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Chapter 18. Markov Chains: Homework

MARKOV CHAINS

Exercise 1.

Is the matrix given below a transition matrix for a Markov chain? Explain.

  1. _autogen-svg2png-0001.png

  2. _autogen-svg2png-0002.png

  1. No

  2. No

Exercise 2.

A survey of American car buyers indicates that if a person buys a Ford, there is a 60% chance that their next purchase will be a Ford, while owners of a GM will buy a GM again with a probability of .80. The buying habits of these consumers are represented in the transition matrix below.

This matrix depicts the buying habits of GM and Ford customers.
Figure 18.0. 

Find the following probabilities:

  1. The probability that a present owner of a Ford will buy a GM as his next car.

  2. The probability that a present owner of a GM will buy a GM as his next car.

  3. The probability that a present owner of a Ford will buy a GM as his third car.

  4. The probability that a present owner of a GM will buy a GM as his fourth car.

Exercise 3.

Professor Hay has breakfast at Hogee's every morning. He either orders an Egg Scramble, or a Tofu Scramble. He never orders Eggs on two consecutive days, but if he does order Tofu one day, then the next day he can order Tofu or Eggs with equal probability.

  1. Write a transition matrix for this problem.

  2. If Professor Hay has Tofu on the first day, what is the probability he will have Tofu on the second day?

  3. If Professor Hay has Eggs on the first day, what is the probability he will have Tofu on the third day?

  4. If Professor Hay has Eggs on the first day, what is the probability he will have Tofu on the fourth day?

  1. _autogen-svg2png-0003.png

  2. 1/2

  3. 1/2

  4. 3/4

Exercise 4.

A professional tennis player always hits cross-court or down the line. In order to give himself a tactical edge, he never hits down the line two consecutive times, but if he hits cross-court on one shot, on the next shot he can hit cross-court with .75 probability and down the line with .25 probability.

  1. Write a transition matrix for this problem.

  2. If the player hit the first shot cross-court, what is the probability that he will hit the third shot down the line?

Exercise 5.

The transition matrix for switching political parties in an election year is given below, where Democrats, Republicans, and Independents are denoted by the letters D, R, and I, respectively.

This matrix shows the tendencies of Democrat, Republicans, and Independents to switch sides during an election year.
Figure 18.0. 

  1. Find the probability of a Democrat voting Republican.

  2. Find the probability of a Democrat voting Republican in the second election.

  3. Find the probability of a Republican voting Independent in the second election.

  4. Find the probability of a Democrat voting Independent in the third election.

  1. 0.3

  2. 0.38

  3. 0.15

  4. 0.175

REGULAR MARKOV CHAINS

Exercise 6.

Determine whether the following matrices are regular Markov chains.

  1. _autogen-svg2png-0010.png

  2. _autogen-svg2png-0011.png

  3. _autogen-svg2png-0012.png

  4. _autogen-svg2png-0013.png

a. No c. No
Exercise 7.

Company I and Company II compete against each other, and the transition matrix for people switching from Company I to Company II is given below.

This matrix shows the tendency of customers to switch between Company I and Company II.
Figure 18.0. 

Find the following.

  1. If the initial market share is 40% for Company I and 60% for Company II, what will the market share be after 3 steps?

  2. If this trend continues, what is the long range expectation for the market?

Exercise 8.

Suppose the transition matrix for the tennis player in Exercise 4. is as follows, where C denotes the cross-court shots and D denotes down-the-line shots.

This matrix depicts the tendency of a tennis player to make cross-court shots and down-the-line shots.
Figure 18.0. 

Find the following.

  1. If the player hit the first shot cross-court, what is the probability he will hit the fourth shot cross-court?

  2. Determine the long term shot distribution.

  1. 0.876

  2. _autogen-svg2png-0016.png

Exercise 9.

Professor Hay never orders eggs two days in a row, but if he orders tofu one day, then there is an equal probability that he will order tofu or eggs the next day.

Find the following:

  1. If Professor Hay had eggs on Monday, what is the probability that he will have tofu on Friday?

  2. Find the long term distribution for breakfast choices for Professor Hay.

Exercise 10.

Many Russians have experienced a sharp decline in their living standards due to President Yeltsin's reforms. As a result, in the parliamentary elections held in December 1995, Communists and Nationalists made significant gains, and a new pattern in switching political parties emerged. The transition matrix for such a change is given below, where Communists, Nationalists, and Reformists are denoted by the letters C, N, and R, respectively.

