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In the gambler’s ruin problem, assume that the gambler initial stake is 1 dollar, and assume that her probability of success on any one game is p. Let t be the number of games until 0 is reached (the gambler is ruined). Show that the generating function for t is. H(z) = 1 − √1 − 4pqz2 2pz , and that. Gambler's ruin markov chain. Consider the gambler's ruin problem: at each play of the game the gambler's fortune increases by one dollar with probability 1/2 or decreases by one dollar with probability 1/2. The game is over when the gambler's fortune either reaches 0 or n dollars. Based on the previous definition, we can now define “homogenous discrete time markov chains” (that will be denoted “markov chains” for simplicity in the following). A markov chain is a markov process with discrete time and discrete state space. 2 simulate 1000 trajectories of a gambler’s ruin markov chain with a = 3, p = 2 / 3 and x = 1 (see subsection 5. 2 above for the meaning of these constants). Use the monte carlo method to estimate the probability that the gambler will leave the casino with $3 in her pocket in at most t = 100 time periods. You should be able to classify the states in the gamblers ruin problem as transient/recurrent. You should be able to derive an expression for the probability of ruin in the gamblers run problem. You should be able to calculate the probability of ruin in the limit of equal forward and backward jump probabilities ( p=q p = q) using l'hopital's rule. A player a a plays a sequence of independent games against an opponent b b. At each game either the player wins say $ 1 with probability p p or loses $ 1 with probability q = 1 − p q = 1 − p. We assume that player a a initially has an amount of $ k and his opponent b b has $ (n-k)
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I'm considering a classical problem about markov chains: a gambler has £8 £ 8 and wishes to get to £10 £ 10. A coin is tossed repeatedly : if it comes down tails, the gambler loses his stake, and if it comes down heads, his stake is return and he wins an amount equal to it. 2 simulate 1000 trajectories of a gambler’s ruin markov chain with a = 3, p = 2 / 3 and x = 1 (see subsection 5. 2 above for the meaning of these constants). Use the monte carlo method to estimate the probability that the gambler will leave the casino with $3 in her pocket in at most t = 100 time periods. The gambler’s ruin problem is essentially a markov chain where the sequence of wealth amounts that gambler a has at any point in time determines the underlying structure. The gambler’s ruin problem is one where a player has a probability p of winning and probability q of losing. For example let’s take a skill game where the player x can beat player y with probability 0. 6 by getting closer to target. The game play begins with player x being allotted 5 points and player y allotted 10 points. In the gambler’s ruin problem, assume that the gambler initial stake is 1 dollar, and assume that her probability of success on any one game is p. Let t be the number of games until 0 is reached (the gambler is ruined). Show that the generating function for t is. H(z) = 1 − √1 − 4pqz2 2pz , and that. A player a a plays a sequence of independent games against an opponent b b. At each game either the player wins say $ 1 with probability p p or loses $ 1 with probability q = 1 − p q = 1 − p. We assume that player a a initially has an amount of $ k and his opponent b b has $ (n-k)
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