Find the probability mass function: four-sided die and six-sided die

In an experiment, a four-sided die and a six-sided die are rolled. These dice both have the numbers you would expect on them. Let $$Z$$ be a random variable that represents the absolute value of the difference. (If a $$4$$ is rolled and a $$1$$ is rolled, then $$Z=3$$.)

What is the probability mass function of $$Z$$?

For the four-sided die:

$$f(1)=f(2)=f(3)=f(4)=\dfrac{1}{4}$$

For the six-sided die:

$$f(1)=f(2)=f(3)=f(4)=f(5)=f(6)=\dfrac{1}{6}$$

I don't know where to go from here. I have an example of this problem using only one die, not two.

• There are only $24$ possibilities for the outcomes on the two dice; list them and take the differences. – saulspatz Oct 3 '18 at 14:01

The biggest difference is $$5$$ and the smallest difference is zero.
Let $$X$$ be the outcome of the $$4$$ sided dice and $$Y$$ be the outcome of the $$6$$ sided dice.
\begin{align} Pr(Z=z) &= Pr(|X-Y|=z)\\ &= \sum_{x=1}^4 Pr(|X-Y|=z|X=x)Pr(X=x) \end{align}