Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 05 (a)
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Question B 05 (a) 

We are given the following transition matrix for a random walk What is the probability that a walker starting in location 1 is in location 1 after 2 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 

What do we calculate when we multiply P with the vector ? 
Hint 2 

Since the walker does two steps, you will need to use . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The first and second entry of give the probability that, after k steps, the random walker is in location 1 and location 2, respectively, when the random walker starts in location 1. Therefore we calculate which tells us that the probability of being in state 1 is 1/2. 