Let us solve the problem step by step.

Problem Description

We are analyzing a Markov chain where three boys, A, B, and C, throw a ball to one another with specific rules:

1. A always throws the ball to B.
2. B always throws the ball to C.
3. C is equally likely to throw the ball to either A or B.

We need to:

1. Verify that this process is Markovian.
2. Find the transition matrix.
3. Classify the states.

Step 1: Markovian Property

A stochastic process is Markovian if the next state depends only on the current state and not on the sequence of past states. Here:

• If the ball is with A, the next state is deterministically B (no memory of the past is needed).
• If the ball is with B, the next state is deterministically C.
• If the ball is with C, the next state depends only on C, not how C was reached (the ball is thrown with equal probability to A or B).

Thus, the process satisfies the Markov property.

Step 2: Transition Matrix

We represent the states as $A, B, C$ and compute the probabilities of transitioning from one state to another based on the rules.

• From $A$: A always throws to B. $P(A \to B) = 1, \quad P(A \to A) = 0, \quad P(A \to C) = 0$

• From $B$: B always throws to C. $P(B \to C) = 1, \quad P(B \to A) = 0, \quad P(B \to B) = 0$

• From $C$: C throws the ball equally to A or B. $P(C \to A) = 0.5, \quad P(C \to B) = 0.5, \quad P(C \to C) = 0$

The transition matrix is: