S1 January 2006 Q6
6. For the events \(A\) and \(B\),
\[\mathrm{P}(A \cap B^{\prime}) = 0.32, \ \mathrm{P}(A^{\prime} \cap B) = 0.11 \text{ and } \mathrm{P}(A \cup B) = 0.65.\](a) Draw a Venn diagram to illustrate the complete sample space for the events \(A\) and \(B\). (3)
(b) Write down the value of \(\mathrm{P}(A)\) and the value of \(\mathrm{P}(B)\). (3)
(c) Find \(\mathrm{P}(A \mid B^{\prime})\). (2)
(d) Determine whether or not \(A\) and \(B\) are independent. (3)
| Scheme | Marks |
|---|---|
| Venn Diagram | M1 |
| 0.32, 0.11 & \(A\), \(B\) | A1 |
| 0.22, 0.35 & box | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(A) = 0.32 + 0.22 = 0.54;\ \mathrm{P}(B) = 0.33\) | M1A1ft;A1ft |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(A \mid B^{\prime}) = \dfrac{\mathrm{P}(A \cap B^{\prime})}{\mathrm{P}(B^{\prime})} = \dfrac{32}{67}\) | M1A1 |
| (2) |
Notes
A1 awrt 0.478
| Scheme | Marks |
|---|---|
| For independence \(\mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B)\) For these data \(0.22 \neq 0.54 \times 0.33 = 0.1782\) | M1A1ft |
| (OR \(\mathrm{P}(A \mid B^{\prime}) \neq \mathrm{P}(A)\) for M1A1ft OR \(\dfrac{2}{3} = \mathrm{P}(A \mid B) \neq \mathrm{P}(A) = 0.54\) for M1A1ft) \(\therefore\) NOT independent | A1ft |
| (3) | |
| (11 marks) |