Dice Roll Probability Calculator
Pick the number of dice, sides (d4 / d6 / d8 / d10 / d12 / d20) and a target sum to instantly see the probability of rolling exactly, at least, or at most that total — plus the full sum distribution. Useful for board games, D&D / tabletop RPGs, probability homework and stats teaching.
Enter a valid dice count, sides and target sum.
Exactly this sum
16.67%
6 / 36
At least this sum
58.33%
21 / 36
At most this sum
58.33%
21 / 36
Sum distribution
Assumes fair, independent dice rolls.
Formula
P(sum = t) = ways(n, d, t) / d^n // ways = number of ways n dice of d sides yield sum t Total outcomes d^n are computed exactly with BigInt — e.g. 2d6 → 36, 3d6 → 216, 4d20 → 160,000. Expected sum E[S] = n · (d + 1) / 2 — e.g. 3d6 → 10.5.
- · Computed by dynamic programming (discrete convolution) one die at a time, with BigInt counts — so the figures are exact, not floating-point estimates.
- · Range: up to 20 dice with up to 100 sides each — well beyond the practical needs of D&D, Monopoly, Snakes & Ladders or mahjong opening rolls.
- · Classic 2d6: a sum of 7 is the most likely (6 / 36 = 16.67 %); snake eyes (1+1) and boxcars (6+6) are each 1 / 36.
- · Classic 3d6: sums 10 and 11 tie as the most likely (27 / 216 = 12.5 % each); rolling a 3 or 18 is rare (1 / 216 ≈ 0.46 %).
- · “At least” and “At most” include the target value itself, i.e. P(S ≥ t) and P(S ≤ t).
- · Assumes perfectly fair, independent dice. Real-world manufacturing bias is typically < 1 % — invisible to the naked eye.
- · References: Feller, "An Introduction to Probability Theory", Vol. I, Ch. VI; identical convolution logic to AnyDice and standard tabletop-RPG tooling.
Frequently asked
Why is 7 the most likely sum when rolling 2d6?
Two dice have 6 × 6 = 36 ordered outcomes. Six of them sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). No other total has more than five, so 7 is the unique mode at 6 / 36 ≈ 16.67 %. It is also the centre of the 2–12 range, and the distribution forms a triangle — the further you are from 7, the fewer ways there are to get there.
Does this calculator handle D&D advantage / disadvantage (max or min of two d20)?
Not directly. This tool computes the probability of a target sum; D&D advantage/disadvantage takes the max (or min) of two d20s. The math is different — e.g. P(max ≥ 15 with advantage) = 1 − (14 / 20)² = 51 %, much higher than a single d20's P(≥ 15) = 6 / 20 = 30 %. For max/min mechanics, treat each die individually rather than summing.
What happens to the distribution as you add more dice?
By the Central Limit Theorem, the sum of n dice tends toward a normal (bell-shaped) distribution as n grows. Mean and variance scale linearly with n, but the standard deviation only grows like √n — so most rolls cluster near n(d+1)/2. For example 10d6 has mean 35 and SD ≈ 5.4, so sums of 28–42 cover over 70 % of all rolls.
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