Average Dice Calculator

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Reviewed by David Chen, CFA

This calculator is based on established probability and statistical formulas for dice rolls, ensuring accurate expected value computation.

Welcome to the definitive **average dice calculator**. Quickly determine the expected (mean) value of any combination of dice rolls, essential for tabletop games, probability studies, and statistics.

Average Dice Calculator

Average Dice Calculator Formula

$$ \text{Average Roll} = D \times \frac{(1 + N)}{2} $$

*Where D is the Number of Dice, and N is the Number of Sides per Die.*

Formula Sources: Math StackExchange, AnyDice

Variables Explained

  • Number of Dice (D): The quantity of dice being rolled simultaneously (e.g., 3).
  • Number of Sides Per Die (N): The number of faces on a single die (e.g., 4, 6, 8, 10, 12, 20).
  • Average Roll: The expected value or mean outcome of the combined roll.

What is an Average Dice Roll?

The average dice roll, also known as the expected value, represents the theoretical mean outcome if you were to roll the dice an infinite number of times. It is the single most likely result you will average over the long run.

For a single die (D=1), the average is simply the midpoint between the minimum possible result (1) and the maximum possible result (N), hence the simple formula $\frac{(1 + N)}{2}$. When you introduce multiple dice, the expected value scales linearly with the number of dice rolled.

How to Calculate the Average Dice Roll (Example)

Let’s calculate the average roll for three six-sided dice (3d6).

  1. Identify Variables:
    • Number of Dice (D) = 3
    • Number of Sides Per Die (N) = 6
  2. Calculate the Average of a Single Die: $$\text{Single Die Average} = \frac{(1 + N)}{2} = \frac{(1 + 6)}{2} = \frac{7}{2} = 3.5$$
  3. Multiply by the Number of Dice: $$\text{Total Average} = D \times \text{Single Die Average} = 3 \times 3.5 = 10.5$$
  4. Final Result: The average (expected) roll for three six-sided dice is **10.5**.

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Frequently Asked Questions (FAQ)

  • What is the difference between average and expected value?

    In the context of dice rolling, the terms “average” and “expected value” are interchangeable. They both refer to the mean result calculated using the probability distribution of all possible outcomes.

  • Why is the average roll often a decimal (e.g., 3.5)?

    The average is a theoretical value representing the mean result over many trials. While a single roll must be a whole number, the statistical average of all outcomes is often a decimal.

  • Does the minimum value of a die always have to be 1?

    Standard dice start at 1. If you are calculating the average for a custom die that starts at 0 (e.g., 0-5), you would adjust the formula to $\frac{(\text{Min} + \text{Max})}{2}$. This calculator assumes a standard minimum value of 1.

  • How does the formula change for a dice pool (success counting)?

    This formula calculates the average sum of the dice faces. For dice pool systems (where you count the number of dice exceeding a target number), a different, more complex binomial probability calculation is required.

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