This matrix depicts the tendencies for the different political parties to take power.
Figure 18.0. 

Find the following.

  1. If in this election Communists received 25% of the votes, Nationalists 30%, and Reformists the rest 45%, what will the distribution be in the next election?

  2. What will the distribution be in the third election?

  3. What will the distribution be in the fourth election?

  4. Determine the long term distribution.

a. _autogen-svg2png-0020.png c. _autogen-svg2png-0021.png

ABSORBING MARKOV CHAINS

Exercise 11.

Given the following absorbing Markov chain.

This matrix depicts the probability of moving from one sate to the other.
Figure 18.0. 

Find the following:

  1. Identify the absorbing states.

  2. Write the solution matrix.

  3. Starting from state 4, what is the probability of eventual absorption in state 1?

  4. Starting from state 2, what is the probability of eventual absorption in state 3?

  1. 1 and 3

  2. This matrix shows the probability of absorption.
    Figure 18.0. 

  3. 2 / 3

  4. 1 / 2

Exercise 12.

Two tennis players, Andre and Vijay each with two dollars in their pocket, decide to bet each other $1, for every game they play. They continue playing until one of them is broke.

Do the following:

  1. Write the transition matrix for Andre.

  2. Identify the absorbing states.

  3. Write the solution matrix.

  4. At a given stage if Andre has $1, what is the chance that he will eventually lose it all?

Exercise 13.

Repeat Exercise 12., if the chance of winning for Andre is .4 and for Vijay .6.

  1. Write the transition matrix for Andre.

  2. Write the solution matrix.

  3. If Andre has $3, what is the probability that he will eventually be ruined?

  4. If Vijay has $1, what is the probability that he will eventually triumph?

  1. This matrix shows the probability of winning a dollar for every dollar won.
    Figure 18.0. 

  2. Andre's solution matrix

    This matrix depict the probability of winning a either zero dollars or four dollars.
    Figure 18.0. 

  3. 27 / 65

  4. 27 / 65

Exercise 14.

Repeat Exercise 12., if initially Andre has $3 and Vijay has $2.

  1. Write the transition matrix.

  2. Identify the absorbing states.

  3. Write the solution matrix.

  4. If Andre has $4, what is the probability that he will eventually be ruined?

Exercise 15.

The non-tenured professors at a community college are regularly evaluated. After an evaluation they are classified as good, bad, or improvable. The "improvable" are given a set of recommendations and are re-evaluated the following semester. At the next evaluation, 60% of the improvable turn out to be good, 20% bad, and 20% improvable. These percentages never change and the process continues.

  1. Write the transition matrix.

  2. Identify the absorbing states.

  3. Write the solution matrix.

  4. What is the probability that a professor who is improvable will eventually become good?

  1. The matrix shows the probability of moving from Improvable to Good, Bad, or remaining Improvable.
    Figure 18.0. 

  2. I and II

  3. _autogen-svg2png-0026.png

  4. 0.75

Exercise 16.

A rat is placed in the maze shown below, and it moves from room to room randomly. From any room, the rat will choose a door to the next room with equal probabilities. Once it reaches room 1, it finds food and never leaves that room. And when it reaches room 5, it is trapped and cannot leave that room. What is the probability the rat will end up in room 5 if it was initially placed in room 3?

This figure shows the layout of the rooms that the rat can navigate through.
Figure 18.0. 

Exercise 17.

In Exercise 16., what is the probability the rat will end up in room 1 if it was initially placed in room 2?

(18.1)10/19

CHAPTER REVIEW

Exercise 18.

Is the matrix given below a transition matrix for a Markov chain? Explain.

  1. _autogen-svg2png-0028.png

  2. _autogen-svg2png-0029.png

  1. No

  2. No

Exercise 19.

A survey of computer buyers indicates that if a person buys an Apple computer, there is an 80% chance that their next purchase will be an Apple, while owners of an IBM will buy an IBM again with a probability of .70. The buying habits of these consumers are represented in the transition matrix below.

This matrix shows the probable buying habits of consumers from Apple to IBM.
Figure 18.0. 

Find the following probabilities:

  1. The probability that a present owner of an Apple will buy an IBM as his next computer.

  2. The probability that a present owner of an Apple will buy an IBM as his third computer.

  3. The probability that a present owner o

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