jmc

Let's name the two numbers $a$ and $b.$ We want the probability that $ab>a+b,$ or $(a-1)(b-1)>1$ using Simon's Favorite Factoring Trick. This inequality is satisfied if and only if $a\ne 1$ or $b\ne 1$ or $a\ne 2\ne b$. There are a total of $16$ combinations such that $a\ne 1$ and $b\ne 1$. Then, we subtract one to account for $(2,2)$, which yields $15$ total combinations out of a total of 25, for a probability of $\boxed{\frac{3}{5}}$