Janet and Samantha play a zero-sum game.

The game is represented by the following pay-off matrix for Janet.

Samantha
\begin{tabular}{c|c|c|c}
 & Strategy & $S_1$ & $S_2$ & $S_3$ \\
\hline
\multirow{4}{*}{Janet} & $J_1$ & 2 & 7 & 6 \\
 & $J_2$ & 5 & 5 & 1 \\
 & $J_3$ & 4 & 3 & 8 \\
 & $J_4$ & 1 & 6 & 4 \\
\end{tabular}

\begin{enumerate}[label=(\alph*)]
\item Explain why Janet should never play strategy $J_4$
[1 mark]

\item Janet wants to maximise her winnings from the game.

She defines the following variables.

$p_1 = $ the probability of Janet playing strategy $J_1$

$p_2 = $ the probability of Janet playing strategy $J_2$

$p_3 = $ the probability of Janet playing strategy $J_3$

$v = $ the value of the game for Janet

Janet then formulates her situation as the following linear programming problem.

Maximise $P = v$

subject to $2p_1 + 5p_2 + 4p_3 \geq v$
$7p_1 + 5p_2 + 3p_3 \geq v$
$6p_1 + p_2 + 8p_3 \geq v$

and $p_1 + p_2 + p_3 \leq 1$
$p_1, p_2, p_3 \geq 0$

\begin{enumerate}[label=(\roman*)]
\item Complete the initial Simplex tableau for Janet's situation in the grid below.
[2 marks]

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $P$ & $v$ & $p_1$ & $p_2$ & $p_3$ & & & value \\
\hline & & & & & & & \\
\hline & & & & & & & \\
\hline & & & & & & & \\
\hline & & & & & & & \\
\end{tabular}

\item Hence, perform one iteration of the Simplex algorithm, showing your answer on the grid below.
[2 marks]

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline $P$ & $v$ & $p_1$ & $p_2$ & $p_3$ & & & value \\
\hline & & & & & & & \\
\hline & & & & & & & \\
\hline & & & & & & & \\
\hline & & & & & & & \\
\end{tabular}
\end{enumerate}

\item Further iterations of the Simplex algorithm are performed until an optimal solution is reached.

The grid below shows part of the final Simplex tableau.

\begin{tabular}{|c|c|c|}
\hline $p_1$ & $p_2$ & value \\
\hline 1 & 0 & $\frac{1}{12}$ \\
\hline 0 & 1 & $\frac{1}{2}$ \\
\end{tabular}

Find the probability of Janet playing strategy $J_3$ when she is playing to maximise her winnings from the game.
[1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 3 Discrete 2024 Q9 [6]}}