### Understanding Markov Chains through a MazeConsider the illustration above, which represents a maze with four numbered rooms connected by paths. The diagram shows:- **Nodes (Rooms):** Labeled as 1, 2, 3, and 4.- **Paths (Edges):** - Room 1 connects to Rooms 2, 3, and 4. - Room 2 connects to Rooms 1, 3, and 4. - Room 3 connects to Rooms 1 and 2. - Room 4 connects to Rooms 1 and 2.#### Problem ScenarioSuppose a group of robots is traversing this maze. At every step, each robot chooses a path to move along, with equal probability of selecting any available path. Robots cannot remain in place. At the end of each move, a robot resides in one of the four rooms.#### Tasks**Part (a):** Construct the transition matrix for the Markov chain based on this scenario. This matrix will represent the probabilities of moving from one room to another.**Part (b):** Determine the steady-state probability vector, which reflects the long-term probabilities of finding a robot in each room.This exercise aids in understanding the basics of Markov chains and their applications in modeling random processes.

First, construct the state transition diagram and then find the transition matrix. For the Markov…

Answered: Suppose that a group of robots is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Aileen, a Scottish spy, has three fake identities that she uses to get information. The process is really quite involved, but she uses a Markov chain process to make it more difficult for her Irish adversaries to track her: "Hope" (state 1), "Trixie"(state 2), and "Fiona" (state 3). The transition matrix is .5 .5 P = .2 .2 .6 .4 .1 .5 On the first observation, Aileen used "Trixie" as her identity. Which of the three fake identities is she most likely to have used on the third observation? O Hope and Fiona are equally likely O Hope and Trixie are equally likely Fiona О Норе O None of the others are correct O Trixie
Central theme: Markov chainsA supermarket chain sells Alfa and Beta brand bread. Currently, Alfa has 50% of the chain's daily bread sales.of the chain's daily bread sales. To increase sales, Alfa launches a new advertising campaign. The bakerybelieves that the change in bread sales in the chain will be based on the transition matrix: A) In the long run, by what percentage can alpha expect to increase its current sales in the chain? Assumethat total daily sales of bread remain constant.
Long-Run Properties of Markov Chains The leading brewery on the West Coast (labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst -believes that brand switching can be modeled as a Markov chain using three states, with states A and B representing customers drinking beer produced from the aforementioned breweries and state C representing all other brands. Data are taken monthly, and the analyst has constructed the following (one-step) transition -matrix from past data. What are the steady-state market shares for the two major -breweries?
A certain television show airs weekly. Each time it airs, its ratings are categorized as "high," "average," or "low," and transitions between these categorizations each week behave like a Markov chain. If the ratings are high this week, they will be high, average, or low next week with probabilities 40%, 10%, and 50% respectively. If the ratings are average this week, they will be high, average, or low next week with probabilities 60%, 20%, and 20% respectively. If the ratings are low this week, they will be high, average, or low next week with probabilities 70%, 20%, and 10% respectively. Write all answers as integers or decimals. If the ratings are average this week, what is the probability that they will be low two weeks from now? If the ratings are average the first week, what is the probability that they will be low the second week? If the ratings are low the first week, what is the probability that they will be high the third week?
DO NOT COPY OTHER SOLUTIONS. USE A TRANSITION PROBABILITY MATRIX WHEN SOLVING. Consider the following Markov chain. Assuming that the process starts in state 4, what is the probability that we eventually reach the recurrent class {1,2}?
Employment Employment Last Year This Year Percentage Industry Industry 80 Small Business 10 Self-Employed 10 Small Business Industry 10 Small Business 60 Self-Employed 30 Self-Employed Industry 10 Small Business 80 Self-Employed 10 Assume that state 1 is Industry, that state 2 is Small Business, and that state 3 is Self-Employed. Find the transition matrix for this Markov process. P =
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4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
Shakira's concerts behave like a Markov chain. If the current concert gets cancelled, then there is an 90% chance that the next concert will be cancelled also. However, if the current concert does not get cancelled, then there is only a 50% chance that the next concert will be cancelled. What is the long-run probability that a concert will not be cancelled? a. 1/4 b. 1/10 c. 1/6 d. 1/2 e. 5/6 f. None of the others are correct
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