jmc

We have $2$ dice with $2$ evens and $4$ odds on each die. For the sum to be even, the 2 rolls must be $2$ odds or $2$ evens. Ways to roll $2$ odds (Case $1$): The total number of ways to obtain $2$ odds on 2 rolls is $4\ast 4=16$, as there are $4$ possible odds on the first roll and $4$ possible odds on the second roll. Ways to roll $2$ evens (Case $2$): Similarly, we have $2\ast 2=4$ ways to obtain 2 evens. Probability is $\frac{20}{36}=\frac{5}{9}$, or $\boxed{\frac{5}{9}}$.