5 Robbie received a new computer game for Christmas. He has already worked through several levels of the game but is now stuck at one of the levels in which he is playing against a character called Conan. Robbie has played this particular level several times.

Each time Robbie encounters Conan he can choose to be helped by a dwarf, an elf or a fairy. Conan chooses between being helped by a goblin, a hag or an imp. The players make their choices simultaneously, without knowing what the other has chosen.

Robbie starts the level with ten gold coins. The table shows the number of gold coins that Conan must give Robbie in each encounter for each combination of helpers (a negative entry means that Robbie gives gold coins to Conan). If Robbie's total reaches twenty gold coins then he completes the level, but if it reaches zero the game ends. This means that each attempt can be regarded as a zero-sum game.

\begin{center}
\begin{tabular}{ l | l | c | c | c | }
\multicolumn{5}{c}{Conan} \\
\cline { 2 - 5 }
 &  & Goblin & Hag & Imp \\
\cline { 2 - 5 }
 & Dwarf & - 1 & - 4 & 2 \\
\cline { 2 - 5 }
Robbie & Elf & 3 & 1 & - 4 \\
\cline { 2 - 5 }
 & Fairy & 1 & - 1 & 1 \\
\cline { 2 - 5 }
 &  &  &  &  \\
\cline { 2 - 5 }
\end{tabular}
\end{center}

(i) Find the play-safe choice for each player, showing your working. Which helper should Robbie choose if he thinks that Conan will play-safe?\\
(ii) How many gold coins can Robbie expect to win, with each choice of helper, if he thinks that Conan will choose randomly between his three choices (so that each has probability $\frac { 1 } { 3 }$ )?

Robbie decides to choose his helper by using random numbers to choose between the elf and the fairy, where the probability of choosing the elf is $p$ and the probability of choosing the fairy is $1 - p$.\\
(iii) Write down an expression for the expected number of gold coins won at each encounter by Robbie for each of Conan's choices. Calculate the optimal value of $p$.

Robbie's girlfriend thinks that he should have included the possibility of choosing the dwarf. She denotes the probability with which Robbie should choose the dwarf, the elf and the fairy as $x , y$ and $z$ respectively. She then models the problem of choosing between the three helpers as the following LP.

$$\begin{aligned}
\text { Maximise } & M = m - 4 , \\
\text { subject to } & m \leqslant 3 x + 7 y + 5 z \\
& m \leqslant 5 y + 3 z \\
& m \leqslant 6 x + 5 z \\
& x + y + z \leqslant 1 , \\
\text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 .
\end{aligned}$$

(iv) Explain how the expression $3 x + 7 y + 5 z$ was formed.

Robbie's girlfriend uses the Simplex algorithm to solve the LP problem. Her solution has $x = 0$ and $y = \frac { 2 } { 7 }$.\\
(v) Calculate the optimal value of $M$.

\hfill \mbox{\textit{OCR D2 2010 Q5 [16]}}