jmc

The sum of two prime numbers is greater than $2$, so if this sum is to be prime, it must be odd. Therefore, one of the primes in the sum must be $2$, and the other must be odd. The first eight prime numbers are $2,3,5,7,11,13,17,$ and $19$. Of the odd ones, only $3,5,11$, and $17$ added to $2$ give a prime number. Therefore, there are $4$ possible pairs whose sum is prime. The total number of pairs is $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$. So the probability is $\frac{4}{28}=\boxed{\frac{1}{7}}$.