Find $q$.

Find $p$.

Write down the probability of drawing three blue marbles.

Explain why the probability of drawing three white marbles is $\frac{1}{6}$.

The bag contains a total of ten marbles of which $w$ are white. Find $w$.

Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.

correct substitution into ${\text{E}}(X)$ formula     (A1)

eg$\,\,\,\,\,$$0(p) + 1(0.5) + 2(0.3) + 3(q) = 1.2$

$q = \frac{1}{{30}}$, 0.0333     A1     N2

[2 marks]

evidence of summing probabilities to 1     (M1)

eg$\,\,\,\,\,$$p + 0.5 + 0.3 + q = 1$

$p = \frac{1}{6},{\text{ }}0.167$     A1     N2

[2 marks]

${\text{P (3 blue)}} = \frac{1}{{30}},{\text{ }}0.0333$     A1     N1

[1 mark]

valid reasoning     R1

eg$\,\,\,\,\,$${\text{P (3 white)}} = {\text{P(0 blue)}}$

${\text{P(3 white)}} = \frac{1}{6}$     AG     N0

[1 mark]

valid method     (M1)

eg$\,\,\,\,\,$${\text{P(3 white)}} = \frac{w}{{10}} \times \frac{{w - 1}}{9} \times \frac{{w - 2}}{8},{\text{ }}\frac{{_w{C_3}}}{{_{10}{C_3}}}$

correct equation     A1

eg$\,\,\,\,\,$$\frac{w}{{10}} \times \frac{{w - 1}}{9} \times \frac{{w - 2}}{8} = \frac{1}{6},{\text{ }}\frac{{_w{C_3}}}{{_{10}{C_3}}} = 0.167$

$w = 6$     A1     N2

[3 marks]

recognizing one prize in first seven attempts     (M1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right),{\text{ }}{\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6}$

correct working     (A1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right){\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6},{\text{ }}0.390714$

correct approach     (A1)

eg$\,\,\,\,\,$$\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right){\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6} \times \frac{1}{6}$

0.065119

0.0651     A1     N2

[4 marks]