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