S1 January 2012 Q2
2.
The events \(A\) and \(B\) are independent with \(\mathrm{P}(A) = \dfrac{1}{4}\) and \(\mathrm{P}(A \cup B) = \dfrac{2}{3}\)
Find
| Scheme | Marks |
|---|---|
| (\(R\) and \(S\) are mutually) exclusive. | B1 |
| (1) |
Notes
B1 for ‘(mutually) exclusive’ or ‘cannot occur at the same time’ seen or equivalent.
‘Intersection is zero’ or ‘no overlaps’ without further explanation is B0.
| Scheme | Marks |
|---|---|
| \(\dfrac{2}{3} = \dfrac{1}{4} + \mathrm{P}(B) - \mathrm{P}(A \cap B)\) use of Addition Rule | M1 |
| \(\dfrac{2}{3} = \dfrac{1}{4} + \mathrm{P}(B) - \dfrac{1}{4} \times \mathrm{P}(B)\) use of independence | M1 A1 |
| \(\dfrac{5}{12} = \dfrac{3}{4}\,\mathrm{P}(B)\) \(\mathrm{P}(B) = \dfrac{5}{9}\) | A1 |
| (4) |
Notes
M1 for use of Addition Formula, including an intersection, with at least one probability substituted. Intersection must be explicitly considered for this mark.
Accept \(\dfrac{2}{3} = \dfrac{1}{4} + \mathrm{P}(B) - 0\) for M1.
M1 for \(\mathrm{P}(A \cap B) = \dfrac{1}{4}\mathrm{P}(B)\)
A1 for completely correct equation or equivalent.
A1 for \(\dfrac{5}{9}\) or exact equivalent..
Venn Diagram with 2 overlapping closed curves and correct values possibly without \(\dfrac{1}{3}\), award M1M1A1.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(A' \cap B) = \dfrac{3}{4} \times \dfrac{5}{9} = \dfrac{15}{36} = \dfrac{5}{12}\) | M1A1ft |
| (2) |
Notes
M1 for \(\dfrac{3}{4}\) x ‘their P(\(B\))’ or ‘their P(\(B\))’- \(\mathrm{P}(A \cap B)\) or \(\mathrm{P}(A \cup B) - \mathrm{P}(B) = \dfrac{2}{3} - \dfrac{1}{4}\)
Or \(\mathrm{P}(A' \cap B) = \mathrm{P}(A') +\) ‘their P(\(B\))’ \(- \mathrm{P}(A' \cup B) = \dfrac{3}{4} + \dfrac{5}{9} - \dfrac{8}{9}\)
A1 for \(\dfrac{5}{12}\) or follow through from their method. Accept exact equivalent.
Correct answer only with no working M1A1 but must be clearly labelled (c).
For part (c) follow through their stated values; do not follow through incorrectly labelled regions on a Venn Diagram.
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(B' \mid A) = \dfrac{(1 - \text{(b)}) \times 0.25}{0.25}\) or P(\(B'\)) or \(\dfrac{\frac{1}{9}}{\frac{1}{4}}\) | M1 |
| \(= \dfrac{4}{9}\) | A1 |
| (2) | |
| (9 marks) |
Notes
M1 for using 1-‘their P(\(B\))’ or \((\mathrm{P}(A \cup B) - \mathrm{P}(A))/\mathrm{P}(A)\) or \((\mathrm{P}(A) - \mathrm{P}(A \cap B))/\mathrm{P}(A)\) with a correct attempt at the numerator and denominator. If mutually exclusive is assumed then the last option gives \(\dfrac{\frac{1}{4}}{\frac{1}{4}}\) for M1.
A1 for \(\dfrac{4}{9}\) or exact equivalent.
Throughout the question we require probabilities between 0 and 1 for method marks.
Venn Diagram: