## Review Exercises for Finite MathematicsApplied to the Real World Topic: Markov Systems

Chapter 9 Summary
True/False Quiz
Index of Review Exercises
Everything for Calculus
Everything for Finite Math
Everything for Finite Math & Calculus
Utility: 4-State Markov System Simulation
Utility: Matrix Algebra Tool

1. Give the transition matrix for each of the following.

P=

P=

2. In each of the following, you are given a transition matrix P and an initial distribution vector v. Compute the quantities asked for.

(a) P=
 0.2 0.8 0.8 0.2
,   v = [100   0]

Probability of going from state 2 to state 1 in two steps =

Distribution vector after two steps is [   ]

(b) P=
 1/2 1/4 1/4 0 1 0 1/2 1/2 0
,   v = [0   160   0]

The two-step transition matrix is
The four-step transition matrix is
Probability of going from state 3 to state 1 in four steps =

Distribution vector after four steps is [     ]

3. Reading Habits Jason Sauter is an avid reader of science fantasy, Proust novels and calculus texts. He visits the bookstore every week and buys one book. (He reads a book every week.) If he is presently reading Proust, there is a 50% chance that he will switch to calculus the next week, and he never reads two Proust volumes in a row. After a week of science fantasy, there is a 75% chance that he will switch to Proust the next week, and a 25% percent chance that he will switch to calculus. He always reads Proust after reading calculus. Assuming this trend continues indefinitely, * what fraction of his library will consist of science fiction 17,000 weeks from now?
science fiction

* Also assume that there is an inexhaustible supply of science fiction, Proust, and calculus books.

4. Jones Beach Jones Beach has a jetty on its westernmost edge, a swimming area is on its easternmost edge, and a volleyball play area in between. One sunny day, you notice that one in six of the people in the volleyball area stroll to the jetty every hour, while two in six stroll over to the swim area. Due to the bad jellyfish infestation, half the beach goers at the swim area meander over to the volleyball area every hour, while the rest of them stay put. Nobody leaves the jetty.

(a) At noon, the entire beach population of 1,200 was in the swimming area. How crowded was the jetty by 2 pm.?
people

(b) What fraction of the people in the volleyball play area at noon will be at the jetty by 2 pm.?
at the jetty

5. Malls Following is a plan of the "Petit Mall." As an potential store owner, you have been offered a lease at various store locations in the mall (the colored areas show the store locations), and are deciding which location to choose. Part of your decision will be based on a knowledge of shopper traffic in different parts of the mall (areas A through F).

During the first part of the morning hours, very few shoppers leave the mall through tghe exit door and the movie theater is closed. Further, shoppers tend to move randomly from one area to an adjacent area. For instance, a shopper in area B has a 1/3 chance of moving to A, C, or D, which a shopper in C is certain to move to B.

(a) If a shopper begins in location A when the mall opens, what is the probability of that shopper being found in location C after two changes of location (two time steps)?

(b) Calculate P2, P3, P4, and P5 by hand. On average, what percentage of shoppers can be found where they started after 5 changes of location?

6. Sports Researchers have modeled the results of cricket games between England and Australia by the following Markov system, in which a time step corresponds to one game. (The states represent wins, losses and draws for England.) ††

 To Win Loss Draw From Win 0.4 0.4 0.2 Loss 0.3 0.4 0.3 Draw 0.3 0.3 0.4

In the long term, England can expect to win of the time, to lose of the time, and to draw of the time.

†† Figures are rounded to one decimal place. Source: Derek Colwell, Brian Jones and Jack Gillet, "A Markov Chain in Cricket," the Mathematical Gazette, June, 1991.

7. Heating (Use of technology recommended. Try the the on-line matrix algebra tool for a direct computaiton, or the Pivot and Gauss-Jordan Tool to solve the associated system of equations.)
According to research by Gulf Oil Corporation, switching between oil, gas and electric heat in US homes followed the following pattern over a one-year period. Ý

 To Oil Gas Electric From Oil 0.825 0.175 0 Gas 0.060 0.919 0.021 Electric 0.049 0 0.951

What are the long term implications of these trends? (Round all percentages to two decimal places.)

% oil   % gas   % electric

Ý Source: Ali Ezzarti, "Forecasting Market Shares of Alternative Home Heating Units," Management Science, Vol. 21, no. 4, Dec. 1974. Copyright (c) 1974 by The Institute of Management Sciences.

8. Neural Networks Consider the following model of a small neural network in which an impulse can travel from node to node with probabilities shown in the diagram.

(a) Write down the transition matrix P, and compute the steady state transition vector v.
(b) Is the given system regular?

9. Airplane Safety Assuming that there is a 1 in 10,000 chance of a plane crashing, use a Markov system to show that one can expect a plane crash on the 10,000th trip (on average).

10. In the Markov system represented by the following transition diagram, determine how long, on average, it will take to reach the absorbing state, assuming that the system starts in State 2.

10. Petit Mall This continues the above question on the "Petit Mall."

During the evening hours, a shopper in Area C has a 0.5 probability of leaving the mall through Area E, while a shopper in area D has a 0.5 probability of going to the movies. Since the mall will be closed by the time the movie ends, regard areas E and F as absorbing states.

(a) Calcuate the fundamental matrix Q .

(b) How many visits will a shopper entering the mall in Area A be expected to make to Area D?

(c) Which area or areas will be visited the most frequently by a shopper entering the mall at Area A?

(d) What fraction of shoppers in Area D will ultimately go to the movies?

Last Updated: August, 2000