jmc

There are $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$ pairs of plates possible in all. Exactly $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ of these pairs are both red plates and $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\frac{16}{36}=\boxed{\frac{4}{9}}$.