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Question
A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be
(i) blue followed by red.
(ii) blue and red in any order.
(iii) of the same colour.
Solution
\[\text{ It is given that the bag contains 3 blue and 5 red marbles.}\]
\[\left( i \right) P\left( \text{ blue followed by red } \right)\]
\[ = \frac{3}{8} \times \frac{5}{8}\]
\[ = \frac{15}{64}\]
\[\left( ii \right) P\left( \text{ red and blue in any order} \right) = P\left( \text{ blue followed by red } \right) + P\left( \text{ red followed by blue } \right)\]
\[ = \frac{3}{8} \times \frac{5}{8} + \frac{5}{8} \times \frac{3}{8}\]
\[ = \frac{15}{64} + \frac{15}{64}\]
\[ = \frac{30}{64} = \frac{15}{32}\]
\[\left( iii \right) P\left( \text{ same colour } \right) = P\left( \text{ both red } \right) + P\left( \text{ both blue } \right)\]
\[ = \frac{5}{8} \times \frac{5}{8} + \frac{3}{8} \times \frac{3}{8}\]
\[ = \frac{25}{64} + \frac{9}{64}\]
\[ = \frac{34}{64}\]
\[ = \frac{17}{32}\]
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