Question 1205112
**1. Identify the Communicating Classes**

* **Class 1: {1, 6}** 
    * State 1 transitions to state 6 with probability 1. 
    * State 6 transitions to state 1 with probability 1. 
    * This forms a closed, irreducible, and recurrent class.

* **Class 2: {2, 3, 4, 5}** 
    * States within this class transition only to other states within the class. 
    * This forms another closed, irreducible, and recurrent class.

**2. Long-Term Behavior**

* In the long run, a Markov Chain will tend to spend most of its time in the recurrent classes.

**3. Calculate Probabilities**

* **(a) P(X₃₀₀₀₀₀₀ = 2 | X₀ = 1)**
    * Since state 1 belongs to Class 1 and state 2 belongs to Class 2, it's impossible to reach state 2 from state 1 in any number of steps. 
    * **P(X₃₀₀₀₀₀₀ = 2 | X₀ = 1) = 0**

* **(b) P(X₃₀₀₀₀₀₁ = 2 | X₀ = 1)**
    * Even after one step, it's still impossible to reach state 2 from state 1 directly. 
    * **P(X₃₀₀₀₀₀₁ = 2 | X₀ = 1) = 0**

* **(c) P(X₃₀₀₀₀₀₂ = 2 | X₀ = 1)**
    * After two steps:
        * State 1 transitions to state 6 with probability 1.
        * State 6 transitions to state 1 with probability 1. 
    * It's still impossible to reach state 2 from state 1 in two steps.
    * **P(X₃₀₀₀₀₀₂ = 2 | X₀ = 1) = 0**

**In summary:**

* Due to the structure of the transition probability matrix and the distinct communicating classes, it's impossible to reach state 2 from state 1 in any number of steps. Therefore, all the requested probabilities are 0.

**Key Concepts**

* **Communicating Classes:** A set of states is a communicating class if every state in the set can be reached from every other state in the set.
* **Irreducible Class:** A communicating class is irreducible if it cannot be further divided into smaller communicating classes.
* **Recurrent Class:** A state is recurrent if, starting from that state, the probability of eventually returning to that state is 1.
* **Long-Term Behavior:** In the long run, a Markov Chain will tend to spend most of its time in its recurrent classes.

I hope this explanation is helpful! Let me know if you have any further questions.