write the transition probability matrix of this markov chain" + "inspection"
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A Markov chain is usually shown by a state transition diagram. Consider a Markov chain with three possible states $1$, $2$, and $3$ and the following transition probabilities \begin{equation} \nonumber P = \begin{bmatrix} \frac{1}{4} & \frac{1}{2} & \frac{1}{4} \\[5pt] \frac{1}{3} & 0 & \frac{2}{3} \\[5pt] \frac{1}{2} & 0 & \frac{1}{2} \end .
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