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You will need to know how to multiply matrices to tackle the later stages of this problem. <br></br>
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The diagram below shows three nodes, numbered $1,2,$ and $3$. If I am at node $1$, I am equally likely to visit node $2$ or $3$ next. If I am at node $3$ I am equally likely to visit node $1$ or $2$ next. If I am at node $2$ I will remain there. These probabilities are shown on the diagram by the arrows linking the nodes.<br></br>
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<div style="text-align: center;"><mdo:image alt="Diagram showing probability of travelling between nodes" height="249" src="prob%20matrix.png" width="240"></mdo:image></div>
I can also represent this diagram using a matrix:<br></br>
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<div>$$A= \left( \begin{array}{ccc} 0 &0.5 &0.5 \\ 0 &1 &0 \\ 0.5 &0.5 &0 \end{array}\right)$$</div>
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<div>Each entry in this matrix $a_{ij}$ gives the probability of travelling from vertex $i$ to vertex $j$. So for example the entry in row $1$ column $2$ gives the probability of travelling from vertex $1$ to vertex $2$.</div>
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<div>What can you say about the total of the entries in each row of a matrix such as this one?</div>
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The probability of being at vertex $2$ after two moves if I started from vertex $1$ is $0.75$. Work out the other entries for the matrix representing the probability of moving from vertex to vertex in two moves.<br></br>
How is this matrix related to the original matrix?<br></br>
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$$A^2=\left(\begin{array}{ccc} 0.25 &0.75 &0 \\ 0 &1 &0 \\ 0 &0.75 &0.25 \end{array} \right)$$<br></br>
$$A^3=\left(\begin{array}{ccc} 0 &0.875 &0.125 \\ 0 &1 &0 \\ 0.125 &0.875 &0 \end{array} \right)$$<br></br>
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Can you explain the zero entries in these two matrices?<br></br>
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Make a suggestion of the form the matrix takes showing the probabilities after $n$ moves. You may wish to consider $n$ odd and even separately. Can you prove that the form you have suggested is correct?<br></br>
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What happens as $n$ gets very large? How can this be explained by the diagram?<br></br>
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Probability Matrix