Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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Question
This is for a Linear Algebra class. Please provide explanations for the answer (3) In the casino game of roulette, a ball is spun around a wheel with 38 slots, 18 of which are colored black. If you bet on black, then you gain a profit of $1 if the ball lands on one of the 18 black slots, but you lose $1 if it lands on one of the other 20 slots (a) Write out the transition matrix for a Markov Chain representing how much money you have in your pocket before and after a single play of roulette assuming that you will start with less than $10 and stop once you get to either $0 or $10. For example, if you have $4 at some point in the night, then you will either have $5 or S3 after the next round depending on whether black comes up or not, but if you get to $0 or $10 at any point, you stop playing (b) Calculate the eigenvalues of your transition matrix and identify all possible sta- tionary distributions. Discuss convergence (unique for any initial? fail to converge? etc. (c) Suppose that you start the night with S5. Calculate the probability that you double your money before losing it all (i.e. probability that you end the night with S10 instead of $0)
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Transcribed Image Text:This is for a Linear Algebra class. Please provide explanations for the answer (3) In the casino game of roulette, a ball is spun around a wheel with 38 slots, 18 of which are colored black. If you bet on black, then you gain a profit of $1 if the ball lands on one of the 18 black slots, but you lose $1 if it lands on one of the other 20 slots (a) Write out the transition matrix for a Markov Chain representing how much money you have in your pocket before and after a single play of roulette assuming that you will start with less than $10 and stop once you get to either $0 or $10. For example, if you have $4 at some point in the night, then you will either have $5 or S3 after the next round depending on whether black comes up or not, but if you get to $0 or $10 at any point, you stop playing (b) Calculate the eigenvalues of your transition matrix and identify all possible sta- tionary distributions. Discuss convergence (unique for any initial? fail to converge? etc. (c) Suppose that you start the night with S5. Calculate the probability that you double your money before losing it all (i.e. probability that you end the night with S10 instead of $0)
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