Dice Roller Simulator
Overview
The Dice Roller Simulator provides a comprehensive virtual dice rolling experience with detailed statistical analysis. Roll single dice or simulate thousands of rolls to understand probability distributions and dice mechanics.
Features
- Multiple Dice Types: d4, d6, d8, d10, d12, d20, d100, and custom sizes
- Flexible Rolling: Roll any number of dice with modifiers
- Special Options: Drop lowest/highest, keep highest (advantage/disadvantage)
- Visual Dice Display: Animated dice showing individual results
- Statistical Analysis: Mean, median, mode, standard deviation
- Distribution Visualization: Histograms and cumulative probability curves
- Probability Tables: Detailed breakdown of results
Dice Notation
Standard dice notation (XdY+Z): - X: Number of dice - d: Stands for “dice” - Y: Sides per die - +Z: Modifier (optional)
Examples: - 2d6: Two six-sided dice (like in Monopoly) - 1d20: One twenty-sided die (common in D&D) - 3d6+2: Three six-sided dice plus 2 - 4d6 drop lowest: Roll four d6, drop the lowest (D&D ability scores)
Common Dice Types
Standard Gaming Dice
- d4: Tetrahedral (pyramid), range 1-4
- d6: Cube (standard die), range 1-6
- d8: Octahedron, range 1-8
- d10: Pentagonal trapezohedron, range 1-10
- d12: Dodecahedron, range 1-12
- d20: Icosahedron, range 1-20 (iconic in RPGs)
- d100: Percentile die, range 1-100
Special Roll Modes
Drop Lowest - Roll all dice, discard the lowest - Common in RPGs: 4d6 drop lowest for ability scores - Increases average roll (higher stats)
Drop Highest - Roll all dice, discard the highest - Less common, used for special mechanics
Keep Highest (Advantage) - Roll multiple dice, keep only the highest - D&D 5e advantage: roll 2d20, keep highest - Significantly improves success chance
Probability Mathematics
Single Die
For a single fair die with n sides: - Each outcome has probability 1/n - Mean (expected value) = (n + 1) / 2 - For d6: mean = 3.5
Multiple Dice
When rolling multiple dice: - Results cluster around the mean (Central Limit Theorem) - More dice = narrower, more predictable distribution - Extreme values (very low or very high) become rare
Common Distributions
2d6 (like in Monopoly, Settlers of Catan): - Range: 2-12 - Mean: 7 - Most likely: 7 (16.67% chance) - Least likely: 2 or 12 (2.78% each)
3d6: - Range: 3-18 - Mean: 10.5 - Distribution approaches normal (bell curve)
1d20 (D&D): - Range: 1-20 - Mean: 10.5 - Uniform distribution (each outcome equally likely)
Applications
Gaming
- Tabletop RPGs (D&D, Pathfinder, etc.)
- Board games (Monopoly, Risk, etc.)
- Wargaming and miniature games
- Online gaming mechanics
Education
- Teaching probability concepts
- Demonstrating the Central Limit Theorem
- Understanding expected values
- Exploring combinatorics
Decision Making
- Random selection with weighted outcomes
- Fair dispute resolution
- Procedural generation in games
Statistics
- Monte Carlo simulations
- Testing random number generators
- Probability modeling
Understanding the Statistics
Mean: Average result over many rolls. For XdY: mean = X × (Y+1)/2
Median: Middle value. For symmetric distributions, approximately equal to mean.
Mode: Most frequent result. Single peak for multiple dice.
Standard Deviation: Spread of results. Larger SD = more variable outcomes.
Distribution Shape: Single die = uniform. Multiple dice = bell curve (normal distribution).
Tips for RPG Players
- Advantage (2d20 keep highest): ~66% chance to roll 11+ (vs 50% normally)
- Disadvantage (2d20 keep lowest): ~66% chance to roll 10- (vs 50% normally)
- 4d6 drop lowest: Average ~12.24 (vs 10.5 for 3d6)
- Critical hits (nat 20): 5% chance on 1d20