6 Rowena and Colin play a game in which each chooses a letter. The table shows how many points Rowena wins for each combination of letters. In each case the number of points that Colin wins is the negative of the entry in the table. Both Rowena and Colin are trying to win as many points as possible.

Colin's letter

\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Rowena's letter}
\begin{tabular}{ c | r | r | r | r }
 & $N$ & $P$ & $Q$ & $T$ \\
\hline
$W$ & 4 & - 1 & 1 & - 2 \\
\hline
$X$ & 1 & 3 & - 1 & 1 \\
\hline
$Y$ & 5 & 1 & 2 & - 1 \\
\hline
$Z$ & 0 & - 1 & 1 & 1 \\
\hline
\end{tabular}
\end{center}
\end{table}

(i) Write down Colin's play-safe strategy, showing your working. What is the maximum number of points that Colin can win if he plays safe?\\
(ii) Explain why Rowena would never choose the letter $W$.

Rowena uses random numbers to choose between her three remaining options, so that she chooses $X , Y$ and $Z$ with probabilities $x , y$ and $z$, respectively. Rowena then models the problem of which letter she should choose as the following LP.

$$\begin{array} { c l } 
\text { Maximise } & M = m - 1 \\
\text { subject to } & m \leqslant 2 x + 6 y + z , \\
& m \leqslant 4 x + 2 y , \\
& m \leqslant 3 y + 2 z , \\
& m \leqslant 2 x + 2 z , \\
& x + y + z \leqslant 1 \\
\text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0
\end{array}$$

(iii) Show how the expression $2 x + 6 y + z$ was formed.

The Simplex algorithm is used to solve the LP problem. The solution has $x = 0.3 , y = 0.2$ and $z = 0.5$.\\
(iv) Show that the optimal value of $M$ is 0.6 .

Colin then models the problem of which letter he should choose in a similar way. When Rowena plays using her optimal solution, from above, Colin finds that he should never choose the letter $N$. Letting $p , q$ and $t$ denote the probabilities that he chooses $P , Q$ and $T$, respectively, Colin obtains the following equations.

$$- 3 p + q - t = - 0.6 \quad - p - 2 q + t = - 0.6 \quad p - q - t = - 0.6 \quad p + q + t = 1$$

(v) Explain how the equation $- 3 p + q - t = - 0.6$ is obtained.\\
(vi) Use the third and fourth equations to find the value of $p$. Hence find the values of $q$ and $t$.

\hfill \mbox{\textit{OCR D2 2012 Q6 [13]}}