\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\).
Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]
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\(\mathrm{P}(B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)} = \frac{1/4}{3/4} = \frac{1}{3}\) B1
Using independence of \(A\) and \(B\).
Either:
Answer Marks
Guidance
\(\mathrm{P}(A^c \cap B) = \mathrm{P}(A') \times \mathrm{P}(B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}\) M1, A1
Allow the use of a Venn diagram or a tree diagram. Product using independence or partition of \(B\). c.a.o.
Or:
Answer Marks
Guidance
\(\mathrm{P}(A^c \cap B) = \mathrm{P}(B) - \mathrm{P}(A \cap B) = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}\) M1, A1
c.a.o.
Either:
Answer Marks
Guidance
\(\mathrm{P}(A^c \cap B') = \mathrm{P}(A') \times \mathrm{P}(B') = \frac{1}{4} \times \frac{2}{3} = \frac{1}{6}\) M1, A1
Product of complements or complement of union or partition of \(A'\). c.a.o.
Or:
Answer Marks
Guidance
\(\mathrm{P}(A^c \cap B') = 1 - \mathrm{P}(A \cup B) = 1 - \left(\frac{3}{4} + \frac{1}{3} - \frac{1}{4}\right) = \frac{1}{6}\) M1, A1
c.a.o.
Or:
Answer Marks
Guidance
\(\mathrm{P}(A^c \cap B') = \mathrm{P}(A') - \mathrm{P}(A^c \cap B) = \frac{1}{4} - \frac{1}{12} = \frac{1}{6}\) M1, A1
c.a.o.
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$\mathrm{P}(B) = \frac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)} = \frac{1/4}{3/4} = \frac{1}{3}$ | B1 | Using independence of $A$ and $B$.
**Either:**
$\mathrm{P}(A^c \cap B) = \mathrm{P}(A') \times \mathrm{P}(B) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$ | M1, A1 | Allow the use of a Venn diagram or a tree diagram. Product using independence or partition of $B$. c.a.o.
**Or:**
$\mathrm{P}(A^c \cap B) = \mathrm{P}(B) - \mathrm{P}(A \cap B) = \frac{1}{3} - \frac{1}{4} = \frac{1}{12}$ | M1, A1 | c.a.o.
**Either:**
$\mathrm{P}(A^c \cap B') = \mathrm{P}(A') \times \mathrm{P}(B') = \frac{1}{4} \times \frac{2}{3} = \frac{1}{6}$ | M1, A1 | Product of complements or complement of union or partition of $A'$. c.a.o.
**Or:**
$\mathrm{P}(A^c \cap B') = 1 - \mathrm{P}(A \cup B) = 1 - \left(\frac{3}{4} + \frac{1}{3} - \frac{1}{4}\right) = \frac{1}{6}$ | M1, A1 | c.a.o.
**Or:**
$\mathrm{P}(A^c \cap B') = \mathrm{P}(A') - \mathrm{P}(A^c \cap B) = \frac{1}{4} - \frac{1}{12} = \frac{1}{6}$ | M1, A1 | c.a.o. | [5]
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$A$ and $B$ are independent events. $P(A) = \frac{3}{4}$ and $P(A \cap B) = \frac{1}{4}$.
Find $P(A' \cap B)$ and $P(A' \cap B')$. [5]
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q6 [5]}}