jmc

Let's name the two numbers $a$ and $b.$ We want the probability that $ab<a+b,$ $\Rightarrow ab-a-b<0$, or $(a-1)(b-1)<1$ (applying Simon's Favorite Factoring Trick). This inequality is satisfied if and only if $a=1$ or $b=1.$ When $a=1$, $b$ can equal $1$ through $5$, and when $b=1$ and $a\ne 1,$ $a$ can equal $2$ through $5$. There are $5^2=25$ ways to choose $a$ and $b,$ so the probability is $\frac{5+4}{25}=\boxed{\frac{9}{25}}.$