jmc

We note that $xy-x-y$ is very close to the expansion of $(x-1)(y-1)$. (This is basically a use of Simon's Favorite Factoring Trick.)

If $xy-x-y$ is even, then $xy-x-y+1=(x-1)(y-1)$ is odd. This only occurs when $x-1$ and $y-1$ are both odd, so $x$ and $y$ must be even. There are $\left(\genfrac{}{}{0ex}{}{5}{2}\right)$ distinct pairs of even integers, and $\left(\genfrac{}{}{0ex}{}{10}{2}\right)$ distinct pairs of integers, so the probability is $\frac{\left(\genfrac{}{}{0ex}{}{5}{2}\right)}{\left(\genfrac{}{}{0ex}{}{10}{2}\right)}=\boxed{\frac{2}{9}}$.