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

Question

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.

Answer 1

Guide:

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}