# Probability of obteining sum "S" if one roll dice "N" times

#### Rafael

We have fair "m" sided "n" dice.
What is the probability (or/and probability density function) for obtaining the sum of numbers on dice "S"
after rolling them (together) "N" times. I need probability dependence on "N". "m","n" and "S" are parameters.

#### Plato

MHF Helper
We have fair "m" sided "n" dice.
What is the probability (or/and probability density function) for obtaining the sum of numbers on dice "S"
after rolling them (together) "N" times. I need probability dependence on "N". "m","n" and "S" are parameters.
I have no idea what "We have fair "m" sided "n" dice" could mean.
Please state clearly what it means. How many sides? How many dice? How many times are they tossed?

• 1 person

#### Rafael

We have "n" dice.
Each have "m" sides.
They are tossed "N" times.

#### Plato

MHF Helper
We have "n" dice.
Each have "m" sides.
They are tossed "N" times.

We must assume that the sides are numbered $$\displaystyle 1,~2,\cdots,~m$$.

If you expand the polynomial $$\displaystyle \left( {\sum\limits_{k = 1}^m {x^k } } \right)^n$$ the term $$\displaystyle Jx^H$$ tells us if we toss n m-sided dice then there are $$\displaystyle J$$ ways to get the sum $$\displaystyle H$$.

Look at this webpage.
You see that models tossing five nine sided dice. The term $$\displaystyle 330x^{38}$$ tells us that there are 330 ways that the sum of the five will equal 38.

So $$\displaystyle \mathcal{P}(S=38)=\frac{330}{9^5}$$.

From there on it is a simple matter of using binomial distribution on N trials.

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• 1 person

#### Rafael

The PDF independent of number of trials will be
Dice - Wikipedia, the free encyclopedia last expression in the section of "Probability"
Is that right? Is there a closed form expression which will also include number of trials "N"?