### RPG Dice Standard notation

Probability mass function of a roll of 3d6 and probability density function of the normal distribution with the same mean and standard deviation.

In most tabletop role-playing games, die rolls required by the system are given in the form AdX. *A* and *X* are variables, separated by the letter *d*, which stands for *die* or *dice*. The letter *d* is most commonly lower-case, but some forms of notation use upper-case *D* (non-English texts can use the equivalent form of the first letter of the given language’s word for “dice”, but also often use the English “d”).

*A*is the number of dice to be rolled (usually omitted if 1).*X*is the number of faces of each dice.

For example, if a game calls for a roll of d4 or 1d4, it means “roll one 4-sided die.”

If the final number is omitted, it is typically assumed to be a six, but in some contexts, other defaults are used.

3d6 would mean “roll three six-sided dice.” Commonly, these dice are added together, but some systems could direct the player use them in some other way, such as choosing the best die rolled.

To this basic notation, an additive modifier can be appended, yielding expressions of the form AdX+B. The plus sign is sometimes replaced by a minus sign (“−”) to indicate subtraction. *B* is a number to be added to the sum of the rolls. So, 1d20−10 would indicate a roll of a single 20-sided die with 10 being subtracted from the result. These expressions can also be chained (e.g. 2d6+1d8), though this usage is less common. Additionally, notation such as AdX−L is not uncommon, the *L* (or *H*, less commonly) being used to represent “the lowest result” (or “the highest result”). For instance, 4d6−L means a roll of 4 six-sided dice, dropping the lowest result. This application skews the probability curve towards the higher numbers, as a result a roll of 3 can only occur when all four dice come up 1 (probability 1/1,296), while a roll of 18 results if *any* three dice are 6 (probability 21/1,296 = 7/432).

Rolling three or more dice gives a probability distribution that is approximately Gaussian, in accordance with the central limit theorem.

### Tabletop Role-Playing Game

A **tabletop role-playing game** (typically abbreviated as **TRPG** or **TTRPG**), also known as a **pen-and-paper role-playing game**, is a form of role-playing game (RPG) in which the participants describe their characters’ actions through speech. Participants determine the actions of their characters based on their characterization, and the actions succeed or fail according to a set formal system of rules and guidelines. Within the rules, players have the freedom to improvise; their choices shape the direction and outcome of the game. Unlike other types of role-playing games, tabletop RPGs are often conducted like radio drama: only the spoken component of a role is acted.

### Probability Distribution

In probability theory and statistics, a **probability distribution** is the mathematical function that gives the probabilities of occurrence of different possible **outcomes** for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

For instance, if *X* is used to denote the outcome of a coin toss (“the experiment”), then the probability distribution of *X* would take the value 0.5 (1 in 2 or 1/2) for *X* = heads, and 0.5 for *X* = tails (assuming that the coin is fair). Examples of random phenomena include the weather condition in a future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc.

### Normal Distribution

In probability theory, a **normal** (or **Gaussian** or **Gauss** or **Laplace–Gauss**) **distribution** is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

The parameter μ is the mean or expectation of the distribution (and also its median and mode), while the parameter σ is its standard deviation. The variance of the distribution is σ 2. A random variable with a Gaussian distribution is said to be **normally distributed**, and is called a **normal deviate**.

### Standard Deviation

In statistics, the **standard deviation** is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

A plot of normal distribution (or bell-shaped curve) where each band has a width of 1 standard deviation – See also: 68–95–99.7 rule.

### 68-95-99.7

In statistics, the **68–95–99.7 rule**, also known as the **empirical rule**, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.

The Bell Curve of Life.