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]