jmc

To obtain this probability, we want to take the number of double pairings over the total number of pairings. Because each integer is paired with each other integer exactly once, we must be careful when counting how many integer pairings there are. That is, $0$ can be paired with $10$ other numbers, $1$ can be paired with $9$ other numbers (not $0$, because we've already paired $0$ and $1$), $2$ can be paired with $8$ other numbers, etc. So, there are $10+9+\dots +1=55$ pairings. Ten of these pairings are doubles ($00$, $11$, etc.). Thus, the probability of choosing a double is $\frac{10}{55}$, which simplifies to $\boxed{\frac{2}{11}}$.