This problem involves conditional probability.

Consider two dice $A$ and $B$, where the largest number on $A$ is $N$. Then $P(A< B)$ is
$$
P(A< B) = \sum^N_{m=1}P(A< B|A=m)P(A=m)
$$
Of course, in this expression $P(A=m)$ is zero if the integer $m$ is not present on the die.

With problems such as these, don't be afraid to start with a period of experimentation: just choose any numbers to begin with and explore their properties. The structure of the problem will soon start to emerge.