Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 24

MATH152 April 2013

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Question A 24

Consider a random walk with three possible states and where the random walker (possibly) changes from state to state once every day according to the transition matrix P. Answer the following questions using the martix P and some of its powers given below. ${\begin{aligned}P={\begin{bmatrix}0.0284&0.5938&0.1626\\0.9207&0.1579&0.1634\\0.0509&0.2483&0.6740\end{bmatrix}}\\P^{10}={\begin{bmatrix}0.2832&0.2907&0.2875\\0.3843&0.3753&0.3786\\0.3325&0.3340&0.3339\end{bmatrix}}\\P^{20}={\begin{bmatrix}0.2874&0.2875&0.2875\\0.3791&0.3790&0.3790\\0.3335&0.3335&0.3335\end{bmatrix}}\end{aligned}}$ What is the probability that a walker that is in location 3 when 10 days have passed from the beginning is in location 1 when 20 days have passed from the beginning?

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 moving from location $\displaystyle i$ to location $\displaystyle j$ in $\displaystyle k$ time steps is $\displaystyle (P^{k})_{j,i}$.

Hint 2
How does the fact that the walker is in location 3 after 10 days affect the question?

Hint 3
Given that the walker is in position 3 after 10 days, does it matter where he was coming from? How many steps does he take after day 10?

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. We are given that the walker is in location 3 when 10 days have passed and want to know the probability that the walker is in location 1 when 20 days have passed. This is identical to the situation if we are given that the walker is in location 3 when 0 days have passed and want to know the probability that the walker is in location 1 when 10 days have passed. The probability of moving from location 3 to location 1 in 10 time steps is $\displaystyle (P^{10})_{1,3}=0.2875$

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