized as Light, Moderate, and Heavy. The following stochastic matrix describes the day-to-day transitions:(a) Interpret the numbers in the second column of the matrix. (b) In the long run, what percent of
A mental-health facility rates patients on their ability to live on their own. The state of a person's health is "able to work and considered cured (C)," or "long-term hospitalization or death (L)," or "group home (G)," or "short term hospital care (S)." The stochastic matrix shown here describes
The contents of a reservoir depend on the available rainfall in the region and the demands on the water supply. Suppose that a reservoir holds up to 4 units of water (a unit might be a million gallons), and policy for the use of the water for irrigation and drinking water never allows the contents
1. Explain why the entries in each column of a transition matrix must add up to 1.2. Suppose that T is a transition matrix andInterpret the number .26. 3. Explain why A2 must be a stochastic matrix if A is a stochastic matrix. 4. True or False Every entry in a stochastic matrix is a conditional
Find the stable distribution for the regular stochastic matrix?
Find the stable matrix for the absorbing stochastic matrix?
In a certain community, currently, 10% of the people are H (high income), 60% are M (medium income), and 30% are L (low income). Studies show that for the children of H parents, 50% also become H, 40% become M, and 10% become L. Of the children of M parents, 40% become H, 30% become M, and 30%
Give another example of a 2 × 2 doubly stochastic matrix. (a) Is your matrix symmetric; that is, does it equal its own transpose? (b) Prove that every 2 × 2 doubly stochastic matrix is symmetric?
1. Give another example of a 3 × 3 doubly stochastic matrix. Is your matrix symmetric? Are all 3 × 3 doubly stochastic matrices symmetric? 2. Give another example of a 4 × 4 doubly stochastic matrix. Is your matrix symmetric? Are all 4 × 4 doubly stochastic matrices symmetric?
Show that A is a doubly stochastic n × n matrix if and only if AEn = En and EnA = En?
1. Show that if A and B are doubly stochastic n Ã— n matrices, then AB is also a doubly stochastic n Ã— n matrix. Show that (AB) En = En and En (AB) = En?2. Show that if A is a doubly stochastic n Ã— n matrix, thenIs a stable distribution for A. First show that A
Let A and B be n Ã— n doubly stochastic matrices and 0because A and B are doubly stochastic.) Similarly, the sum of the jth column of C is Therefore, C is doubly stochastic, so the set of all n Ã— n doubly stochastic matrices is a convex set?
Determine whether or not the matrix is a regular stochastic matrix?1.2. 3. 4. 5.
Refer to Exercise 21 of Section 8.1. In the long run, what percentage of the customers will be High users? Refer to Exercise 21, A cell phone provider classifies its customers as Low users (less than 400 minutes per month) or High users (400 or more minutes per month). Studies have shown that 80%
Refer to Exercise 24 of Section 8.1. In the long run, what percentage of the governors will be Democrats? Refer to Exercise 24, For a certain group of states, it was observed that 70% of the Democratic governors were succeeded by Democrats and 30% by Republicans. Also, 40% of the Republican
Refer to Exercise 25 of Section 8.1. What percentage of the mice will be going to the left after many days?Refer to Exercise 25,Each day, mice are put into a T-maze (a maze shaped like a "T"; Fig. 7). In this maze, they have the choice of turning to the left (are rewarded with cheese) or to the
A certain university has a computer room with 219 terminals. Each day, there is a 3% chance that a given terminal will break and a 70% chance that a given broken terminal will be repaired. In the long run, about how many terminals in the room will be working?
Suppose that 60% of people who own a General Motors car buy a GM car as their next car and 90% of people who own a non-GM car buy a non-GM car as their next car. What will General Motors' market share be in the long run?
