Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 23
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Question A 23 

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.
If you are the random walker and you want to maximize the probability of getting to location 2 when 10 days have passed from the beginning, what location should you start from? 
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 

The probability of moving from location to location in time steps is . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The probability of moving from location to location in time steps is . We know , , and , so to maximize the probability of getting to location 2, you should start from location 1. 