# Question 1:

## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B$ and $C$ or "band" and "choir" but NOT $P(B)$ and $P(C)$ | B1 (1) | Allow other non-trivial pairs e.g. $B$ and $L \cap C$ but not $L$ and $L'$ |

## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[L$ and $C$ independent implies$]$ $P(L \cap C) = P(L) \times P(C) = 0.4 \times 0.3$ | M1 | For clear attempt to use the rule for independence. Rule stated and one correct substitution. |
| $p = \mathbf{0.12}$ | A1 (2) | For $0.12$ (either labelled $p$ or part (b) or correctly placed on Venn diagram) |

## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $q = 0.4 - 0.13 - \text{their } p = \mathbf{0.15}$ | B1ft | For $0.15$ or a correct $q$ allowing ft of their $p$. The ft requires all values concerned to be probabilities. |
| $r = 0.3 - \text{their } p = \mathbf{0.18}$ | B1ft | For $0.18$ or a correct $r$ allowing ft of their $p$ |
| $s = 1-(0.4+0.3-\text{their } p)$ or $1-(0.4+\text{their } r) = \mathbf{0.42}$ | B1ft (3) | For $0.42$ or a correct $s$ allowing ft of their $p$ or $r$. (Labelled on or on Venn diagram) |

## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(L \mid B \cup C)$ or $\dfrac{P(L \cap [B \cup C])}{P(B \cup C)} = \dfrac{0.13 + \text{"0.12"}}{0.13+0.3}$ | M1; A1ft | For a correct probability expression (letters and symbols) **and** any ratio of probabilities (num < denom). May be implied by a correct (or correct ft) probability ratio. 1st A1ft for a correct (or correct ft) probability ratio (num < denom) |
| $= \dfrac{25}{43}$ | A1 (3) **[9 marks]** | 2nd A1 for $\dfrac{25}{43}$ or exact equivalent. NB completed Venn diagram. (If answers conflict the script takes preference over diagram) |

# Question 1:

**Venn Diagram probabilities:**

$q = 0.15$, $p = 0.12$, $r = 0.18$, $B = 0.13$, $S = 0.42$ | (diagram) | Accept probabilities in any exact form e.g. $\frac{3}{20}$ for 0.15 or $\frac{3}{25}$ for 0.12

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