Number of times a dice must be thrown to get $6$ ten times before getting $5$ eight times

Question

Well, the title says it all. How are we supposed to start this?

I have calculated a recurrence relation for this $E(n, m)=\frac{4}{6}(1+E(n, m))+\frac{1}{6}(1+E(n-1, m))+\frac{1}{6}(1+E(n, m-1))$.

Is this the correct way and is there a better way to solve this?