A wandering ghost travels between four regions of the island where it lives: Toronbo Shores, Koholint Prairie, Tabahl Wasteland, and Goponga Swamp. It moves each day according to a Markov chain as given in the diagram. (From a given region, each possible move is shown and is equally likely.) Goponga Swamp Koholint Prairie Tabah/ Wasteland Toronbo Shores Whenever the ghost is at Toronbo Shores, what is the probability it will be in Koholint Prairie in two days? Whenever the ghost is at Toronbo Shores, what is the probability it will be in Koholint Prairie in three days?

A wandering ghost travels between four regions of the island where it lives: Toronbo Shores, Koholint Prairie, Tabahl Wasteland, and Goponga Swamp. It moves each day according to a Markov chain as given in the diagram. (From a given region, each possible move is shown and is equally likely.) Goponga Swamp Koholint Prairie Tabah/ Wasteland Toronbo Shores Whenever the ghost is at Toronbo Shores, what is the probability it will be in Koholint Prairie in two days? Whenever the ghost is at Toronbo Shores, what is the probability it will be in Koholint Prairie in three days?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 12EQ: 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction...

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Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.
12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)
Fifty-four players (including Gabe Kaplan and JamesGarner) participated in the 1980 World Series of Poker.Each player began with $10,000. Play continued until oneplayer had won everybody else’s money. If the World Seriesof Poker were to be modeled as a Markov chain, how manyabsorbing states would the chain have?
Fifty-four players participated in the 1980 World Series of Poker. Each player began with $10,000. Play continued until one player had won everybody else’s money. If the World Series of Poker were to be modeled as a Markov chain, how many absorbing states would the chain have?
Assuming that the climate in the city of Metepec behaves according to a Markov chain, consider the following data. The probability of rain on any following day is 70% if it rained the day before, and 20% if it did not rain the day before; Likewise, the probability that the next day will be dry is 80% if the day before was dry, and 30% if it rained the day before. From this information: C. Find the probability that Friday is a dry day, given that it rained today, Tuesday. D. Suppose you want to calculate the probability that it will rain in exactly six months, describe two different strategies you could use to perform this calculation. Would it really matter to consider what the weather was today?
Suppose that a basketball player’s success in free-throw shooting can be described with a Markov chain. If the player made the last free throw, then she is four times more likely to make the next free throw as miss it. If the player missed her last free throw, then she is equally likely to make or miss the next free throw. If she misses her first free throw, what is the probability she also misses her third and fifth free throw?
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Anne and Barry take turns rolling a pair of dice, with Anne going first. Anne’s goal is to obtain a sum of 3, while Barry’s goal is to obtain a sum of 4. The game ends when either player reaches his goal,and the one reaching the goal is the winner. Define a Markov Chain to model the problem.
a) Suppose that whether or not it rains, today edpends on previous weather conditions through the last three days. Show how this system may be analyzed by using a Markov chain. How many states are needed? b) Suppose that if it has rained for the past three days, then it will rain today with probability 0.8; if it did not rain for any of the past three days, then it will rain today with probability 0.2; and in any other case the weather today will, with probibility 0.6, be the same as the weather yesterday. Determine P for this Markov chain.
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Each item is inspected and is declared to either pass or fail. The machine can work in automatic or manual mode. If it outputs two failed items in a row in automatic mode, it is switched to manual. Once it produces two passing items in a row in manual mode, it is switched back to automatic. Sup- pose that failure rate is a in automatic and b in manual. You modeled the system as a Markov chain with a diagram given below, where states represent the mode and the status of the previously man- ufactured item, so for example, state “manual-1 success” represents that the machine is in manual mode and the previous item passed.
At Community College, 10% of all business majors switched to another major the next semester, while the remaining 90% continued as business majors. Of all nonbusiness majors, 20% switched to a business major the following semester, while the rest did not. Set up these data as a Markov transition matrix, and calculate the probability that a business major will no longer be a business major in the long run