4

4. [Maximum mark: 6]
   A species of bird can nest in two seasons: Spring and Summer. The probability of nesting in Spring is $k$. The probability of nesting in Summer is $\frac{k}{2}$. This is shown in the following tree diagram.

(a) Complete the tree diagram to show the probabilities of not nesting in each season. Write your answers in terms of $k$.
It is known that the probability of not nesting in Spring and not nesting in Summer is $\frac{5}{9}$.

(b) (i) Show that $9k^2 - 27k + 8 = 0$.

(ii) Both $k = \frac{1}{3}$ and $k = \frac{8}{3}$ satisfy $9k^2 - 27k + 8 = 0$. State why $k = \frac{1}{3}$ is the only valid solution.