# Question 4:

## Part (a)
| $\text{P}(B \cap R') = \mathbf{0}$ | B1 | |

## Part (b)
| $\text{P}(B) = 0.27 + 0.33 = 0.6$, $\text{P}(D) = 0.27 + 0.15 + t$, $\text{P}(B \cap D) = 0.27$ | M1 | Attempting 3 suitable probabilities, one involving $t$ (at least 2 correct) |
| $[\text{P}(B) \times \text{P}(D) = \text{P}(B \cap D)]$: $0.6 \times (0.42 + t) = 0.27$ | M1 | Using independence to form linear equation in $t$ |
| $0.42 + t = \frac{0.27}{0.6}$ or $0.6t = 0.018$ | A1 | Solving to correct equation |
| $t = \mathbf{0.03}$ | A1 | For 0.03 or exact equivalent |

## Part (c)
| $u = 1 - (0.6 + 0.15 + t)$ | M1 | Correct expression for $u$ |
| $u = \mathbf{0.22}$ | A1ft | For 0.22 or ft their $t$; i.e. $u = 0.25 - t$ provided $u$ and $t$ are probabilities |

## Part (d)(i)
| $\frac{\text{P}(D \cap R \cap B)}{\text{P}(R \cap B)} = \frac{0.27}{0.27 + 0.33}$ or $\text{P}(D|R \cap B) = \text{P}(D|B) = \text{P}(D)$ | M1 | Correct numerical ratio of probabilities |
| $= \mathbf{0.45}$ | A1 | For 0.45 or exact equivalent |

## Part (d)(ii)
| $\frac{\text{P}(D \cap [R \cap B'])}{\text{P}(R \cap B')} = \frac{0.15}{0.15 + u}$ | M1 | Correct numerical ratio of probabilities, ft their $u$ provided $u$ is a probability |
| $= \frac{15}{37}$ | A1 | For $\frac{15}{37}$ or $0.40\overline{5}$ or awrt 0.41 |

## Part (e)
| $40 \times \text{"0.45"}$ and $37 \times \text{"}\frac{15}{37}\text{"}$ | M1 | Correct method for both, using their 0.45 and $\frac{15}{37}$ provided both in $[0,1]$; NB $\text{P}(D) \times 77$ is M0 |
| $= \mathbf{33}$ | A1 | For 33 only |

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