# Question 2:

## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| 6 | B1 | 6 |

## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The total number of points for each combination is 10, subtracting 5 from each entry gives a total of 0 for each entry | B1 | Total $= 10$ changes to total $= 0$, or subtracting 5 gives total $= 0$ for every cell |

## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Pay-off matrix written out with row minima and column maxima | M1 | Writing out pay-off matrix for zero-sum game (or explaining the given matrix gives same play safes) |
| Play-safe for R is Philip | B1 | P, cao; row minima need not be seen |
| Play-safe for C is Mike | A1 | M, cao; col maxima need not be seen; accept any reasonable identification |
| Not stable since $-1 \neq 0$ | B1 | Any equivalent reasoning; their row maximin $\neq$ their col minimax |
| If Team R plays safe then Team C should choose Liam | B1 | 'Liam' or 'L', or follow through their choice of play safe for Team R |

## Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| If entry for row P column L is increased, col max for Liam is at least as big as at present so column M is still the column minimax; and row min for Philip is at least as big as at present so row P is still the row maximin | M1 | Using either original or augmented values; a reasonable explanation of either part |
| Correct explanation of both | A1 | A correct explanation of both; in play safe row and not in play safe column, without further explanation $\Rightarrow$ M1, A0 |

## Part (v):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sanjiv's scores are dominated by Philip's; Sanjiv scores fewer hits than Philip for each choice of captains from Team C | B1 | Identifying dominance by $P$ and explaining it or showing the three comparisons |

## Part (vi):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4p + 6(1-p)$ or $-1p + 1(1-p) + 5$ $= 6-2p$ | M1 | Using original or reduced values correctly |
| Achieving given expression from valid working | A1 | |
| M: $5p + 5(1-p)$ or $0(p) + 0(1-p) + 5 = 5$ | | |
| N: $6p + 3(1-p)$ or $1p + -2(1-p) + 5 = 3p+3$ | B1 | 5 and $3p+3$, cao |

## Part (vii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Graph with $E = 6-2p$ drawn correctly with appropriate scales | M1 | Appropriate scales and line $E = 6-2p$ drawn correctly |
| Other lines drawn correctly | A1 | (Their) other lines drawn correctly |
| $3p + 3 = 6 - 2p \Rightarrow p = 0.6$ | B1 | Solving for their $p$ or from graph |
| Expect at least 4.8 hits | B1 | Their $E$ for chosen value of $p$ or from graph |

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