I'm trying to calculate the probability distribution for what I'm calling a 'super roll'. A super roll is made by throwing all seven of the standard DnD dice and adding up their values.
The standard dice are:
- 1d4
- 1d6
- 1d8
- 1d10
- 1d10 × 10 [10, 20, 30... 90, 100]
- 1d12
- 1d20
I can use this equation to find $P(X_n=p)$ for a single die. $$P(p,n,s) = \frac{1}{s^n} \sum_{k=0}^{\lfloor(p-n)/s\rfloor} (-1)^k {n \choose k} {p-sk-1 \choose n-1}$$
But I don't know how to go about getting a combined distribution when I add new dice with different sides, s. I also imagine the 1d10 × 10 will cause a problem too and I imagine the above equation will need to be modified somehow.
I've tried playing around with some algebra but I'm not really getting anywhere. Can anyone help me in getting me started with this problem?
You compute the convolution of all these discrete uniform laws. Convolution is explained, for example, in this wikipedia article.