Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 05 (c)
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Question B 05 (c) 

Suppose four towns are connected by roads in the configuration shown below. A random driver wakes up every morning and flips a coin. If the coin is heads, she stays where she is for the day. If the coin is tails, she drives to the next town, choosing one of the roads with no preference. (For example, if she leaves Town A, she is equally likely to go to Town B or Town C, but she will not go to Town D that day.) (c) What is the equilibrium probability of the system? 
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Hint 

The the equilibrium probability satisfies . The sum of each component in is . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We can find the equilibrium probability distribution by solving for below:
We can simplify these equations to
Equating (1) and (4), we get
Substracting (3) from (2) we get
Hence,
Plugging this into equation (1), we get
Using the last condition, that the sum of all variables is equal to 1 we get: answer: 