jmc

The total number of ways to choose 6 numbers is $\left(\genfrac{}{}{0ex}{}{10}{6}\right)=210$. Assume $3$ is the second-lowest number. There are $5$ numbers left to choose, $4$ of which must be greater than $3$, and $1$ of which must be less than $3$. This is equivalent to choosing $4$ numbers from the $7$ numbers larger than $3$, and $1$ number from the $2$ numbers less than $3$.$$\left(\genfrac{}{}{0ex}{}{7}{4}\right)\left(\genfrac{}{}{0ex}{}{2}{1}\right)=35\times 2$$. Thus, $\frac{35\times 2}{210}=\boxed{\frac{1}{3}}$.