Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (a)
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Question B 04 (a) 

Suppose in the year 2020, 50 million people live in cities and 50 million in the suburbs. Every year, 10% of city residents move to the suburbs and 20% of the residents of the suburbs move to cities. (a) Write down the probability transition matrix for this problem, using the ordering (1) city and (2) suburbs. 
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 set of states is given and each state can move to the other states or stay the same state on each step. Then, we define transition probability by the probability that the current state move to the state in the next step.

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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let city and suburbs be two states as in the hint and 1 year be one step. Since 10% of city residents move to the suburbs every year, 90% of city residents remain in the city. Using definition in the Hint, this implies that and . On the other hand, 20% of the residents of the suburbs move to cities every year, so that 80% of the suburbs residents stay in the suburbs. Then, we have and . To sum, the transition probability matrix is
