Determine the value of ${\text{P}}(A \cup B)$ when

(i)     $A$ and $B$ are mutually exclusive;

(ii)     $A$ and $B$ are independent.

Determine the range of possible values of ${\text{P}}\left( {A|B} \right)$.

(i)     use of ${\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B)$     (M1)

${\text{P}}(A \cup B) = 0.2 + 0.5$

$= 0.7$     A1

(ii)     use of ${\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A){\text{P}}(B)$     (M1)

${\text{P}}(A \cup B) = 0.2 + 0.5 - 0.1$

$= 0.6$     A1

[4 marks]

${\text{P}}\left( {A|B} \right) = \frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}}$

${\text{P}}\left( {A|B} \right)$ is a maximum when ${\text{P}}(A \cap B) = {\text{P}}(A)$

${\text{P}}\left( {A|B} \right)$ is a minimum when ${\text{P}}(A \cap B) = 0$

$0 \le {\text{P}}\left( {A|B} \right) \le 0.4$     A1A1A1

Note:     A1 for each endpoint and A1 for the correct inequalities.

[3 marks]

Total [7 marks]

This part was generally well done.

Disappointingly, many candidates did not seem to understand the meaning of the word ‘range’ in this context.