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Let's represent the brands with the following notation: - SS: Sorey State Boogie Boards - CT: C & T Super Professional Boards We can define the transition matrix P as follows: P = \(\begin{bmatrix} P(SS \to SS) & P(SS \to CT) \\ P(CT \to SS) & P(CT \to CT) \end{bmatrix}\) Where each entry represents the probability of a user switching from one brand to another or staying with the same brand. Based on the problem statement, we know: - Half of the SS users switched to CT, so P(SS → SS) = 1 - 0.5 = 0.5 and P(SS → CT) = 0.5. - Three quarters of CT users remained with CT, so P(CT → CT) = 0.75 and P(CT → SS) = 0.25. Now, we can substitute these values into the transition matrix: P = \(\begin{bmatrix} 0.5 & 0.5 \\ 0.25 & 0.75 \end{bmatrix}\)

To find the transition matrix for 2 seasons, we need to multiply the matrix P by itself. This is called matrix squaring. Let P² denote the transition matrix after 2 seasons: P² = P × P = \(\begin{bmatrix} 0.5 & 0.5 \\ 0.25 & 0.75 \end{bmatrix}\) × \(\begin{bmatrix} 0.5 & 0.5 \\ 0.25 & 0.75 \end{bmatrix}\) Compute the matrix product: P² = \(\begin{bmatrix} (0.5×0.5 + 0.5×0.25) & (0.5×0.5 + 0.5×0.75) \\ (0.25×0.5 + 0.75×0.25) & (0.25×0.5 + 0.75×0.75) \end{bmatrix}\) Simplify, and obtain the result: P² = \(\begin{bmatrix} 0.375 & 0.625 \\ 0.3125 & 0.6875 \end{bmatrix}\)

Recall that we are interested in the probability that an SS user will still be using the same brand two seasons later. This is given by the value P(SS → SS) in the P² matrix: P(SS → SS; 2 seasons) = 0.375 So, the probability that an SS user will still be using the Sorey State Board two seasons later is 37.5%.