Virtual Dice Roller

Tabletop RPG Polyhedral Probability & Combinatorics

Polyhedral dice probability follows discrete uniform distributions:

• Single Die Expected Value (Mean μ): μ_1 = (S + 1) / (2)

• d4: μ = 2.5

• d6: μ = 3.5

• d8: μ = 4.5

• d10: μ = 5.5

• d12: μ = 6.5

• d20: μ = 10.5

• d100: μ = 50.5

• Multi-Die Sum: μ_total = (N × μ_1) + Modifier

• D&D 5e Ability Stat Rolling (4d6 Drop Lowest): Rolling four 6-sided dice and discarding the lowest single value raises the expected average ability score from 10.5 (on 3d6) to 12.24, significantly reducing the risk of sub-optimal stats.

How to Use This Calculator

Enter your Number of Dice (1 to 20) (e.g. 1), Die Type (Number of Sides) — choose from d20 (20-sided / D&D standard), d6 (6-sided / Classic cube), d4 (4-sided / Tetrahedron), Roll Modifier / Bonus (+/-) (e.g. 5), Advantage / Drop Rule — choose from Keep All (Standard roll), Advantage / Drop Lowest 1 (4d6 stat rolls), Disadvantage / Drop Highest 1, and Tabletop Gaming Benchmark — choose from Custom Dice Notation, D&D 5e D20 Check (1d20 + 5 modifier), D&D Ability Stat (4d6 drop lowest). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Conversion factors (to SI base unit or reference unit):

sides: 20

mod: 5

d4: 4

d6: 6

d8: 8

d10: 10

d12: 12

d20: 20

d100: 100

Worked Example

A D&D wizard casting an 8d6 Fireball spell rolls [5, 6, 4, 3, 6, 2, 5, 4] for a total of 35 fire damage (close to the theoretical expected mean of 28.0).

Understanding Your Result

This calculation gives a precise result based on the inputs you provided. For practical decisions, consider whether the result changes meaningfully with slight variations in your inputs — if a 5–10% change in input causes a significant output change, gather more precise input data before deciding. Use this tool to quickly compare multiple scenarios side by side.