Game

[pug]

I'm trying to make a game where you have 2 6 sided dice, each with 3 different colors on them, and the goal of the game is to roll as many unique combinations of rolls within 9 rolls. I know there is a very low chance of only rolling one unique possibility or rolling all 9 different combinations of rolls, but I can't figure out the chance of rolling 2 unique rolls to 8 unique rolls and everything between. Is there some kind of calculator I can use to solve this? How do I figure this out?

 

[pka]

I'm trying to make a game where you have 2 6 sided dice, each with 3 different colors on them, and the goal of the game is to roll as many unique combinations of rolls within 9 rolls. I know there is a very low chance of only rolling one unique possibility or rolling all 9 different combinations of rolls

I am not clearly understanding the setup. There are two dice, say their faces are coloured RED, ORANGE & GREEN (two each).
I get this table of outcomes: \(\begin{array}{*{20}{c}} {}&|&R&O&G \\ \hline R&|&{R,R}&{R,O}&{R,G} \\ O&|&{O,R}&{O,O}&{O,G} \\ G&|&{G,R}&{G,O}&{G,G} \end{array}\)
There are nine elementary events. Thus each has probability \(\dfrac{1}{9}\).
You wrote "I can't figure out the chance of rolling 2 unique rolls to 8 unique rolls and everything between. Is there some kind of calculator I can use to solve this? How do I figure this out?" I don't understand that question? Is my model correct?

 

[Dr.Peterson]

One important question is whether the dice both have the same three colors, or if there are six distinct colors involved. In the latter case, you are right that there are 9 possible outcomes from rolling both dice at once. In the former case, there are only 6 distinguishable outcomes, which are not equally likely. (E.g. RO and OR are indistinguishable.)

I would use the word "distinct" where you use "unique", if I'm understanding you correctly.

I think you are asking for the probability distribution of the random variable X = number of distinct outcomes among 9 rolls.

 

[pka]

One important question is whether the dice both have the same three colors, or if there are six distinct colors involved. In the latter case, you are right that there are 9 possible outcomes from rolling both dice at once. In the former case, there are only 6 distinguishable outcomes, which are not equally likely.

While I wondered the same about different colours, I doubt that we receive an answer. Look at this:

Add to that the students who ask for help, receive it, and never return to see it.

Prof Kurtz voiced one of my pet peeves. I think that this will be one of those cases.
That said, I just assumed that these dice were as regular dice were two identically marked die.
So each die has two faces coloured one of three distinct colours.
With a regular set of two dice there are three ways to toss a sum of ten: \((5,5),~(6,4),~\&~(4,6)\) .
We do see that while \((6,4),~\&~(4,6)\) are different pairs the sum is the same.
So \(\mathcal{P}(S=10)=\dfrac{3}{36}\).