| Date | November 2016 | Marks available | 6 | Reference code | 16N.1.hl.TZ0.10 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Hence and Show that | Question number | 10 | Adapted from | N/A |
Question
Consider two events \(A\) and \(A\) defined in the same sample space.
Given that \({\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}\) and \({\text{P}}(B|A') = \frac{1}{6}\),
Show that \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)\).
(i) show that \({\text{P}}(A) = \frac{1}{3}\);
(ii) hence find \({\text{P}}(B)\).
Markscheme
METHOD 1
\({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)\) M1
\( = {\text{P}}(A) + {\text{P}}(A \cap B) + {\text{P}}(A' \cap B) - {\text{P}}(A \cap B)\) M1A1
\( = {\text{P}}(A) + {\text{P}}(A' \cap B)\) AG
METHOD 2
\({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)\) M1
\( = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A|B) \times {\text{P}}(B)\) M1
\( = {\text{P}}(A) + \left( {1 - {\text{P}}(A|B)} \right) \times {\text{P}}(B)\)
\( = {\text{P}}(A) + {\text{P}}(A'|B) \times {\text{P}}(B)\) A1
\( = {\text{P}}(A) + {\text{P}}(A' \cap B)\) AG
[3 marks]
(i) use \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)\) and \({\text{P}}(A' \cap B) = {\text{P}}(B|A'){\text{P}}(A')\) (M1)
\(\frac{4}{9} = {\text{P}}(A) + \frac{1}{6}\left( {1 - {\text{P}}(A)} \right)\) A1
\(8 = 18{\text{P}}(A) + 3\left( {1 - {\text{P}}(A)} \right)\) M1
\({\text{P}}(A) = \frac{1}{3}\) AG
(ii) METHOD 1
\({\text{P}}(B) = {\text{P}}(A \cap B) + {\text{P}}(A' \cap B)\) M1
\( = {\text{P}}(B|A){\text{P}}(A) + {\text{P}}(B|A'){\text{P}}(A')\) M1
\( = \frac{1}{3} \times \frac{1}{3} + \frac{1}{6} \times \frac{2}{3} = \frac{2}{9}\) A1
METHOD 2
\({\text{P}}(A \cap B) = {\text{P}}(B|A){\text{P}}(A) \Rightarrow {\text{P}}(A \cap B) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\) M1
\({\text{P}}(B) = {\text{P}}(A \cup B) + {\text{P}}(A \cap B) - {\text{P}}(A)\) M1
\({\text{P}}(B) = \frac{4}{9} + \frac{1}{9} - \frac{1}{3} = \frac{2}{9}\) A1
[6 marks]
