Dice Roll Probability Calculator

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Expertly Reviewed by David Chen, CFA • Updated Oct 24, 2023

Calculate the exact probability of rolling specific sums with multiple dice. Whether you are playing Dungeons & Dragons, Settlers of Catan, or analyzing statistical variations, this dice roll probability calculator provides precise mathematical odds instantly.

Dice Roll Probability Calculator

Probability Result
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Dice Roll Probability Calculator Formula

P(S = k) = (1 / s^n) * Σ [(-1)^i * C(n, i) * C(k – s*i – 1, n – 1)]

Source: Wolfram MathWorld – Dice Probability Formulas

Variables:

  • n: The number of dice being rolled simultaneously.
  • s: The number of sides on each individual die (e.g., 6 for a standard cube).
  • k: The target total sum you are calculating the probability for.
  • P(S): The resulting probability, usually expressed as a percentage or fraction.

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What is a Dice Roll Probability Calculator?

A dice roll probability calculator is a statistical tool used to determine the likelihood of achieving a specific outcome when rolling one or more dice. While a single six-sided die is simple (16.67% per face), multiple dice create a “normal distribution” or bell curve effect where middle sums like 7 are much more common than extremes like 2 or 12.

This calculator uses combinatorial mathematics to solve for the number of ways a sum can be achieved, divided by the total possible outcomes ($s^n$). Understanding these odds is crucial for game design, risk assessment, and professional gaming strategies.

How to Calculate Dice Roll Probability (Example)

Let’s calculate the probability of rolling exactly a 4 using 2 six-sided dice:

  1. Determine Total Outcomes: For 2 dice with 6 sides, total outcomes = $6^2 = 36$.
  2. Identify Successful Combinations: To get a sum of 4, the possible rolls are (1,3), (2,2), and (3,1). That is 3 combinations.
  3. Apply Formula: Probability = $3 / 36$.
  4. Convert to Percentage: $1 / 12 \approx 8.33\%$.

Frequently Asked Questions (FAQ)

What is the most common sum on two 6-sided dice?

The sum of 7 is the most frequent result, occurring in 6 out of 36 possible combinations (16.67% chance).

Does the order of dice matter?

In terms of calculating probability, yes. Rolling a (1,2) is considered a distinct outcome from rolling a (2,1) even if the dice look identical.

What is a “Fair Die”?

A fair die is one where every side has an equal probability ($1/s$) of landing face up, indicating it is not weighted or physically biased.

Can I use this for D20 rolls?

Yes, simply change the “Sides per Die” input to 20 to calculate odds for Dungeons & Dragons mechanics.

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