Commuters can get into town by car or bus. Surveys have shown that, for those taking their car on a particular day, 20% take their car the next day and 80% take a bus. Also, for those taking a bus on a particular day, 50% take their car the next day and 50% take a bus. In the long run, what
The changes in weather from day to day on the planet Xantar form a regular Markov process. Each day is either rainy or sunny. If it rains one day, there is a 90% chance that it will be sunny the following day. If it is sunny one day, there is a 60% chance of rain the next day. In the long run, what
Refer to the stochastic matrix in Example 6 of Section 8.1. In the long run, what percentage of American women will work outside of the home? Refer to Example 6, In 1960, census figures showed that 40% of American women worked outside of the home. Use the stochastic matrix of Example 5 to determine
The Day-by-Day car rental agency rents cars only on a daily basis. Rented cars can be returned at the end of the day to any of the agency's three locations-A, B, or C. Figure 4 shows the percentages of cars returned to each of the locations on the basis of where they were picked up. Assume that all
Refer to Exercise 28 of Section 8.1. In the long run, what percentage of the people will have a strenuous workout on a particular day? Refer to Exercise 28, A group of physical fitness devotees works out in the gym every day. The workouts vary from strenuous to moderate to light. When their
With respect to a certain gene, geneticists classify individuals as dominant, recessive, or hybrid. In an experiment, individuals are crossed with hybrids, then their offspring are crossed with hybrids, and so on. For dominant individuals, 50% of their offspring will be dominant and 50% will be a
The day-to-day changes in weather for a certain part of the country form a Markov process. Each day is sunny, cloudy, or rainy. If it is sunny one day, there is a 70% chance that it will be sunny the following day, a 20% chance that it will be cloudy, and a 10% chance of rain. If it is cloudy one
Explain why this fact does not contradict the main premise of this section with regard to the uniqueness of the stable distribution?Show thatAnd
As shown in Example 2,is not a regular stochastic matrix. Show that acts like a stable distribution for this matrix, and explain why this fact does not contradict the main premise of this section?
Refer to Exercise 33 of Section 8.1. In the long run, what fraction of female babies will have a High birth weight? An Average birth weight?Refer to Exercise 33,Birth weights can be classified as Low (less than 6 pounds), Average (between 6 and 8 pounds), and High (greater than 8 pounds). Birth
Figure 5 describes the migration pattern of a species of bird from year to year among three habitats: I, II, and III.Figure 5,(a) Set up the stochastic matrix that displays these transitions. (b) If there are 1000 birds in each habitat at the beginning of a year, how many will be in each habitat at
Consider the stochastic matrix A, whereApproximate the stable matrix of A by raising A to a high power. Then find the exact stable distribution by solving an appropriate system of linear equations. Check your answer by forming the product of A and the stable distribution?
Repeat Exercise 29 for each of the matrices in Exercises 30-32.Refer to Exercise 29,Consider the stochastic matrix A, whereApproximate the stable matrix of A by raising A to a high power. Then find the exact stable distribution by solving an appropriate system of linear equations. Check your answer
Find the stable distribution for the given regular stochastic matrix?1.2. 3. 4.
Determine whether the transition diagram corresponds to an absorbing stochastic matrix.1.2. 3. 4.
The matrices in Exercises 1-2 are absorbing stochastic matrices in standard form. In each, identify R and S and compute the fundamental matrix and the stable matrix?1.2. 3.
Exercises 19 and 20 refer to Example 7. 1. Interpret the entry .79 in the fundamental matrix. 2. If the gambler begins with $2, what is the expected number of times that he will play before quitting? Refer Example 7, Consider a game of chance with the following characteristics: A person repeatedly
The lawyers at a law firm are either associates or partners. At the end of each year, 30% of the associates leave the firm, 20% are promoted to partner, and 50% remain associates. Also, 10% of the partners leave the firm at the end of each year. Assume that a lawyer who leaves the firm does not
Colleges have been rapidly making broadband Internet service available in their residence halls. Of the colleges that offer no broadband Internet service, each year 10% introduce FIOS Internet service, 30% introduce cable Internet service, and 60% continue to offer no broadband Internet service.
