Explanation Given: The next day after a fair day would be fair with a probability of 0.6 and cloudy with a probability of 0.25. The next day after a rainy day would be cloudy with a probability of 0.4 and rainy with a probability of 0.25. The next day after a cloudy day would be cloudy with a probability of 0.35 and rainy with a probability of 0.25. Explanation: The probabilities for the weather of the next day can be represented by a transition matrix. Three classifications of weather would imply a 3 × 3 transition matrix. Define the following variables: F : fair weather C : cloudy weather R : rainy weather Use the provided information to set up the transition matrix P . F C R P = F C R [ 0.6 0.25 a 13 a 21 0.35 0.25 a 31 0.4 0.25 ] For a transition matrix, the sum of all entries for each row needs to be 1. Use this information to obtain the remaining values as follows: 0.6 + 0.25 + a 13 = 1 a 13 = 1 − 0.6 − 0.25 = 1 − 0.85 = 0.15 a 21 + 0.35 + 0.25 = 1 a 21 = 1 − 0.35 − 0.25 = 1 − 0.6 = 0.4 a 31 + 0.4 + 0.25 = 1 a 31 = 1 − 0.4 − 0.25 = 1 − 0.65 = 0.35 This gives the following transition matrix: F C R P = F C R [ 0.6 0.25 0.15 0.4 0.35 0.25 0.35 0.4 0.25 ] The long-range probability for the weather of the next day can be computed as the equilibrium vector for the transition matrix P . A vector v is said to be an equilibrium vector for a transition matrix P if the following condition holds: v P = v Let the vector v be defined as follows, so that the multiplication holds: [ v 1 v 2 v 3 ] Then, v P = v [ v 1 v 2 v 3 ] [ 0.6 0.25 0.15 0.4 0.35 0.25 0.35 0.4 0.25 ] = [ v 1 v 2 v 3 ] Simplify the left-hand side to obtain [ 0.6 v 1 + 0.4 v 2 + 0.35 v 3 0.25 v 1 + 0.35 v 2 + 0.4 v 3 0.15 v 1 + 0.25 v 2 + 0.25 v 3 ] = [ v 1 v 2 v 3 ] This leads to the following system of equations: 0.6 v 1 + 0.4 v 2 + 0.35 v 3 = v 1 0.25 v 1 + 0.35 v 2 + 0.4 v 3 = v 2 0.15 v 1 + 0.25 v 2 + 0.25 v 3 = v 3 This can be simplified to − 0.4 v 1 + 0.4 v 2 + 0.35 v 3 = 0 0.25 v 1 − 0.65 v 2 + 0.4 v 3 = 0 0.15 v 1 + 0.25 v 2 − 0.75 v 3 = 0 The system involving these four equations can be solved with the help of the Gauss–Jordan method. The augmented matrix for the system is [ − 0.4 0

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 10.2, Problem 40E

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The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

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Chapter 10.2, Problem 39E

Chapter 10.2, Problem 41E

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Chapter 10 Solutions

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

Ch. 10.1 - Decide whether each matrix could be a...Ch. 10.1 - Decide whether each matrix could be a probability...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Decide whether each matrix could be a probability...Ch. 10.1 - Decide whether each matrix could be a...Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Decide whether each matrix could be a transition...Ch. 10.1 - Decide whether each matrix could be a...

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - In Exercises and 16, write each transition diagram...Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Find the first three powers of each transition...Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Insurance An insurance company classifies its...Ch. 10.1 - Insurance The difficulty with the mathematical...Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - 32. Land Use In one state, a Board of Realtors...Ch. 10.1 - Business The change in the size of businesses in a...Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Housing Patterns In a survey investigating changes...Ch. 10.1 - Migration A study found that the way people living...Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.2 - Which of the following transition matrices are...Ch. 10.2 - Which of the following transition matrices are...Ch. 10.2 - Which of the following transition matrices are...Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Find the equilibrium vector for each transition...Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Find the equilibrium vector for each transition...Ch. 10.2 - Prob. 16E Ch. 10.2 - Find the equilibrium vector for each...Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Business and Economics Quality Control The...Ch. 10.2 - 26. Quality Control Suppose improvements are made...Ch. 10.2 - (a) Dry Cleaning Using the initial probability...Ch. 10.2 - Mortgage Refinancing In 2009, many homeowners...Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - Prob. 32E Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - Migration As we saw in the last section, a study...Ch. 10.2 - 36. Criminology A study male criminals in...Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Prob. 41E Ch. 10.2 - 42. 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How can we calculate the expected total...Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Business and Economics 23. Solar Energy In...Ch. 10.3 - 24. Company Training Program A company with a...Ch. 10.3 - Contagion Under certain conditions, the...Ch. 10.3 - 26. Medical Prognosis A study using Markov chains...Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Gambler's Ruin (a) Write a transition matrix tor a...Ch. 10.3 - 32. Tennis Consider a game of tennis when each...Ch. 10.3 - Professional Football In Exercise 40 of the first....Ch. 10 - 1. 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The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

Solution Summary: The author explains how three classifications of weather would imply a 3times 3 transition matrix.

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