jmc

There are $\left(\genfrac{}{}{0ex}{}{5}{3}\right)$ possible groups of cards that can be selected. If $4$ is the largest card selected, then the other two cards must be either $1$, $2$, or $3$, for a total $\left(\genfrac{}{}{0ex}{}{3}{2}\right)$ groups of cards. Then, the probability is just $\frac{\left(\genfrac{}{}{0ex}{}{3}{2}\right)}{\left(\genfrac{}{}{0ex}{}{5}{3}\right)}=\boxed{\frac{3}{10}}$.