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

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. (a) Write a stochastic matrix that represents the behaviour of the game. Label the columns with the corresponding “states” of the game. Hint: Once you win, you win forever; and once you lose, you lose forever; i.e., if you reach state 1 at some point you will continue to stay at state 1 foreverafter. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2009/Question 08 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since the matrix is stochastic, all columns should sum to 1. We see that we have a 1/3 chance of moving to square 1 and a 2/3 chance of moving to square 3 from square 2, since the probability of rolling any specific number on the die is 1/6 and the probabilities are independent, so ; similarly, we have a 1/3 chance of moving to square 2 and a 2/3 chance of moving to square 4 from square 3. If we get to squares 1 or 4, then we stay there, since the game is over. So let column 1 represent the first square of the game, column 2 the second, column 3 the third, and column 4 the fourth. Then the stochastic matrix representing the behaviour of the game is 