A mouse is placed in one of the compartments of the maze shown in Fig. 7. After each minute of looking unsuccessfully for food, the mouse exits through one of the doors at random and moves to an adjacent compartment. The Exit door is one way; that is, the mouse cannot return after it has exited the
Heather and Blake play a card game in which they take turns drawing a card from a shuffled deck of 52 cards. Heather wins the game if she draws a heart, and Blake wins the game if he draws a black card. When a player doesn't win on his or her turn, the card is returned to the deck, the deck is
Suppose that the following data were obtained from the records of a certain two-year college: Of those who were freshmen (F) during a particular year, 80% became sophomores (S) the next year and 20% dropped out (D). Of those who were sophomores during a particular year, 90% graduated (G) by the
A manufacturer of precise measuring devices carefully tests each device manufactured. Seventy percent of all newly manufactured devices are approved, 20% are sent back to be recalibrated, and 10% are determined to be beyond repair and are destroyed. Eighty percent of the devices that have been
A retailer classifies accounts as having one of four possible states: "paid up," "overdue at most 30 days," "overdue less than 60 days but more than 30 days," and "bad." If no payment is made on an overdue account by the end of the month, the status moves to the next state. When a partial payment
The managers in a company are classified as top managers, middle managers, and first-line managers. Each year, 10% of top managers retire, 10% leave the company, 60% remain top managers, and 20% are demoted to middle managers. Each year, 5% of middle managers retire, 15% leave the company, 10% are
As a variation of the gambler's ruin problem, suppose that on each play, the probability of winning is ½ and the gambler stops playing if he accumulates $4 or goes broke.(a) What is the probability of his eventually going broke if he starts with $1? $2? $3?(b) If the gambler begins with $2,
The concept of a random walk can be applied to a wide variety of fields such as statistical physics, finance, biology, and psychology. The following problem is a simplified version of a random walk: A particle moves on a line and at any time is located at one of the integers 0, 1, 2, 3, 4, 5. If
A soft drink manufacturer puts one of three different quotations on the inside of each bottle cap. Assuming that each of the quotations is equally likely to appear, what is the probability of receiving all three after purchasing five of the soft drinks? What is the expected number of soft drinks
Consider a game of tennis between player A and player B that has reached the state deuce (or 40-40). Continuing from this point, after each rally, the game will be in one of the five states-A wins, B wins, Deuce, Advantage A, or Advantage B. Suppose that the stochastic matrix for the remainder of
Consider the absorbing stochastic matrix A, whereApproximate the stable matrix of A by raising A to a high power. Then find the exact stable distribution by calculating S(I - R)-1.
Repeat Exercise 33 for the matrixRefer to Exercise 33, Consider the absorbing stochastic matrix A, where Approximate the stable matrix of A by raising A to a high power. Then find the exact stable distribution by calculating S(I - R)-1.
Determine whether the given matrix is an absorbing stochastic matrix.1.2. 3. 4.
Convert the absorbing stochastic matrix to standard form?1.2. 3. 4.
Determine the optimal pure strategies for the payoff matrix of the game. If the game is strictly determined, give its value?1.2. 3. 4.
1. (True or False) A strictly determined game can have more than one saddle point.2. (True or False) In a strictly determined game, the value of each saddle point is the same as the value of the game.3. Rosa's Pizzeria and Carlo's Pizzeria compete for customers. They are each considering either
1. Two competing companies, Rigelbucks and Canisbucks, are deciding in which of three malls (A, B, or C) to move their coffee houses. A market research firm has determined that the number of customers each store gains from or loses to the other per day as a result of the location selected is given
1. Suppose that R and C play a game by matching coins. On each play, C pays R the number of heads shown (0, 1, or 2) minus twice the number of tails shown. 2. In the children's game Scissors Paper Stone, each of two children calls out one of the three words. If they both call out the same word,

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