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

$$\begin{gathered} \text{Given: Bag}=\left(4W+5R+7B\right)\text{balls} \\ P\left(\text{atleast 2 white balls}\right)=1-P\left(\text{maximum 1 white ball}\right) \\ =1-\left[P\left(\text{no white}\right)+P\left(\text{exactly one white}\right)\right] \\ =1-\left[\frac{12}{16}\times \frac{12}{16}\times \frac{12}{16}\times \frac{12}{16}+\frac{4}{16}\times \frac{12}{16}\times \frac{12}{16}\times \frac{12}{16}\times 4\right] \\ =1-\left[\frac{81}{256}+\frac{108}{256}\right] \\ =1-\frac{189}{256} \\ =\frac{256-189}{256} \\ =\frac{67}{256} \end{gathered}$$