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Question
A bag contains 4 white, 7 black and 5 red balls. 4 balls are drawn with replacement. What is the probability that at least two are white?
Solution
\[\text{ Given: Bag }= \left( 4W + 5R + 7B \right) \text{ balls} \]
\[P\left( \text{ atleast 2 white balls } \right) = 1 - P\left( \text{ maximum 1 white ball } \right)\]
\[ = 1 - \left[ P\left( \text{ no white } \right) + P\left( \text{ exactly one white } \right) \right]\]
\[ = 1 - \left[ \frac{12}{16} \times \frac{12}{16} \times \frac{12}{16} \times \frac{12}{16} + \frac{4}{16} \times \frac{12}{16} \times \frac{12}{16} \times \frac{12}{16} \times 4 \right]\]
\[ = 1 - \left[ \frac{81}{256} + \frac{108}{256} \right]\]
\[ = 1 - \frac{189}{256}\]
\[ = \frac{256 - 189}{256}\]
\[ = \frac{67}{256}\]
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