Gamblers Ruin person repeatedly bets $1 each day. If he wins, he wins $1 (he receives

. Find the inverse of the matrix (p. 137) B 2 -3 -1 5R

Use a graphing utility to find the inverse of the matrix (p. 139) H 1 - 1 2 0 0 - 1 2 1 - 1 2 0 0 - 1 2 1 - 1 2 0 0 - 1 2 1 X

True or False The transition matrix P of an absorbing Markov chain has at least one entry equal to 1.

The transition matrix P of an absorbing Markov chain can be subdivided into __________ submatrices.

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = C S 1 0 1 2 1 2

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = C S 1 3 2 3 0 1

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = C S 0 1 1 2 1 2

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = C 1 3 2 3 1 0 P = C S

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = D U 100 1 3 1 3 1 3 001

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = E U 1 4 1 2 1 4 010 1 3 1 3 1 3

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = E U 100 0 1 2 1 2 0 2 5 3 5

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = E 1 3 0 2 3 010 5 8 0 3 8

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = D U 010 1 3 1 3 1 3 100 T

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = E V 1 4 1 2 1 4 100 2 3 1 3 0

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = F V 1 4 1 4 1 4 1 4 0100 0010 1 3 1 3 1 3 0

In Problems 516, determine if the Markov chain, whose transition matrix P is given, is absorbing. If it is, identify the absorbing state(s).P = F 1000 1 2 0 1 2 0 1 3 1 3 0 1 3 0001

In Problems 1722, subdivide the transition matrix of each absorbing Markov chain.P = D U 100 1 3 1 3 1 3 001 T

In Problems 1722, subdivide the transition matrix of each absorbing Markov chain.P = E 1 4 1 2 1 4 010 1 3 1 3 1 3 P = D U

In Problems 1722, subdivide the transition matrix of each absorbing Markov chain.P = F 1 4 1 4 1 4 1 4 0100 1 2 0 1 2 0 0001 V

In Problems 1722, subdivide the transition matrix of each absorbing Markov chain.P = F X 1 3 1 3 0 1 3 1 2 0 1 2 0 0010 0001 V

In Problems 1722, subdivide the transition matrix of each absorbing Markov chain.P = H X 10000 1 2 0 1 4 0 1 4 00100 0 1 2 0 1 2 0 0 0 1 2 0 1 2

In Problems 1722, subdivide the transition matrix of each absorbing Markov chain.P = H 0 1 4 1 4 0 1 2 0 1000 0 0100 1 2 0 0 1 2 0 0 1 2 1 2 0 0

Suppose that for a certain absorbing Markov chain the fundamental matrix T is found to be (a) What is the expected number of times a person will have $3, given that she started with $1? With $2? (b) If a player starts with $3, how many games can she expect to play before absorption?

Use the matrix T from Problem 23 to answer the following questions: (a) What is the expected number of times a person will have $2 given that she started with $1? With $2? (b) If a player starts with $1, how many games can she expect to play before the game ends?

Gamblers Ruin Problem A person repeatedly bets $1 each day. If he wins, he wins $1 (he receives his bet of $1 plus winnings of $1). He stops playing when he goes broke, or when he accumulates $3. His probability of winning is 0.4 and of losing is 0.6. What is the probability of eventually accumulating $3 if he starts with $1? With $2?

The following data were obtained from the admissions office of a 2-year junior college. Of the first-year class (F), 75% became sophomores (S) the next year and 25% dropped out (D). Of those who were sophomores during a particular year, 90% graduated (G) by the following year and 10% dropped out. (a) Set up a Markov chain with states D, F, G, and S that describes the situation. (b) How many states are absorbing? (c) Determine the matrix T. (d) Determine the probability that an entering first-year student will eventually graduate.

Gamblers Ruin Problem Colleen wants to purchase a $4000 used car, but has only $1000 available. Not wishing to finance the purchase, she makes a series of wagers in which the winnings equal whatever is bet. The probability of winning is 0.4 and the probability of losing is 0.6. In a daring strategy Colleen decides to bet all her money or at least enough to obtain $4000 until she loses everything or has $4000. That is, if she has $1000, she bets $1000; if she has $2000, she bets $2000; if she has $3000, she bets $1000. (a) What is the expected number of wagers placed before the game ends? (b) What is the probability that Colleen is wiped out? (c) What is the probability that Colleen wins the amount needed to purchase the car?

(a) Answer the questions in Problem 27 if the probability of winning is 0.5. (b) Answer the questions in Problem 27 if the probability of winning is 0.6.

Military Strategy Three armored cars, A, B, and C, are engaged in a three-way battle. Armored car A has probability of destroying its target, B has probability of destroying its target, and C has probability of destroying its target. The armored cars fire at the same time and each fires at the strongest opponent not yet destroyed. Using as states the surviving cars at any round, set up a Markov chain and answer the following questions: (a) How many states are in this chain? (b) How many are absorbing? (c) Find the expected number of rounds fired. (d) Find the probability that A survives.

Stock Price Model In the model discussed on pages 606608 a stock currently showing no change would, on the average, stay how long in that state?

Stock Price Model Refer to the model discussed on pages 606608. By making state I an absorbing state, compute the average number of days elapsed before a stock currently decreasing would start to increase again.

In Problems 3233, use a graphing utility or Excel.uppose that for a certain absorbing Markov chain the transition matrix P is given by Determine the matrices and Q. Find the fundamental matrix T. Determine the product

In Problems 3233, use a graphing utility or Excel.Consider the maze with five rooms shown in the figure. The system consists of the maze and a mouse. We assume the following learning pattern exists: If the mouse is in room 2, 3, or 4, it moves with equal probability to any room the maze permits; if it is in room 1, it gets caught in the trap and remains there; if it is in room 5, it remains in that room with the cheese forever. (a) Construct the transition matrix P. (b) How many rooms represent an absorbing state for this Markov chain? (c) If a mouse begins in room 3, what is the probability that it will find the cheese in 5 steps? Determine the powers of the matrix P. (d) Subdivide the transition matrix. (e) Determine the matrices and Q. (f) Find the fundamental matrix T. (g) Determine the product (h) How many steps, on the average, will the mouse make in the maze before it ends up in the trap?