Assuming that the birth of a boy or a girl is equally likely, what is the probability that the three children in a family include at least one boy and one girl? Express your answer as a common fraction.
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 $1-\frac{1}{8}-\frac{1}{8}=\boxed{\frac{3}{4}}$.