| Answer | Marks | Guidance |
|--------|-------|----------|
| For each pair of strategies, whatever one player wins, the other person loses. | E1 | Must see this statement oe and Row gain + Col gain = 0 oe |
| Row min -5, -5, -3 [Max value -3]; Col max -1, 4, 0 [Min value -1]; Monica [plays] $C$ and Vladimir [plays] $D$ | M1, A1 | All 6 values correct; Must be in context |
| Row $C$ dominates Row $B$ [Monica plays $A$ with probability $p$ plays $C$ with probability $1 - p$] [Vladimir plays] | E1 | Row $B$ is dominated by row $C$ |
| | | |
| $D$. Monica wins $-p - 2(1-p) = p - 2$ | M1 | One expression correct (unsimplified) |
| $E$. Monica wins $-5p + 4(1-p) = 4 - 9p$ | A1 | All 3 correct (unsimplified) |
| $F$. Monica wins $-3(1-p) = 3p - 3$ | | |
| Must have exactly three straight lines | M1 | |
| All correct (eg 4 to -5, -2 to -1, -3 to 0) with numbers on vertical axes shown | A1 | |
| Correct equation | m1 | PI by correct value for $p$ |
| $p = \frac{3}{5}$ | | |
| Monica plays $A$ [with probability] $\frac{3}{5}$ oe | A1 | Both statements needed (condone omission of 'play $B$ [with probability] zero') |
| Monica plays $C$ [with probability] $\frac{2}{5}$ oe | | |
| Value of game $= \frac{3}{5} - 2$ or $4 - 9 \times \frac{3}{5}$ or $= -1.4$ or $-\frac{7}{5}$ | A1 | Must see correct substitution of $p = 0.6$ |
| AG | | Must include statement and no errors seen (condone $V =$) |

# Question 4 (continued)

| Answer | Marks | Guidance |
|--------|-------|----------|
| [Monica plays] [A, Vladimir loses] $-p - 5q$ [C, Vladimir loses] $-2p + 4q - 3(1-p-q)$ | M1 | Either expression correct |
| $-p - 5q = -1.4$ and $-2p + 4q - 3(1-p-q) = -1.4$ | A1 | Both equations correct (or simplified versions eg $p + 7q = 1.6$) |
| $q = 0.1$ | A1 | Either $p$ or $q$ correct |
| $p = 0.9$ | | |
| $(1-p-q) = 0$ | | |
| Vladimir plays $D$ [with probability] 0.9, plays $E$ [with probability] 0.1, plays $F$ [with probability] 0 (or, never plays $F$) | E1 | Must have all 3 probabilities |
| Or, [A, Vladimir loses] $-p - 5(1-p)$ [C, Vladimir loses] $-2p + 4(1-p)$ | (M1) | Either expression correct, but must have discounted F, here, or on final line |
| Equating to $-1.4$ | (A1) | Or, equating to each other |
| $p = 0.9$ | (A1) | |
| Vladimir plays $D$ [with probability] 0.9, plays $E$ [with probability] 0.1, **plays $F$ [with probability] 0** (or, never plays $F$) | (E1) | Must have all 3 probabilities |