jmc

There are only two ways for a family of three to not have at least one boy and at least one girl: either the family is all boys, or it is all girls. The probability that a family is all boys is ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$, and the probability that a family is all girls is also $\frac{1}{8}$. Therefore, the probability that a family of three is neither all girls nor all boys is $1-\frac{1}{8}-\frac{1}{8}=\boxed{\frac{3}{4}}$.