To analyze this Markov chain with transition matrix $Q$ and states $S = \{0, 1, 2, 3\}$, let’s go through each part of the problem carefully.

Problem Setup

The transition matrix $Q$ for this Markov chain is:

$Q = \begin{bmatrix} \frac{1}{2} & \frac{1}{6} & 0 & \frac{1}{3} \\ 0 & 0 & \frac{3}{4} & \frac{1}{4} \\ 0 & \frac{1}{4} & \frac{1}{3} & \frac{5}{12} \\ 0 & 1 & 0 & 0 \end{bmatrix}$

Each entry $Q_{ij}$ represents the probability of moving from state $i$ to state $j$.

Now, we’ll solve each part of the problem.

Part (i): Calculate $f_{01}^n$ and $f_{01}$

Step 1: Understanding $f_{01}^n$ and $f_{01}$

• $f_{01}^n$ represents the probability that the chain, starting from state 0, first hits state 1 at the $n$-th step.
• $f_{01}$ is the total probability of ever hitting state 1, starting from state 0, and is given by $f_{01} = \sum_{n=1}^{\infty} f_{01}^n$.

To compute $f_{01}^n$, we’ll analyze the paths leading from state 0 to state 1. We then use recurrence relations for first passage probabilities.

Step 2: Set Up and Solve Recurrence for $f_{01}$

To find $f_{01}$, let's set up the recurrence relation using the transition probabilities. Define $f_{01} = P(\text{starting from 0, hit state 1 for the first time})$:

1. Case 1: From state 0, go directly to state 1 with probability $Q_{01} = \frac{1}{6}$.
2. Case 2: From state 0, go to state 2 with probability $Q_{02} = 0$, or go to state 3, then eventually return to state 1, calculating this iteratively.

Further detailed recursive calculations