Write down the probability that the first disc selected is red.

Let $X$ be the number of red discs selected. Find the smallest value of $m$ for which ${\text{Var}}(X{\text{ }}) < 0.6$.

${\text{P(red)}} = \frac{5}{{15 + m}}$     A1     N1

[1 mark]

recognizing binomial distribution     (M1)

eg$\,\,\,\,\,$$X \sim B(n,{\text{ }}p)$

correct value for the complement of their $p$ (seen anywhere)     A1

eg$\,\,\,\,\,$$1 - \frac{5}{{15 + m}},{\text{ }}\frac{{10 + m}}{{15 + m}}$

correct substitution into ${\text{Var}}(X) = np(1 - p)$     (A1)

eg$\,\,\,\,\,$$4\left( {\frac{5}{{15 + m}}} \right)\left( {\frac{{10 + m}}{{15 + m}}} \right),{\text{ }}\frac{{20(10 + m)}}{{{{(15 + m)}^2}}} < 0.6$

$m > 12.2075$     (A1)

$m = 13$     A1     N3

[5 marks]