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.
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.