Q3) Three white and three black balls are distributed in two urns in such a way that each urn contains three balls. We say that the system is in state i, i = 0,1,2,3, if the first urn contains i white balls initially. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. The system is in state j,j = 0,1,2,3, if the first urn contains j white balls after the exchange. Let Xn denote the state of the system after the nth step. a) Explain why {Xn, n = 0,1,2,...} is a Markov chain. b) Calculate its transition probability matrix P.

Q3) Three white and three black balls are distributed in two urns in such a way that each urn contains three balls. We say that the system is in state i, i = 0,1,2,3, if the first urn contains i white balls initially. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. The system is in state j,j = 0,1,2,3, if the first urn contains j white balls after the exchange. Let Xn denote the state of the system after the nth step. a) Explain why {Xn, n = 0,1,2,...} is a Markov chain. b) Calculate its transition probability matrix P.

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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Question

Q3) Three white and three black balls are distributed in two urns in such a way that each urn contains
three balls. We say that the system is in state i, i = 0,1,2,3, if the first urn contains i white balls initially.
At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second,
and conversely with the ball from the second urn. The system is in state j,j = 0,1,2,3, if the first urn
contains j white balls after the exchange. Let Xn denote the state of the system after the nth step.
a) Explain why {X, n = 0,1,2,...} is a Markov chain.
%3D
b) Calculate its transition probability matrix P.

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