Consider the Markov chain having the following transition probability matrix.

i) Draw the diagram of a Markov chain.

ii) Classify the states of a Markov chain, i.e., persistent, transient, non-null and a periodic state. Also check the irreducibility of Markov chain.

iii) Find the closed sets.

iv) Find the probability of absorption to the closed classes. Also find the mean time up to absorption from transient state 3 to 4.

i) To draw the diagram of the Markov chain, you can represent each state as a node, and draw arrows between nodes to represent the transitions. The thickness of the arrows represents the transition probabilities. Here's the diagram:

         [1] --- 1/3 ---> [1]
        /  |            /   |
   2/3 /   | 2/3   1/4 /    |
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