**(a)** $P(S) = 0.31 + p$, $P(D) = 0.35$, $P(S \cap D) = 0.14$ | M1 (4) |
$(0.31 + p)(0.35) = 0.14$ oe | M1 |
$P(S) = 0.4$ or $0.31 + p = 0.4$ or $0.35p = 0.0315$ | A1 |
$p = 0.09$ | A1 |

**(b)** $P(S \cup M \cup D) = 1$ so $q = 1 - (0.17 + 0.10 + 0.15 + 0.06 + 0.04) - p$ or $0.48 - p$ | M1 (2) |
$q = 0.39$ | A1ft |

**(c)(i)** $[P(D | S \cap M) =] \frac{P(D \cap S \cap M)}{P(S \cap M)} = \frac{0.10}{0.27}$ | M1 (4) |
$= \frac{10}{27}$ or awrt $0.370$ | A1 |

**(c)(ii)** $[P(D | S' \cap M) =] \frac{P(D \cap S' \cap M)}{P(S' \cap M)} = \frac{0.15}{0.54}$ | M1 (4) |
$= \frac{5}{18}$ or awrt $0.278$ | A1 |

**(d)** 27 order $S \cap M$ so expect $27 \times \frac{10}{27}$ $D$ or 36 order $S' \cap M$ so expect $36 \times \frac{5}{18}$ $D$ | M1 (2) |
So expect **20 (desserts)** | A1cao | [12]

**Notes:**

(a) 1st M1 for attempting $P(S), P(D)$ and $P(S \cap D)$ with at least 2 correct. These may be seen in a conditional probability. 2nd M1 using the independence condition and their values to form a suitable equation for $p$ or $P(S)$. 1st A1 for $P(S) = 0.4$ or $0.31 + p = 0.4$ or $0.35p = 0.0315$ (i.e. one move from $p = ...$). NB $P(S|D) = \frac{0.14}{0.35}$ and $P(D|S) = \frac{0.14}{0.31+p}$

(b) M1 for using sum of probabilities = 1 and ft their $p$. A1ft for $0.48 - \text{"their } p\text{"} (provided $0 < \text{their } p < 0.48$)

(c) 1st M1 for correct ratio of probabilities or correct ratio expression with at least one correct probability substituted. (M0 if numerator is $P(D) \times P(S \cap M)$ or numerator > denominator). 1st A1 for $\frac{10}{27}$ or awrt 0.370. 2nd M1 for correct ratio of probabilities or correct ratio expression with at least one correct probability substituted. (M0 if numerator is $P(D) \times P(S' \cap M)$ or numerator > denominator). 2nd A1 for $\frac{5}{18}$ or awrt 0.278.

(d) M1 for at least one correct calculation ft their probabilities from (c), i.e. either $27 \times \text{their (c)(i)}$ or $36 \times \text{their (c)(ii)}$.

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