Solution — 2016 Question 8 Probability

$$\begin{array}{cc}& \mathrm{P}\left(\text{both balls are red}\right)\\ & =\frac{1}{2}\left(\frac{3}{10}\right)\left(\frac{2}{9}\right)\\ & =\frac{1}{30}∎\end{array}$$

$$\begin{array}{cc}& \mathrm{P}\left(\text{same colour}\right)\\ & =\frac{1}{30}+\frac{1}{2}\left(\frac{5}{10}\right)\left(\frac{4}{9}\right)+\frac{1}{2}\left(\frac{2}{10}\right)\left(\frac{1}{9}\right)+\frac{1}{2}\left(\frac{4}{6}\right)\left(\frac{3}{5}\right)+\frac{1}{2}\left(\frac{2}{6}\right)\left(\frac{1}{5}\right)\\ & =\frac{7}{18}\end{array}$$

$$\begin{array}{cc}& \mathrm{P}\left(\text{different colours}\right)\\ & =1-\mathrm{P}\left(\text{same colour}\right)\\ & =1-\frac{7}{18}\\ & =\frac{11}{18}∎\end{array}$$

$$\begin{array}{cc}& \mathrm{P}(\text{both red}|\text{same colour})\\ & =\frac{\mathrm{P}\left(\text{both red and same colour}\right)}{\mathrm{P}\left(\text{same colour}\right)}\\ & =\frac{\frac{1}{30}}{\frac{7}{18}}\\ & =\frac{3}{35}∎\end{array}$$