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

MATH307 April 2009

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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 forever-after.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. 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 $P(AorB)=P(A)+P(B)$; 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 ${\begin{bmatrix}1&{\frac {1}{3}}&0&0\\0&0&{\frac {1}{3}}&0\\0&{\frac {2}{3}}&0&0\\0&0&{\frac {2}{3}}&1\end{bmatrix}}$