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:


For the six-sided die:


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

  • 1
    $\begingroup$ There are only $24$ possibilities for the outcomes on the two dice; list them and take the differences. $\endgroup$ – 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}


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