Dice Roll Probability Calculator
Calculate probability of rolling any sum with multiple dice.
Supports d4, d6, d8, d10, d12, d20, and custom dice for tabletop RPGs and board games.
Dice probability calculations determine the likelihood of outcomes when rolling one or more dice. Probability is expressed as a fraction, decimal, or percentage — all equivalent ways of showing how likely an event is.
Single die (d6) probabilities: A standard six-sided die has 6 equally likely outcomes: 1, 2, 3, 4, 5, 6.
- Probability of any single number = 1/6 ≈ 16.7%
- Probability of rolling at least 4 = 3/6 = 50%
- Probability of rolling an even number = 3/6 = 50%
Two dice — sum probabilities: With two d6 dice, there are 6 × 6 = 36 possible combinations.
- Sum of 7: 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) → 6/36 = 16.7% (most likely)
- Sum of 2: 1 way (1+1) → 1/36 = 2.8%
- Sum of 12: 1 way (6+6) → 1/36 = 2.8%
General formula for multiple dice: Expected value of n dice, each with s sides: Mean = n × (s + 1) / 2 Variance = n × (s² − 1) / 12
Example — rolling 3d6 (three six-sided dice): Mean = 3 × 7/2 = 10.5 Variance = 3 × 35/12 = 8.75, Standard deviation = 2.96
Probability of rolling at least one 6 on multiple dice: P(at least one 6 on 4 dice) = 1 − (5/6)⁴ = 1 − 0.482 = 51.8%
RPG applications:
- d20 systems: each number 1–20 has a 5% chance
- Advantage (roll 2d20, take highest): mean rises from 10.5 to 13.83
- Disadvantage (take lowest): mean drops to 7.17
Expected value in gambling: If a game pays £10 for rolling a 6 but you pay £2 to play: Expected value = (1/6 × £10) + (5/6 × £0) − £2 = £1.67 − £2 = −£0.33 per game You lose on average — the house has a 33p edge per roll.