Solution — 2016 Question 8 Probability

Tree diagram for 2016 Q8

(a)

\begin{matrix} P (both balls are red) \\ = \frac{1}{2} (\frac{3}{10}) (\frac{2}{9}) \\ = \frac{1}{30} ∎ \end{matrix}

(b)

\begin{matrix} P (same colour) \\ = \frac{1}{30} + \frac{1}{2} (\frac{5}{10}) (\frac{4}{9}) + \frac{1}{2} (\frac{2}{10}) (\frac{1}{9}) + \frac{1}{2} (\frac{4}{6}) (\frac{3}{5}) + \frac{1}{2} (\frac{2}{6}) (\frac{1}{5}) \\ = \frac{7}{18} \end{matrix}

\begin{matrix} P (different colours) \\ = 1 - P (same colour) \\ = 1 - \frac{7}{18} \\ = \frac{11}{18} ∎ \end{matrix}

(c)

\begin{matrix} P (both red | same colour) \\ = \frac{P (both red and same colour)}{P (same colour)} \\ = \frac{\frac{1}{30}}{\frac{7}{18}} \\ = \frac{3}{35} ∎ \end{matrix}