Coin Flip Simulator
Flip a virtual coin one or multiple times and track heads-versus-tails counts and percentages for quick experiments or games.
Dice Roll Simulator
Roll virtual dice with 4, 6, 8, 10, 12, or 20 sides, combine multiple dice, and review totals, averages, and distributions.
Planning notes, formulas, and examples
About the Dice Roll Simulator
The Dice Roll Simulator lets you roll virtual dice of any type — standard d6, d20 for tabletop RPGs, or custom-sided dice. Roll one or many dice and see individual results, totals, and statistics.
Dice rolling is essential in board games, tabletop roleplaying games (D&D, Pathfinder), probability education, and randomized decision-making. This simulator supports d4, d6, d8, d10, d12, d20, and custom dice.
Roll large numbers of dice to explore probability distributions. The sum of multiple dice follows a roughly bell-shaped distribution, illustrating the central limit theorem in action.
When This Page Helps
No physical dice needed. Roll any combination of dice, track totals, and explore probability distributions without setup.
How to Use the Inputs
- Select the type of die (d4, d6, d8, d10, d12, d20, or custom).
- Enter the number of dice to roll.
- Click Roll to simulate.
- View individual results and the total.
- Roll repeatedly to see distribution patterns.
Formula used
For a fair d-sided die:
P(any face) = 1/d
Expected value per die = (d+1)/2
Variance per die = (d²−1)/12Example Calculation
Result: e.g. 3 + 5 = 8
Two six-sided dice rolled. Expected sum = 7 (average of 3.5 per die). Sums range from 2 to 12, with 7 being the most common.
Tips & Best Practices
- The expected value of a d6 is 3.5, so the average sum of 2d6 is 7.
- d20 is the iconic die for D&D attack rolls and saving throws.
- The sum of multiple dice approximates a normal (bell curve) distribution.
- 2d6 has 36 possible outcomes, not 12; the sums are not equally likely.
- 7 is the most common sum with 2d6 (6 ways out of 36).
- Advantage in D&D: roll 2d20 and take the higher — average is about 13.8.
Probability Distributions of Dice
A single die has a uniform distribution. The sum of two dice forms a triangular distribution (2d6 peaks at 7). As more dice are added, the distribution becomes increasingly bell-shaped.
Dice in Gaming
Tabletop RPGs use polyhedral dice sets (d4 through d20). Wargames use handfuls of d6. Casino games (craps) are built entirely around 2d6 probability.
Dice Notation
"3d8+5" means roll 3 eight-sided dice and add 5 to the total. This concise notation is standard in tabletop gaming and game design documentation.
Consistent practice with varied problems builds computational fluency and deepens conceptual understanding that transfers across many technical fields.
Sources & Methodology
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Frequently Asked Questions
D&D uses d4, d6, d8, d10, d12, and d20. The d20 is used for most ability checks and attack rolls. Damage dice vary by weapon.
There are 6 ways to roll a 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) out of 36 possible outcomes. That is 16.7%, the highest probability for any sum.
As you add more dice, their sum's distribution approaches a bell curve (normal distribution), regardless of the individual die shape. This is the central limit theorem.
A loaded or weighted die has unequal probabilities for each face. This simulator uses fair dice with equal probability for all faces.
Roll 2d20 and take the higher (advantage) or lower (disadvantage). Advantage increases the average from 10.5 to about 13.8; disadvantage lowers it to about 7.2.
Yes, the custom option lets you set any number of sides. However, physical dice with more than 120 faces are impractical; virtual dice have no such limit.
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Generate random numbers within a custom range, choose integers or decimals, and control count for testing, sampling, raffles, or classroom use.