Dice Roller

1. **Select the number of sides (n):** Choose a standard die type (d4, d6, d8, d10, d12, d20, d100) or enter a custom number of sides for a non-standard die.
2. **Set the number of dice (k):** Enter how many dice you want to roll simultaneously (e.g., 2 for a classic 2d6 roll).
3. **Add a modifier (optional):** Enter a positive or negative integer modifier to add to the total (common in tabletop RPGs like D&D 5e).
4. **Click 'Roll':** The calculator instantly generates a random result for each individual die and displays both the individual values and the grand total.
5. **Read your results:** Review each die's result and the summed total. Re-roll as many times as you need — each roll is completely independent and fair.

S = X₁ + X₂ + … + Xₖ,  where each Xᵢ ~ DiscreteUniform(1, n)

Expected Value: E[S] = k × (n + 1) / 2
Minimum: S_min = k
Maximum: S_max = k × n

• S — The total sum of all dice rolled in a single throw.
• k — The number of dice being rolled (e.g., k = 3 means three dice).
• n — The number of sides on each die (e.g., n = 6 for a standard d6, n = 20 for a d20).
• Xᵢ — The individual outcome of the i-th die, a random integer uniformly distributed between 1 and n (inclusive).
• E[S] — The expected (average) total when rolling k dice each with n sides, equal to k × (n + 1) / 2.

Die 1 rolls: 4
Die 2 rolls: 6
Die 3 rolls: 2

Total S = 4 + 6 + 2 = 12

Expected value check: E[S] = 3 × (6 + 1) / 2 = 3 × 3.5 = 10.5
Range: minimum = 3, maximum = 18

Result: Total Roll = 12 | Individual results: [4, 6, 2]

The result of 12 means the sum of all three d6 dice was 12, which falls above the average of 10.5 for a 3d6 roll. In Dungeons & Dragons, this would represent a solid ability score — not exceptional, but above average. Individual die results are shown so you can apply any game rule that targets specific dice (e.g., 'drop the lowest').

Understanding Dice Probability

Uniform Distribution

A fair die is a model of a discrete uniform distribution: every face from 1 to n has exactly a 1/n probability of appearing on any given roll. A standard d6 has a 1-in-6 (≈16.67%) chance for each face.

Rolling Multiple Dice — The Sum Distribution

When you roll two or more dice and add the results, the distribution of the total is no longer uniform. It becomes a roughly bell-shaped (normal-like) distribution due to the Central Limit Theorem. For example, with 2d6 the most likely result is 7 (six ways to make it out of 36 total combinations), while a 2 or 12 each have only a 1-in-36 chance.

Common Dice in Tabletop RPGs

| Die | Sides | Min | Max | Average | |-----|-------|-----|-----|---------| | d4 | 4 | 1 | 4 | 2.5 | | d6 | 6 | 1 | 6 | 3.5 | | d8 | 8 | 1 | 8 | 4.5 | | d10 | 10 | 1 | 10 | 5.5 | | d12 | 12 | 1 | 12 | 6.5 | | d20 | 20 | 1 | 20 | 10.5 | | d100| 100 | 1 | 100 | 50.5 |

Modifiers

Many RPG rules add a flat modifier to a roll (e.g., '+3 Strength modifier'). The modifier shifts the entire distribution up or down by a constant — it does not change the spread (standard deviation) of the roll.

Standard Deviation of a Single Die

The variance of a single die with n sides is (n² − 1) / 12, so the standard deviation is √((n² − 1) / 12). For a d6, SD ≈ 1.71; for a d20, SD ≈ 5.77.

Fairness and Randomness

Digital dice rollers use pseudorandom number generators (PRNGs). High-quality implementations use cryptographically secure PRNGs to ensure results are statistically indistinguishable from a truly fair die.

• **Confusing the number of dice with the number of sides:** '2d6' means 2 dice each with 6 sides — not 26-sided dice or 2 dice with different side counts.
• **Forgetting the minimum is not zero:** A d6 rolls 1–6, not 0–5. Always remember the minimum result is 1 per die, so k dice have a minimum sum of k, not 0.
• **Assuming multiple dice produce a uniform distribution:** Rolling 2d6 does NOT give each total equal probability. The total of 7 is six times more likely than a total of 2 or 12.
• **Applying modifiers incorrectly:** A '+5 modifier' adds 5 to the final sum, not to each individual die. Don't multiply the modifier by the number of dice.
• **Re-rolling old results instead of fresh ones:** Each roll is fully independent; past results have no influence on future rolls (the 'gambler's fallacy').

• Probability and Statistics for Engineering and the Sciences — Cengage Learning — Jay L. Devore (textbook reference for discrete uniform distribution)
• Discrete Uniform Distribution — Wolfram MathWorld — Wolfram Research / MathWorld

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