October 2020 Paper 2 Q7
7 You are given that \(\mathrm{P}(A) = 0.6,\ \mathrm{P}(B) = 0.5\) and \(\mathrm{P}(A \cup B)^{\prime} = 0.2\).
(a) Find \(\mathrm{P}(A \cap B)\). [2]
(b) Find \(\mathrm{P}(A \mid B)\). [2]
(c) State, with a reason, whether \(A\) and \(B\) are independent. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(0.6 + 0.5 - \mathrm{P}(A \cap B) = 1 - 0.2\) oe soi | M1 | 1.1a |
| \(= 0.3\) | A1 | 1.1 |
| [2] |
Notes
M1: or M1 for probabilities in bold correct in table or marked correctly on Venn diagram
| \(A\) | \({\sim}A\) | ||
|---|---|---|---|
| \(B\) | 0.3 | 0.2 | 0.5 |
| \({\sim}B\) | 0.3 | 0.2 | 0.5 |
| 0.6 | 0.4 | 1 |
NB 0.3 from \(0.6 \times 0.5\) does not score
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{0.3}{0.5}\) | M1 | 1.1 |
| \(= 0.6\) | A1 | 1.1 |
| [2] |
Notes
M1: \(\dfrac{\text{their } \mathrm{P}(A \cap B)}{0.5}\)
| Scheme | Marks | AO |
|---|---|---|
| independent since \(\mathrm{p}(A) = \mathrm{p}(A/B)\) oe | B1 | 2.4 |
| [1] |
Notes
B1: or \(0.6 \times 0.5 = 0.3\) or \(\mathrm{P}(A \cap B) = \mathrm{P}(A) \times \mathrm{P}(B)\)
FT their values with correct argument