Question 1.16.2 (Markov chains, Part IB, 1991, 307D)Three girls A, B and C are playing table tennis. In each game, two of the girls playagainst each other and the third girl does not play. The winner of any given gamen plays again in game n + 1. The probability that girl x will beat girl y in any gamethat they play against each other is sx/(Sx+Sy) for x, y = {A,B,C}, x‡y, where sa,SB, SC represent the playing strengths of the three girls.In what proportion of games does each girl play, in the long run?

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Answered: Question 1.16.2 (Markov chains, Part…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose a person is thinking about entering a MarioKart competition at theirlocal videogame store. The competition costs 20 dollars to enter. The competition consists of 10 matches. The person knows that their chance of winning each match is 65% and that each match is independent of the others. The payout for the competition is: WIN ALL 10 Matches: 500 Dollar Prize WIN 9 Matches: 250 Dollar PrizeWIN 8 Matches: 50 Dollar Prize OTHERWISE: Nothing 1) Suppose that this person enters the competition 5 times.What is the probability they break even?(Hint: To break even they would need to win the $50 prize twice and lose the other three times. Use a modified binomial to calculate this probability.)
A spacecraft navigation system consists of components of two types - A and B - connected in a series-parallel arrangement, as in Figure 1.2. A B A B B Figure 1.2 Illustration of series-parallel system The system will function as long as at least one component of each type functions. The probability that a type A component will function correctly throughout the planned lifetime of the spacecraft is 0-9; the corresponding probability for a component of type B is 0-8. Components may be assumed to fail independently of one another. The numbers of components of the two types may be varied in order to adjust the probability of failure of the navigation system. (a) Show that a requirement that the probability of failure of the entire system should be less than 1% may be met by using three components of each type (as in the diagram above). (b) If components of type A weigh 1 kg, while those of type B weigh 2 kg, what is the composition of the most reliable navigation system weighing 11 kg or…
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QUESTION 2 Three local textile distributors (K, L, and M) are competing for a contract to supply textiles to Company XYZ. The probabilities that textile distributors K, L, and M will win the contract are 0.4, 0.3, and 0.3, respectively. If textile distributors K, L, and M win the contract, the probabilities that they will make profits are 0.65, 0.85, and 0.45, respectively. a) Draw a tree diagram for the above information. b) Calculate the probability that the contract was awarded to textile distributor K given that the contract is found to be unprofitable.
ne bets $1. 0.45, he wins the game and with probability 1 - p = 0.55, he loses the game. His goal is to increase his capital to $3, and as soon as he does, the game is over. The game is also over if his capital is reduced to zero. Construct an absorbing Markov chain and answer the following questions. • What is the expected duration of the game? • What is the probability that he goes broke?
7
(Ch. 2.1) A library has 5 copies of a certain book in stock. Two copies (1 and 2) are first printings, and the other three (3,4 and 5) are second printings. A student finds these copies on a shelf and begins to examine in random order, stopping when he finds a second printing of the book. For example, one possible outcome is (5), and another is (2,1,3) (a) List the outcomes in the sample space 8. (b) Let A denote the event that exactly one book must be examined. What outcomes are in A? (c) Let B be the event that book 4 is the one selected. What outcomes are in B?
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?
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?

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