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Math

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.

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.

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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