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Joe Marshel

Joe M.

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Patha Sharma verified

Numerade educator

A production line comprises three machines working independently of each other. For each machine, the probability of working through the day is p ∈ (0, 1) and may breakdown during a day with probability q = 1 - p, independently of its previous history. There is a single repairman who, if he has to, repairs exactly one machine overnight. If there is a machine not working by the end of the day, then the repairman works during the night. Let Xn denote the number of working machines at the end of day before the repairman begins any over-night repair. a-) Specify the state space of (Xn : n ≥ 0). b-) Determine the transition matrix in terms of p. c-) Given that 2 machines are idle tonight, what is the probability of one idle machine 3 nights later, if p = 0.8 (answer 6 decimal places).

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Patha Sharma verified

Numerade educator

Consider the experiment of tossing a coin until the pattern THT is obtained. Assume that P(H) = p and P(T) = q = 1 - p. a-) Construct the Engel diagram for the experiment. You must include in your graph the direction and the probabilities. b-) Obtain the equations solution of which will help find the expected waiting time for the THT pattern. Please show all working in detail step by step, thanks.

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Patha Sharma verified

Numerade educator

A second year mathematics unit at a university in a foreign country has unpredictable patterns of administering weekly tests. There will be either no test, or one test, and if there is one, it is either a 15 minute minor test or a 30 minute major test. For any given week, if there was no test in the previous week, the probability that there will be a minor test is 0.6, and the probability that there will be a major test is 0.3. For a week where there was a minor test in the previous week, the probability of minor and major tests are 0.4 and 0.2 respectively. If there was a major test in the previous week, this week there will be a minor test with probability 0.3 and no test with probability 0.7. Let Xn be the Markov chain for the situation described above, with state space {0, 1, 2}, where 0 indicates no test, 1 stands for a minor test, and 2 indicating a major test. a-) Write down the transition matrix for the Markov chain. b-) Find the two-step transition probability matrix for the Markov chain. c-) Given that there was no test this week, find the probability that there is a test in two weeks time. d-) Compute P(X5 = 2|X3 = 1, X1 = 0).

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INSTANT ANSWER

A second year mathematics unit at a university in a foreign country has unpredictable patterns of administering weekly tests. There will be either no test, or one test, and if there is one, it is either a 15 minute minor test or a 30 minute major test. For any given week, if there was no test in the previous week, the probability that there will be a minor test is 0.6, and the probability that there will be a major test is 0.3. For a week where there was a minor test in the previous week, the probability of minor and major tests are 0.4 and 0.2 respectively. If there was a major test in the previous week, this week there will be a minor test with probability 0.3 and no test with probability 0.7. Let Xn be the Markov chain for the situation described above, with state space {0, 1, 2}, where 0 indicates no test, 1 stands for a minor test, and 2 indicating a major test. a-) Write down the transition matrix for the Markov chain. b-) Find the two-step transition probability matrix for the Markov chain. c-) Given that there was no test this week, find the probability that there is a test in two weeks time. d-) Compute P(X5 = 2|X3 = 1, X1 = 0).

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