Write down an expression for ${\text{P}}(C \cap D)$ in terms of $k$.

Given that ${\text{P}}(C \cap D) = 0.162$ find $k$.

Find ${\text{P}}(C'|D)$.

${\text{P}}(C \cap D) = 2k \times 3{k^2}$     (A1)

${\text{P}}(C \cap D) = 6{k^3}$     A1     N2

[2 marks]

their correct equation     (A1)

eg$\;\;\;2k \times 3{k^2} = 0.162,{\text{ }}6{k^3} = 0.162$

$k = 0.3$     A1     N2

METHOD 1

finding their ${\text{P}}(C' \cap D)$ (seen anywhere)     (A1)

eg  $0.4 \times 0.27,0.27 - 0.162,0.108$

correct substitution into conditional probability formula     (A1)

eg$\;\;\;{\text{P}}(C'|D) = \frac{{{\text{P}}(C' \cap D)}}{{0.27}},{\text{ }}\frac{{(1 - 2k)(3{k^2})}}{{3{k^2}}}$

${\text{P}}(C'|D) = 0.4$     A1     N2

METHOD 2

recognizing ${\text{P}}(C'|D) = {\text{P}}(C')$     A1

finding their ${\text{P}}(C') = 1 - {\text{P}}(C)$ (only if first line seen)     (A1)

eg$\;\;\;1 - 2k,{\text{ }}1 - 0.6$

${\text{P}}(C'|D) = 0.4$     A1     N2

[3 marks]

Total [7 marks]