What is the probability distribution when a fair die is rolled once?

When a fair die is rolled once, the possible outcomes for the up-turned face are the integers from 1 to 6.
The sample space $S = \{1, 2, 3, 4, 5, 6\}$.
The total number of possible outcomes is 6.

Let $X$ be the random variable representing the number on the up-turned face. The possible values for $X$ are $x \in \{1, 2, 3, 4, 5, 6\}$.

Since the die is fair, each outcome has an equal probability of occurring.
The probability of any specific outcome $x$ is given by:

For each value of $x$ from 1 to 6, there is only one favorable outcome (e.g., rolling a 1, rolling a 2, etc.).
Therefore, for each $x \in \{1, 2, 3, 4, 5, 6\}$:

The probability distribution can be shown in a table:

The sum of all probabilities is $\frac{1}{6} \times 6 = 1$, which confirms it is a valid probability distribution.

The probability distribution for the up-turned faces is: