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4 PROBLEM FOUR Suppose XXX is gambling in a Casino with 3 dollars in his pocket,...
The Nero Place casino has a new, exciting gambling machine: the multiplying bandit. How does it...
The Nero Place casino has a new, exciting gambling machine: the multiplying bandit. How does it work? The bandit has a lever or “arm” that the player may depress up to 10 times. After each pull, a H (heads) or a T (tails) appears, each with probability 1/2 . The game is over when heads appears for the first time, or when the player has pulled the arm 10 times. The player wins 2k dollars if heads appears after k...
In the casino gambling game of American Roulette the wheel has 38 pockets numbered 00,0,1, ....
In the casino gambling game of American Roulette the wheel has 38 pockets numbered 00,0,1, . . . ,36. Half of the numbers from 1 and 36 are painted black, while the others are painted red. The numbers 00 and 0 are painted green. A ball is equally likely to land in any pocket. Listed below are several of the many possible bets on where the ball lands, together with their winning payouts based on a$1 stake. In each case...
(20 points) Suppose a small local hospital has 3 nurses. On any given day, independent of...
(20 points) Suppose a small local hospital has 3 nurses. On any given day, independent of the number of nurses that are working, either one of the working nurses becomes unavailable with probability 1 - q > 0, or all of the nurses are available with probability q. If a nurse becomes unavailable, with certainty s/he will be available next day (we make the simplifying assumption that if a nurse becomes unavailable, s/he becomes unavailable at the beginning of the...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov ch...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov chain starts in state 1. What is the probability that it is in state 3 after two steps? (c) (10 points) Caleulate the steady-state distribution (s) for states 1, 2, and 3, respee- tively
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix...
(6(4 pts) A player (Joe) goes to a casino and plays a fair game. The player...
(6(4 pts) A player (Joe) goes to a casino and plays a fair game. The player may wager any amount of money. There is a 0.5 probability of winning. If the player wins, then the player get twice the amount of the bet in winnings. If the player loses, the player gets nothing. Think of betting on a coin toss. If you win you double your money, if you lose you lose your money. This is a "fair" game because...
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At ea...
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At each of these times, the casino can be in one of two states z E1, 2}. When z = 1 the casino uses a fair die, while when z = 2 the die is biased so that rolling a 1 is more simple hidden Markov model (HMM). We observe a "occasionally dishonest casino", where at time likely....
Roulette is one of the most common games played in gambling casinos in Las Vegas and...
Roulette is one of the most common games played in gambling casinos in Las Vegas and elsewhere. An American roulette wheel has slots marked with the numbers from 1 to 36 as well as 0 and 00 (the latter is called "double zero"). Half of the slots marked 1 to 36 are colored red and the other half are black. (The 0 and 00 are colored green.) With each spin of the wheel, the ball lands in one of these...
A Markov chain has the transition matrix P = 1.4.61 L.7 .3 Suppose that on the...
A Markov chain has the transition matrix P = 1.4.61 L.7 .3 Suppose that on the initial observation, the chain is in state 1 with probability 2. What is the probability that the system will be in state 1 on the next observation? 0.38 0.60 0.64 0.36 0.56
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following trans...
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following transition 0 1/3 2/3 1/2 0 1/2 If P(o-: 1)-P(Xo-2-1/4, calculate E(%) (use a computer).
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following transition 0 1/3 2/3 1/2 0 1/2 If P(o-: 1)-P(Xo-2-1/4, calculate E(%) (use a computer).
P is the (one-step) transition probability matrix of a Markov chain with state space {0, 1, 2, 3,...
P is the (one-step) transition probability matrix of a Markov chain with state space {0, 1, 2, 3, 4 0.5 0.0 0.5 0.0 0.0 0.25 0.5 0.25 0.0 0.0 P=10.5 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.0 0.5 0.5/ (a) Draw a transition diagram. (b) Suppose the chain starts at time 0 in state 2. That is, Xo 2. Find E Xi (c)Suppose the chain starts at time 0 in any of the states with...
(20 points) Suppose a small local hospital has 3 nurses. On any given day, independent of the number of nurses that are working, either one of the working nurses becomes unavailable with probability 1 - q > 0, or all of the nurses are available with probability q. If a nurse becomes unavailable, with certainty s/he will be available next day (we make the simplifying assumption that if a nurse becomes unavailable, s/he becomes unavailable at the beginning of the...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov chain starts in state 1. What is the probability that it is in state 3 after two steps? (c) (10 points) Caleulate the steady-state distribution (s) for states 1, 2, and 3, respee- tively
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix...
(6(4 pts) A player (Joe) goes to a casino and plays a fair game. The player may wager any amount of money. There is a 0.5 probability of winning. If the player wins, then the player get twice the amount of the bet in winnings. If the player loses, the player gets nothing. Think of betting on a coin toss. If you win you double your money, if you lose you lose your money. This is a "fair" game because...
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At each of these times, the casino can be in one of two states z E1, 2}. When z = 1 the casino uses a fair die, while when z = 2 the die is biased so that rolling a 1 is more simple hidden Markov model (HMM). We observe a "occasionally dishonest casino", where at time likely....
A Markov chain has the transition matrix P = 1.4.61 L.7 .3 Suppose that on the initial observation, the chain is in state 1 with probability 2. What is the probability that the system will be in state 1 on the next observation? 0.38 0.60 0.64 0.36 0.56
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following transition 0 1/3 2/3 1/2 0 1/2 If P(o-: 1)-P(Xo-2-1/4, calculate E(%) (use a computer).
Problem 5. A Markov chain Xn, n probability matrix: 0 with states 1, 2, 3 has the following transition 0 1/3 2/3 1/2 0 1/2 If P(o-: 1)-P(Xo-2-1/4, calculate E(%) (use a computer).
P is the (one-step) transition probability matrix of a Markov chain with state space {0, 1, 2, 3, 4 0.5 0.0 0.5 0.0 0.0 0.25 0.5 0.25 0.0 0.0 P=10.5 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0 0.0 0.5 0.5/ (a) Draw a transition diagram. (b) Suppose the chain starts at time 0 in state 2. That is, Xo 2. Find E Xi (c)Suppose the chain starts at time 0 in any of the states with...
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