Consider the three competing grocery stores in town, ABC Grocery, 123 Grocery and XYZ Grocery. Consider the transition probabilities TO From abc grocery 123 grocery xyz grocery ABC grocery 0.65 0.15 0.2 123 GROCERY 0.15 0.80 0.05 xyz grocery 0.10 0.15 0.75 a. Compute the steady-state probabilities for this three-state Markov process. b. What market share will each store obtain

Consider the three competing grocery stores in town, ABC Grocery, 123 Grocery and XYZ Grocery. Consider the transition probabilities TO From abc grocery 123 grocery xyz grocery ABC grocery 0.65 0.15 0.2 123 GROCERY 0.15 0.80 0.05 xyz grocery 0.10 0.15 0.75 a. Compute the steady-state probabilities for this three-state Markov process. b. What market share will each store obtain

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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3. Consider an undiscounted Markov decision process with three states 1, 2, 3, with respec- tive rewards -1, -2,0 for each visit to that state. In states 1 and 2, there are two possible actions: a and b. The transitions are as follows: • In state 1, action a moves the agent to state 2 with probability 0.8 and makes the agent stay put with probability 0.2. In state 2, action a moves the agent to state 1 with probability 0.8 and makes the agent stay put with probability 0.2. . In either state 1 or state 2, action b moves the agent to state 3 with probability 0.1 and makes the agent stay put with probability 0.9. Find the optimal policy that minimises the expected total cost and find the corresponding value function.
Draw the transition probability graph and construct the transition probability matrix of the following problems. 2. A police car is on patrol in a neighborhood known for its gang activities. During a patrol, there is a 60% chance of responding in time to the location where help is needed; else regular patrol will continue. chance for cancellation (upon receiving a call, there is a 10% in which case normal patrol is resumed) and a 30% chance that the car is already responding to a previous call. When the police car arrives at the scene, there is a 10% chance that the instigators will have fled (in which case the car returns back to patrol) and a 40% chance that apprehension is made immediately. Else, the officers will search the area. If apprehension occurs, there is a 60% chance of transporting the suspects to the police station; else they are released and the car returns to patrol.
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The following data consists of a matrix of transition probabilities (P) of three competing companies, the initial market share state ℼ(1), and the equilibrium probability states. Assume that each state represents a firm (Company 1, Company 2, and Company 3, respectively) and the transition probabilities represent changes from one month to the next. The market share of Company 2 after two periods is a. 0.283 b. 0.26 c. 0.261 d. 0.47
The purchase patterns for two brands of toothpaste can be expressed as a Markov process with the following transition probabilities. To From Special B MDA Special B 0.90 0.10 MDA 0.02 0.98 (a)Which brand appears to have the most loyal customers? Explain. MDA has the most loyal customers because ( ) %? stay with them and only ( ) %? switch to the other brand, as opposed to Special B where only ( )%? stay with them and ( )%? switch. (b) What are the projected market shares for the two brands? (Enter exact numbers as integers, fractions, or decimals.) Special B?1=? MDA?2=?
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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.
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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
In any given day the air quality in a certain city is either good or bad. Records show that when the air quality is good on one day, then there is a 95% chance that it will be good the next day, and when the air quality is bad on one day, then there is 45% chance it will be bad the next day. a. Give the transition matrix. b. if the air quality is good today, what is the probability it will be good two days from now? c. if the air quality is bad today, what is the probability it will be bad three days from now? d. if the there is 20% chance the air quality is good today, what is the probability it will be good tomorrow?
Given the following Markov Chain diagram. 0.15 On Duty Off 0.85 0.2 Duty 0.8 Given that you were on duty the first day (i.e., P(Day 1=On Duty) = 1 and P(Day %3D 1= Off Duty) = 0), answer the following questions. • What is the probability that you need to report towork on the 2mnd day (i.e., P(Day 2 = On Duty))? Answer: (Please write your answer with 2 decimal points) • What is the probability that you take a leave on the 2nd day (i.e., P(Day 2 = Off Duty))? Answer: (Please write your answer with 2 decimal points) • What is the probability that you go to work on the third day (i.e., P(Day 3=On Duty))? Answer: |(Please write your answer with 4 decimal points) • What is the probability that the third day was a Day Off (i.e., P(Day 3=Day Off)? Answer: (Please write your answer with 4 decimal points)
In a hotel, each employee is in one of three possible job classifications, and changes that classifications (independently) according to a Markov chain with transition probabilities. What percentage of employees are in each classification?