# Permutations for the possible sum values of successful 4 sided dice rolls (with unique values)

Scenario:

• 4 sided dice with values 2.7 - 3.1 - 3.5 - 3.9
• Even probability for value each number.
• Dice is rolled 6 times.

Questions:

1a) If adding the number of the new roll to the number of the previous roll, how many unique sum values are there at each roll? (is this a permutations calculation?)

example 1, Roll 1 = 2.7, Roll 2 = 3.1: One possible Sum after 2 rolls is 5.8

Example 2, Roll 1 = 3.1, Roll 2 = 2.7: Different route but same possible Sum of 5.8

1b) How would I automate the generation of these values, is it simple to do in excel?

Thanks.

• Welcome to Math Stackexchange. "Solve this for me"-questions are not perceived well on this site. Please provide some information on how this problem came up and how you have attempted to solve it so far. You can find more information on how to ask a good question here: math.meta.stackexchange.com/questions/9959/… Commented Apr 24, 2023 at 9:32
• Hello, thanks for the response. I do not have a great understanding of statistics, and any attempt I have made to the above questions have be using the permutations formula (n!/(n-k)!. But I cannot be sure that I have applied the formula correctly as I am confused about if/how this formula will work when applying it to successive dice rolls. I did begin the create tables in excel to essentially simulate the possibilities after every roll, but it quickly became apparent that as the first roll has 4 outcomes and the second has 16 (7 unique), 3 rolls onwards has too many to list manually Commented Apr 24, 2023 at 10:08

We can simplify this problem by applying a transformation, because the values on your die are a simple linear function of the values on a standard die labeled $$1$$ to $$4$$. That is, if you take a die labeled $$[1,2,3,4]$$, and replace each face labeled $$x$$ with $$0.4x+2.3$$, then the resulting labels are $$[2.7,3.1,3.5,3.9]$$. It should be obvious that six copies of a standard die can roll any number between $$6$$ and $$24$$ inclusive. This means that six copies of your die can roll any number of the form $$0.4 x+6\times 2.3$$, where $$x$$ can be any integer between $$6$$ and $$24$$, inclusive.
This implies that for any sum, $$S$$, the number of ways that six of your dice can sum to $$S$$ is equal to the number of ways that a standard die with labels $$[1,2,3,4]$$ can sum to $$2.5(S - 6\times 2.3)$$ All that remains is, how do you compute the number of ways to roll each possible sum with six standard four-sided dice? This can indeed be done in Excel.
• Enter the number $$1$$ into cells $$A5,A6,A7,A8$$.
• Enter the formula =SUM(A1:A4) into cell $$B5$$.
• Click and drag to fill the formula in $$B5$$ to all of the cells between $$B5$$ and $$F28$$.
The column $$F$$ will now contain the number of ways to roll each possible sum with six standard dice. Specifically, row number $$r$$ has the number of ways to roll $$r-4$$. If you then divide each entry in column $$F$$ by $$4^6$$, you get the probability of attaining each roll.