# Question 2 (Paper 4737):

## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4p-(1-p) = 5p-1$ | M1 | For $4p-1(1-p)$ or equivalent, seen or implied |
| $5p-1$ or $-1+5p$ | A1 | cao |
| $-2p+5(1-p) = 5-7p$ | B1 | For any form of this expression cao |
| $4(1-p) = 4-4p$ | B1 | For any form of this expression cao **[4]** |

## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Graph with horizontal axis from 0 to 1 | M1 | For correct structure to graph with a horizontal axis that extends from 0 to 1, but not more than this, and with consistent scales |
| Line $E=5p-1$ plotted from $(0,-1)$ to $(1,4)$ | A1 ft | For line $E=5p-1$ plotted from $(0,-1)$ to $(1,4)$ |
| Line $E=5-7p$ plotted from $(0,5)$ to $(1,-2)$ | A1 ft | For line $E=5-7p$ plotted from $(0,5)$ to $(1,-2)$ |
| Line $E=4-4p$ plotted from $(0,4)$ to $(1,0)$ | A1 ft | For line $E=4-4p$ plotted from $(0,4)$ to $(1,0)$ |
| All three cases correct or ft from (i) | | **[4]** |
| $p=0.5$ | B1 | For this or ft their graph **[1]** |

## Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $5(0.5)-1$ | M1 | For substituting their $p$ into any of their equations (must be seen, cannot be implied from value) |
| $= 1.5$ points per game | A1 | For 1.5 cao |
| Bea may not play her best strategy | B1 | For this or equivalent; describing a mixed strategy that involves $Z$ **[3]** |

## Part (iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| Value $= 1.5$ | B1 ft | Accept $-1.5$, ft from (iii) |
| If Amy plays using her optimal strategy, Bea should never play strategy $Z$ | M1 | For identifying that she should not play $Z$ |
| Assuming Bea knows Amy will make random choice between $P$ and $Q$ so each has probability 0.5, it does not matter how she chooses between $X$ and $Y$ | A1 | For a full description of how she should play. (If candidate assumes Bea does not know then Bea should play $P$ with probability $\frac{7}{12}$ and $Q$ with probability $\frac{5}{12}$) **[3]** |