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State Transition Matrix Model
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Two coffee shops, JavaJoy and BeanBrew, serve a small town. Each month, 15% of JavaJoy’s customers switch to BeanBrew, and 25% of BeanBrew’s customers switch to JavaJoy. Let \(J_n\) and \(B_n\) represent the number of customers at JavaJoy and BeanBrew, respectively, during month \(n\). Which matrix equation correctly models \(J_{n+1}\) and \(B_{n+1}\), the number of customers at each shop during month \(n+1\)?

A

$$\begin{bmatrix} J_{n+1} \\ B_{n+1} \end{bmatrix} = \begin{bmatrix} 0.85 & 0.25 \\ 0.15 & 0.75 \end{bmatrix} \begin{bmatrix} J_n \\ B_n \end{bmatrix}$$

B

$$\begin{bmatrix} J_{n+1} \\ B_{n+1} \end{bmatrix} = \begin{bmatrix} J_n \\ B_n \end{bmatrix} \begin{bmatrix} 0.85 & 0.25 \\ 0.15 & 0.75 \end{bmatrix}$$

C

$$\begin{bmatrix} J_{n+1} \\ B_{n+1} \end{bmatrix} = \begin{bmatrix} 0.85 & 0.15 \\ 0.25 & 0.75 \end{bmatrix} \begin{bmatrix} J_n \\ B_n \end{bmatrix}$$

D

$$\begin{bmatrix} J_{n+1} \\ B_{n+1} \end{bmatrix} = \begin{bmatrix} 0.75 & 0.15 \\ 0.25 & 0.85 \end{bmatrix} \begin{bmatrix} J_n \\ B_n \end{bmatrix}$$

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