Question

A bag contains 4 red and 5 black balls, a second bag contains 3 red and 7 black balls. One ball is drawn at random from each bag, find the probability that the (i) balls are of different colours (ii) balls are of the same colour.

Answer 1

$$\begin{gathered} \text{Given:} \\ \text{Bag A=}\left(\text{4R+5B}\right)\text{balls} \\ \text{Bag B=}\left(\text{3R+7B}\right)\text{balls} \\ \left(\mathrm{i}\right)P\left(\text{balls of different colours}\right)=P\left(\text{red from bag A and black from bag B}\right)+P\left(\text{red from bag B and black from bag A}\right) \\ =\frac{4}{9}\times \frac{7}{10}+\frac{3}{10}\times \frac{5}{9} \\ =\frac{28}{90}+\frac{15}{90} \\ =\frac{43}{90} \\ \left(\mathrm{ii}\right)P\left(\text{balls of same colour}\right)=P\left(\text{both red}\right)+P\left(\text{both black}\right) \\ =\frac{4}{9}\times \frac{3}{10}+\frac{7}{10}\times \frac{5}{9} \\ =\frac{12}{90}+\frac{35}{90} \\ =\frac{47}{90} \end{gathered}$$