Events $A$ and $B$ are such that ${\text{P}}(A) = \frac{2}{5},{\text{ P}}(B) = \frac{{11}}{{20}}$ and ${\text{P}}(A|B) = \frac{2}{{11}}$.

(a)     Find ${\text{P}}(A \cap B)$.

(b)     Find ${\text{P}}(A \cup B)$.

(c)     State with a reason whether or not events $A$ and $B$ are independent.

(a)     ${\text{P}}(A \cap B) = {\text{P}}(A|B) \times P(B)$

${\text{P}}(A \cap B) = \frac{2}{{11}} \times \frac{{11}}{{20}}$     (M1)

$= \frac{1}{{10}}$     A1

[2 marks]

(b)     ${\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)$

${\text{P}}(A \cup B) = \frac{2}{5} + \frac{{11}}{{20}} - \frac{1}{{10}}$     (M1)

$= \frac{{17}}{{20}}$     A1

[2 marks]

(c)     No – events A and B are not independent     A1

EITHER

${\text{P}}(A|B) \ne {\text{P}}(A)$     R1

$\left( {\frac{2}{{11}} \ne \frac{2}{5}} \right)$

OR

${\text{P}}(A) \times {\text{P}}(B) \ne {\text{P}}(A \cap B)$

$\frac{2}{5} \times \frac{{11}}{{20}} = \frac{{11}}{{50}} \ne \frac{1}{{10}}$     R1

Note:     The numbers are required to gain R1 in the ‘OR’ method only.

Note:     Do not award A1R0 in either method.

[2 marks]

Total [6 marks]