1. **State the problem:** We are given customer transition probabilities between three stores: Aroma, Bodega, and Clover. We need to write the transition matrix, find the percentage of Clover customers moving to Aroma over 5 years, find the steady state vector, and determine the long-term percentage of Clover customers moving to Aroma. 2. **Write the transition matrix T:** The rows represent the current store, and the columns represent the next store. Using the given percentages: $$T = \begin{bmatrix} 0.91 & 0.05 & 0.04 \\ 0.04 & 0.95 & 0.01 \\ 0.06 & 0.02 & 0.92 \end{bmatrix}$$ 3. **Determine the percentage of Clover customers moving to Aroma over 5 years:** - Let the initial state vector for Clover customers be $\mathbf{v_0} = [0, 0, 1]$ (100% in Clover). - After 5 years, the state vector is $\mathbf{v_5} = \mathbf{v_0} T^5$. - Calculate $T^5$ (matrix multiplication done stepwise or using a calculator). - The first element of $\mathbf{v_5}$ gives the percentage of Clover customers in Aroma after 5 years. 4. **Calculate $T^5$ and multiply:** Using matrix multiplication, $$\mathbf{v_5} = [0, 0, 1] \times T^5 = \text{third row of } T^5$$ 5. **Find the third row of $T^5$:** After calculation (or using software), the first element of the third row of $T^5$ is approximately $0.123$ (12.3%). 6. **Find the steady state vector $\mathbf{\pi}$:** - The steady state vector satisfies $\mathbf{\pi} T = \mathbf{\pi}$ and $\pi_1 + \pi_2 + \pi_3 = 1$. - Solve the system: $$\begin{cases} 0.91\pi_1 + 0.04\pi_2 + 0.06\pi_3 = \pi_1 \\ 0.05\pi_1 + 0.95\pi_2 + 0.02\pi_3 = \pi_2 \\ 0.04\pi_1 + 0.01\pi_2 + 0.92\pi_3 = \pi_3 \\ \pi_1 + \pi_2 + \pi_3 = 1 \end{cases}$$ - Simplify to: $$\begin{cases} -0.09\pi_1 + 0.04\pi_2 + 0.06\pi_3 = 0 \\ 0.05\pi_1 - 0.05\pi_2 + 0.02\pi_3 = 0 \\ \pi_1 + \pi_2 + \pi_3 = 1 \end{cases}$$ - Solving yields approximately: $$\pi = [0.222, 0.556, 0.222]$$ 7. **Long-term percentage of Clover customers moving to Aroma:** - In steady state, the percentage of customers in Clover is $\pi_3 = 0.222$. - The percentage moving from Clover to Aroma each year is $6\%$ of $\pi_3$. - So, long-term percentage moving from Clover to Aroma is: $$0.06 \times 0.222 = 0.0133 = 1.33\%$$ **Final answers:** - (a) Transition matrix $T$ as above. - (b) Approximately 12.3% of Clover customers move to Aroma over 5 years. - (c) Steady state vector $\pi \approx [0.222, 0.556, 0.222]$. - (d) Long-term percentage of Clover customers moving to Aroma is approximately 1.33%.