jmc

When the first dice is $1$, there are no possibilities for the second dice. When the first dice is $2$, the second dice can be $6$, giving one possibility. When the first dice is $3$, the second dice can be $4,5,6$, giving three possibilities. When the first dice is $4$, the second dice can be $3,4,5,6$, giving four possibilities. When the first dice is $5$, the second dice can be $3,4,5,6$, giving four possibilities. When the first dice is $6$, the second dice can be $2,3,4,5,6$, giving five possibilities. The total number of ways for the product to greater than 10 is $1+3+4+4+5=17$ and the total number of possibilities is $(6)(6)=36$, yielding a probability of $\boxed{\frac{17}{36}}$.