Solution:

This problem involves Markov Chains, and we are tasked with building a state diagram, preparing the state matrix, and calculating probabilities after specific time transitions.

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Step 1: State Diagram

The states $S_1$ and $S_2$ transition based on the following rules:

1. $75\%$ of customers in $S_1$ remain in $S_1$, while $25\%$ move to $S_2$.
2. $65\%$ of customers in $S_2$ remain in $S_2$, while $35\%$ move to $S_1$.

This can be represented by the following diagram:

• $S_1 \xrightarrow{25\%} S_2$, and $S_2 \xrightarrow{35\%} S_1$
• $S_1 \xrightarrow{75\%} S_1$, and $S_2 \xrightarrow{65\%} S_2$

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Step 2: State Matrix

The state matrix $P$ is constructed from the transition probabilities:

P(S_1 \to S_1) & P(S_1 \to S_2) \\ P(S_2 \to S_1) & P(S_2 \to S_2) \end{bmatrix} = \begin{bmatrix} 0.75 & 0.25 \\ 0.35 & 0.65 \end{bmatrix}$$ --- #### Step 3: **Determine Probabilities** ##### (i) $$P(S_1 \to S_2)$$ in the 3rd week To find the probability of transitioning from $$S_1$$ to $$S_2$$ in the 3rd week, we calculate $$P^3$$, the third power of the transition matrix $$P$$. From $$P^3$$, the $$(1,2)$$-entry gives $$P(S_1 \to S_2)$$. ##### (ii) $$P(S_2 \to S_2)$$ in the 2nd week To find the probability of remaining in $$S_2$$ after 2 weeks, we calculate $$P^2$$. The $$(2,2)$$-entry of $$P^2$$ provides $$P(S_2 \to S_2)$$. --- Let’s compute $$P^2$$ and $$P^3$$. ### Matrix Computations We compute $$P^2$$ and $$P^3$$ using the given transition matrix. np # Define the transition matrix P P = np.array([[0.75, 0.25], [0.35, 0.65]]) # Calculate P^2 and P^3 P2 = np.linalg.matrix_power(P, 2) P3 = np.linalg.matrix_power(P, 3) # Extract the required probabilities P_S1_to_S2_3rd_week = P3[0, 1] # (1,2) entry of P^3 P_S2_to_S2_2nd_week = P2[1, 1] # (2,2) entry of P^2 P2, P3, P_S1_to_S2_3rd_week, P_S2_to_S2_2nd### Results: #### Transition Matrix Calculations: 1. $$P^2$$: $$P^2 = \begin{bmatrix} 0.65 & 0.35 \\ 0.49 & 0.51 \end{bmatrix}$$ 2. $$P^3$$: $$P^3 = \begin{bmatrix} 0.61 & 0.39 \\ 0.546 & 0.454 \end{bmatrix}$$ --- #### Specific Probabilities: 1. $$P(S_1 \to S_2)$$ in the 3rd week: $$P(S_1 \to S_2) = 0.39$$ 2. $$P(S_2 \to S_2)$$ in the 2nd week: $$P(S_2 \to S_2) = 0.51$$ --- ### Summary: - The probability of transitioning from $$S_1$$ to $$S_2$$ in the 3rd week is $$39\%$$. - The probability of staying in $$S_2$$ in the 2nd week is $$51\%$$. Do you have any questions or want further explanation? --- ### Expanding Questions: 1. How can we calculate the steady-state probabilities for this Markov chain? 2. What is the long-term probability of being in state $$S_1$$ or $$S_2$$? 3. Can we visualize the transitions using a graph or diagram? 4. How do these probabilities change over a longer timeframe, say 10 weeks? 5. What assumptions are made in this Markov model? ### Tip: Markov Chains are useful in modeling systems where the future state depends only on the current state and not on the sequence of past states (memoryless property).