# Thread: Dice Problem

1. ## Dice Problem

Suppose you roll a red die, a green die, and a blue die. Let X denote the number of different faces showing on the three die. Let Y denote the maximum of the three faces. For example, if you roll 3 with the red die, 5 with the green die, and 3 with the blue die, then X = 2 because there are two different faces (3 and 5) showing on the dice, and Y = 5 (the largest face). Find the joint probability density function for X and Y. Please use a chart to display your pdf.

I am confused as to how this chart helps me find the pdf.

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3. ## Re: Dice Problem

Originally Posted by ballerninja29
[COLOR=#333333][FONT=Aspira]Suppose you roll a red die, a green die, and a blue die. Let X denote the number of different faces showing on the three die. Let Y denote the maximum of the three faces. pdf
It is possible that $X\in\{1,2,3\}~\&~Y\in\{1,2,3,4,5,6\}$
There are $6^3=216$ possible outcomes. Of those only $6$ yield $X=1$ WHY?
Now $\mathscr{P}(Y=1)=\dfrac{1}{6^3}$ BUT $\mathscr{P}(X=3)=\dfrac{6\cdot 5\cdot 4}{6^3}$

What really complicates matters is trying to calculate $\mathscr{P}(X=2,~Y=5)=~?$ All of these would work.
We need at least one five for the maximum, with that we need at least another and one other number or two of the same number less than five.
$\begin{array}{l}5,5,1\\5,5,2\\5,5,3\\5,5,4\\ \vdots \\5,1,1 \end{array}$

You should be able to see by now how very complicated this question is.
If you know anything about generating functions then have a look at this.
It may help you to sort things out??