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

MATH307 April 2009

Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q2 (a)  •  Q2 (b)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q3 (d)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q4 (d)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q7 (a)  •  Q7 (b)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q8 (d)  •  

   Other MATH307 Exams

  •  December 2012  •  April 2013  •  April 2012  •  April 2006  •  December 2008  •  December 2010  •  

[hide] 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 $x_{0}$ is written in terms of the eigenvectors, $v_{i}$, the coefficients $(c_{1},c_{2},c_{3},c_{4})=(0.43,0.57,0.28,0.93)$, 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 $c_{i}$, $\lambda _{i}$, and $v_{i}$.

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!

[show] Hint

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] 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 $v_{1}$ and $v_{2}$ are the end states of the game (landing on the winning and losing square, respectively), $c_{1}$ and $c_{2}$ represent the probabilities of winning and losing, respectively. (They sum to 1, showing that if the game goes on long enough, it will end at one of these two states.) So the probability of winning the game is 0.43. Since two of the $\lambda _{i}$ have magnitude 1 (these being $\lambda _{1}$ and $\lambda _{2}$), the long-term behaviour of this matrix is a linear combination of the corresponding $v_{i}$, with coefficients being the corresponding $c_{i}$ (this depends on $x_{0}$ because there is more than one $\lambda _{i}$ with magnitude 1). In this case, these are $c_{1}$, $v_{1}$, $c_{2}$, and $v_{2}$. Thus the long-term behaviour of the game is $c_{1}*v_{1}+c_{2}*v_{2}$.