Find ${\text{P}}(X < 9)$.

Find the value of $\mu$.

Find $\lambda$.

Given that $Y > 9$, find ${\text{P}}(Y < 13)$.

valid approach     (M1)

eg$\,\,\,\,\,$${\text{P}}(X < \mu ) = 0.5,{\text{ }}0.5 - 0.3$

${\text{P}}(X < 9) = 0.2$ (exact)     A1     N2

[2 marks]

$z =  - 0.841621$ (may be seen in equation)     (A1)

valid attempt to set up an equation with their $z$     (M1)

eg$\,\,\,\,\,$$- 0.842 = \frac{{\mu  - X}}{\sigma },{\text{ }} - 0.842 = \frac{{X - \mu }}{\sigma },{\text{ }}z = \frac{{9 - \mu }}{{2.1}}$

10.7674

$\mu  = 10.8$     A1     N3

[3 marks]

${\text{P}}(X > 9) = 0.8$ (seen anywhere)     (A1)

valid approach     (M1)

eg$\,\,\,\,\,$${\text{P}}(A) \times {\text{P}}(B)$

correct equation     (A1)

eg$\,\,\,\,\,$$0.8 \times {\text{P}}(Y > 9) = 0.4$

${\text{P}}(Y > 9) = 0.5$     A1

$\lambda  = 9$     A1     N3

[5 marks]

finding ${\text{P}}(9 < Y < 13) = 0.373450$ (seen anywhere)     (A2)

recognizing conditional probability     (M1)

eg$\,\,\,\,\,$${\text{P}}(A|B),{\text{ P}}(Y < 13|Y > 9)$

correct working     (A1)

eg$\,\,\,\,\,$$\frac{{{\text{0.373}}}}{{0.5}}$

0.746901

0.747     A1     N3

[5 marks]