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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.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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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