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?

 

Answer 1

\[\text{ Given: Bag }= \left( 4W + 5R + 7B \right) \text{ balls} \]

\[P\left( \text{ atleast 2 white balls } \right) = 1 - P\left( \text{ maximum 1 white ball } \right)\]

\[ = 1 - \left[ P\left( \text{ no white } \right) + P\left( \text{ exactly one white } \right) \right]\]

\[ = 1 - \left[ \frac{12}{16} \times \frac{12}{16} \times \frac{12}{16} \times \frac{12}{16} + \frac{4}{16} \times \frac{12}{16} \times \frac{12}{16} \times \frac{12}{16} \times 4 \right]\]

\[ = 1 - \left[ \frac{81}{256} + \frac{108}{256} \right]\]

\[ = 1 - \frac{189}{256}\]

\[ = \frac{256 - 189}{256}\]

\[ = \frac{67}{256}\]