Solution:

Step 1: Understanding the problem

This problem involves a stochastic process where the ball's position among three boys $A, B,$ and $C$ evolves over time. The process is Markovian if the next state depends only on the current state and not on any past states.

From the problem:

1. $A$ always throws the ball to $B$.
2. $B$ always throws the ball to $C$.
3. $C$ throws the ball either to $A$ or $B$, each with a probability of $\frac{1}{2}$.

We need to:

1. Prove the process is Markovian.
2. Derive the transition matrix.
3. Classify the states.
4. Determine if all states are ergodic.

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Step 2: Prove the process is Markovian

The process is Markovian because:

• The state of the process at time $t+1$ (who has the ball) depends only on the state at time $t$, i.e., who has the ball at $t$.
• The probabilities of transitioning to the next state are independent of the earlier history of the process.

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Step 3: Transition Matrix

The states of the system can be denoted as:

1. $A$: The ball is with $A$.
2. $B$: The ball is with $B$.
3. $C$: The ball is with $C$.

The transition probabilities can be summarized as follows:

• $P(A \to B) = 1$ (A always throws to B).
• $P(B \to C) = 1$ (B always throws to C).
• $P(C \to A) = \frac{1}{2}$, $P(C \to B) = \frac{1}{2}$ (C throws to A or B with equal probability).

Thus, the transition matrix $P$ is:

\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}$$ --- #### **Step 4: Classify the States** 1. **Irreducibility**: - A Markov chain is irreducible if it is possible to go from any state to any other state in a finite number of steps. - In this case, starting from any state ($$A, B, C$$), it is possible to reach any other state due to the transitions. For example: - $$A \to B \to C \to A$$ ensures all states communicate. 2. **Periodicity**: - The period of a state is the greatest common divisor (GCD) of the lengths of all cycles returning to the state. - For $$A$$, the cycle lengths are 3 (e.g., $$A \to B \to C \to A$$). - Similarly, $$B$$ and $$C$$ have cycles of length 3. - All states have a period of 3, so the chain is **periodic**. 3. **Ergodicity**: - A state is ergodic if it is both **irreducible** and **aperiodic**. - Since the chain is periodic, the states are **not ergodic**. --- #### **Step 5: Conclusion** 1. The process is **Markovian** because transitions depend only on the current state. 2. The **transition matrix** is: $$P = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}$$ 3. The states $$A, B, C$$ are **periodic** with period 3. 4. The states are **not ergodic** because of periodicity. --- ### **Further Exploration:** 1. What happens to the transition matrix if $$C$$ throws the ball with probabilities other than $$\frac{1}{2}$$? 2. Can the process become ergodic by modifying the rules for $$C$$'s throws? 3. Compute the steady-state probabilities if the chain were ergodic. 4. Analyze how long it takes on average to return to $$A$$ starting from $$A$$. 5. Investigate what happens if $$A, B,$$ and $$C$$ throw the ball randomly. --- ### **Tip**: When classifying Markov chains, always check for irreducibility and periodicity first. These properties are key to understanding the chain's long-term behavior.