Science:Math Exam Resources/Courses/MATH152/April 2012/Question 07 (a)

MATH152 April 2012

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Question 07 (a)
Consider a random walk that has transition matrix: $P={\begin{bmatrix}1/4&1/4&1/4\\3/4&3/4&1/4\\0&0&1/2\end{bmatrix}}$ If the walker starts in state 3 then find the probability that he will be in state 3 again after two steps.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
The probability of being at the $i^{th}$ state at the $n^{th}$ time step is given by the $i^{th}$ entry in the vector $\mathbf {p} ^{(n)}$, where $\mathbf {p} ^{(n)}=P^{n}\mathbf {p} ^{(0)}$ and $\mathbf {p} ^{(0)}$ is the initial distribution of probability.

Hint 2
In our case, what is $\mathbf {p} ^{(0)}$, when the random walker starts in the third state with probability 1? Which entry $\mathbf {p} ^{(2)}$ is the probability that the random walker is in state 3 after 2 steps?

Checking a solution serves two purposes: helping you if, after having used all the hints, 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.

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. The probability of being at the $i^{th}$ state at the $n^{th}$ time step is given by the $i^{th}$ entry in the vector $\mathbf {p} ^{(n)}$, where $\mathbf {p} ^{(n)}=P^{n}\mathbf {p} ^{(0)}$ and $\mathbf {p} ^{(0)}$ is the initial distribution of probability. In this case, $\mathbf {p} ^{(0)}=[0,0,1]^{T}$ since we are starting in the third state with probability 1. We want to compute the $3^{rd}$ entry of $\mathbf {p} ^{(2)}$, ${\begin{aligned}\mathbf {p} ^{(2)}=P^{2}\mathbf {p} ^{(0)}&={\begin{bmatrix}1/4&1/4&1/4\\3/4&3/4&1/4\\0&0&1/2\end{bmatrix}}^{2}{\begin{bmatrix}0\\0\\1\end{bmatrix}}={\begin{bmatrix}1/4&1/4&1/4\\3/4&3/4&1/2\\0&0&1/4\end{bmatrix}}{\begin{bmatrix}0\\0\\1\end{bmatrix}}={\begin{bmatrix}1/4\\1/2\\{\color {blue}1/4}\end{bmatrix}}\end{aligned}}$ Therefore, the probability of being in the third state at the second step is 1/4.