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7. A student has three routes of getting to his school: A, B, and C. Each day he picks one, and his choice is influenced only by his previous day choice: - if he takes route A today, then he will take route A, B, or C next day with respective 1 1 1 probabilities 2'4'4 - if he takes route B today, then he will take route A, B, or C next day with respective probabilities, 0, - if he takes route C today, then he will take route A, B, or C next day with respective 1 1 5 probabilities 4'12 (a) Formulate this as a Markov chain by defining its states and giving its (one-step) transition matrix. (b) Find the n-step transition matrix P(n)=2 and 20. (c) If he takes route A today, what is the probability that he will take route A in 2 the day after tomorrow?

7. A student has three routes of getting to his school: A, B, and C. Each day he picks one, and his choice is influenced only by his previous day choice: - if he takes route A today, then he will take route A, B, or C next day with respective 1 1 1 probabilities 2'4'4 - if he takes route B today, then he will take route A, B, or C next day with respective probabilities, 0, - if he takes route C today, then he will take route A, B, or C next day with respective 1 1 5 probabilities 4'12 (a) Formulate this as a Markov chain by defining its states and giving its (one-step) transition matrix. (b) Find the n-step transition matrix P(n)=2 and 20. (c) If he takes route A today, what is the probability that he will take route A in 2 the day after tomorrow?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.3: Conditional Probability; Independent Events; Bayes' Theorem
Problem 42E
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7. A student has three routes of getting to his school: A, B, and C. Each day he picks one, and his choice is influenced only by his previous day choice: – if he takes route A today, then he will take route A, B, or C next day with respective 1 1 probabilities, 244 - if he takes route B today, then he will take route A, B, or C next day with respective 1 probabilities, 0, 2 - if he takes route C today, then he will take route A, B, or C next day with respective 1 probabilities 3'4'12' (a) Formulate this as a Markov chain by defining its states and giving its (one-step) transition matrix. (b) Find the n-step transition matrix P(n)=2 and 20. (c) If he takes route A today, what is the probability that he will take route A in 2 the day after tomorrow? (d) The probability that he takes route A today is 0.4. Use the results from part (b) to determine the probability that he will take route B 20 days from now. i=0 (a) Use equations лP = π, Σ²0¹₁ = 1 to find the steady state probabilities. Are the probabilities consistent with the numbers in P (20)?
Transcribed Image Text:7. A student has three routes of getting to his school: A, B, and C. Each day he picks one, and his choice is influenced only by his previous day choice: – if he takes route A today, then he will take route A, B, or C next day with respective 1 1 probabilities, 244 - if he takes route B today, then he will take route A, B, or C next day with respective 1 probabilities, 0, 2 - if he takes route C today, then he will take route A, B, or C next day with respective 1 probabilities 3'4'12' (a) Formulate this as a Markov chain by defining its states and giving its (one-step) transition matrix. (b) Find the n-step transition matrix P(n)=2 and 20. (c) If he takes route A today, what is the probability that he will take route A in 2 the day after tomorrow? (d) The probability that he takes route A today is 0.4. Use the results from part (b) to determine the probability that he will take route B 20 days from now. i=0 (a) Use equations лP = π, Σ²0¹₁ = 1 to find the steady state probabilities. Are the probabilities consistent with the numbers in P (20)?
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