November 6, 2024 at 8:54AM Solve the problem on a separate sheet of paper and submit the work to Grgdescope. SHOW ALL WORK using MatLab. 1. An experiment was conducted by having a group of children solve a puzzle. It was found that of those who had solved the puzzle on a given trial, 30\% could solve it on the next trial. And of those who couldn't solve the puzzle on a given trial, $$ 60 \% $$ could solve it on the next trail. At the present time, 80\% of the children solved the puzzle on the current trial, but 20\% did not. Assume that this process can be modeled as a Markov process. - Write the transition matrix. Label the columns and rows, using S for "able to solve" and NS for "not able to solve. - Write the initial distribution matrix. - What percent of the children will be able to solve the puzzle on the next trial (trial 1)? - What percent of the children will be able to solve the puzzle on trial 2? - After many trials, what percent of the children will fail to solve the puzzle on each trial?

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1. Transition Matrix: $$ P = \begin{bmatrix} 0.30 & 0.60 \ 0.70 & 0.40 \end{bmatrix} $$ 2. Initial Distribution Matrix: $$ \pi_0 = \begin{bmatrix} 0.80 \ 0.20 \end{bmatrix} $$ 3. Percent of Children Able to Solve on Trial 1: Calculated from $$ \pi_1 $$ 4. Percent of Children Able to Solve on Trial 2: Calculated from $$ \pi_2 $$ 5. Percent of Children Failing to Solve in Steady State: Calculated from steady-state distribution

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