# Rolling a die (many dice)

## 3 - Sample Space

, and the random variable of interest is the sum S of the numbers on the two dice, then S is a discrete random variable whose distribution is described by the probability mass function.

## 4 - Random Variable

A discrete random variable can also be used to describe the process of rolling dice and the possible outcomes.

### 4.1 - Sum of two dice

Example if the random variable X is the sum S of the numbers on the two dice.

Given that the sample space for one dice (representing the numbers on one die) is $$n_i = {1, 2, 3, 4, 5, 6}$$ the random variable function maps the pair to the sum: $$X((n_1, n_2)) = n_1 + n_2$$ and its probability mass function (given the dice are fair) is

$$f_X(S) = \frac{\min(S-1, 13-S)}{36}, \text{ for } S \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$$