# Question 2:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4p - (1-p)$ | M1 | For $4p - 1(1-p)$ or equivalent, seen or implied |
| $= 5p - 1$ | A1 | For $5p-1$ or $-1+5p$ 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 |

## Part (ii)
| Answer | Marks | 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 lines correct or ft from (i) | | 4 marks total |
| $p = 0.5$ | B1 | For this or ft their graph |

## Part (iii)
| Answer | Marks | 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 $Z$ |

## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 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 that Amy will make a random choice between $P$ and $Q$ so that each has probability 0.5, it does not matter how she chooses between strategies $X$ 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 $\frac{5}{12}$) | | Alternative answer noted |

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