imc

Note that for this restriction to be true, there must be $3$ balls of each color. There are a total of $2^6=64$ ways to color the balls, and there are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ ways for three balls chosen to be painted white. Thus, the answer is $\frac{20}{64}=\boxed{\frac{5}{16}}$.