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It costs $10 to play a gambling game. If the player wins, then this $10 is returned and an additional $10 is given to the player. If the player loses, then the player loses the $10. The probability of winning is .4. Assume that a player starts with $20 and plays until either losing everything or having a total of $30. This is a gambler's ruin Markov chain. a) Describe the state and draw the transition diagram. b) Write the transition matrix in standard form. c) What is the probability that the player wins, i.e. the player finishes with $30.

It costs $10 to play a gambling game. If the player wins, then this $10 is returned and an additional $10 is given to the player. If the player loses, then the player loses the $10. The probability of winning is .4. Assume that a player starts with $20 and plays until either losing everything or having a total of $30. This is a gambler's ruin Markov chain. a) Describe the state and draw the transition diagram. b) Write the transition matrix in standard form. c) What is the probability that the player wins, i.e. the player finishes with $30.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 16E: Consumer Preference In a population of 100,000 consumers, there are 20,000 users of Brand A, 30,000...
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It costs $10 to play a gambling game. If the player wins, then this $10 is returned and an additional $10 is given to the player. If the player loses, then the player loses the $10. The probability of winning is .4. Assume that a player starts with $20 and plays until either losing everything or having a total of $30. This is a gambler's ruin Markov chain. a) Describe the state and draw the transition diagram. b) Write the transition matrix in standard form. c) What is the probability that the player wins, i.e. the player finishes with $30. d) What is the probability that the player loses, i.e. the player finishes with $0. e) For each transient state, find the expected number of times that we visit that state. f) How many rounds do we expect the game to last?
Transcribed Image Text:It costs $10 to play a gambling game. If the player wins, then this $10 is returned and an additional $10 is given to the player. If the player loses, then the player loses the $10. The probability of winning is .4. Assume that a player starts with $20 and plays until either losing everything or having a total of $30. This is a gambler's ruin Markov chain. a) Describe the state and draw the transition diagram. b) Write the transition matrix in standard form. c) What is the probability that the player wins, i.e. the player finishes with $30. d) What is the probability that the player loses, i.e. the player finishes with $0. e) For each transient state, find the expected number of times that we visit that state. f) How many rounds do we expect the game to last?
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