# Science:Math Exam Resources/Courses/MATH307/April 2009/Question 08 (d)

MATH307 April 2009

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### Question 08 (d)

A player begins a game of chance by placing a marker in box 2, marked start. A die is rolled, and the marker is moved one square to the left if a 1 or 2 is rolled and one square to the right if a 3, 4, 5, or 6 is rolled. This process continues until the marker lands in square 1, in which case the player wins the game, or in square 4, in which case the player loses the game.

(c) Given that when the initial state ${\displaystyle x_{0}}$ is written in terms of the eigenvectors, ${\displaystyle v_{i}}$, the coeﬃcients ${\displaystyle (c_{1},c_{2},c_{3},c_{4})=(0.43,0.57,0.28,0.93)}$, determine the probability of winning the game. Explain your solution by writing a expression for the long-time behaviour of the game in terms of ${\displaystyle c_{i}}$, ${\displaystyle \lambda _{i}}$, and ${\displaystyle v_{i}}$.

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