| Date | November 2014 | Marks available | 3 | Reference code | 14N.1.hl.TZ0.4 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Determine | Question number | 4 | Adapted from | N/A |
Question
Events \(A\) and \(B\) are such that \({\text{P}}(A) = 0.2\) and \({\text{P}}(B) = 0.5\).
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 \({\text{P}}\left( {A|B} \right)\).
Markscheme
(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]
Examiners report
This part was generally well done.
Disappointingly, many candidates did not seem to understand the meaning of the word ‘range’ in this context.
