Dice Probability Calculator

How it works

Sum of n dice with s sides: ways(k) from convolving n uniform 1…s dice, P(k) = ways(k) ÷ sⁿ. At least / at most / between = sum of P(k) over the range. Advantage: P(max ≤ t) = F(t)²; disadvantage: P(min ≥ t) = (1 − F(t − 1))². Keep highest/lowest m of n: exact count over every face combination. Successes: binomial, P(k) = C(n, k)·qᵏ·(1 − q)ⁿ⁻ᵏ with q = matching faces ÷ s. Reroll faces ≤ r once: P(face) = [face > r]/s + r/s².

Adding several dice does not give a flat distribution: totals bunch toward the middle because there are many more ways to make a middling sum than an extreme one. With two six-sided dice a 7 can be made six ways out of 36, while a 2 or a 12 has just one way each. This calculator builds the exact distribution for any number of dice and sides, then answers the question you ask — exactly a total, at least, at most, or anywhere between two numbers — as a percentage, as 1-in-N odds and, for ordinary sums, as a count of favourable outcomes. It also handles the ways tabletop games read dice. A modifier shifts every total, as in a d20 attack roll with +5. Advantage and disadvantage roll the whole set twice and keep the higher or lower total, so a d20 with advantage reaches 11 or more 75% of the time instead of 50%. Keep-highest and keep-lowest cover 4d6 drop the lowest for ability scores, or 3d20 keep the highest. Counting successes treats each die as a hit when it shows a chosen face or better, as in dice-pool systems. In More options, rerolling 1s (or 1s and 2s, like Great Weapon Fighting) once changes each die’s odds before they are added. The table lists every possible result with its exact, at-least and at-most chance, and the chart shows the shape of the distribution.

Worked example

For 2d6 summing to exactly 7: there are 6 favourable combinations out of 36 equally likely rolls, so P = 6 ÷ 36 = 16.67%, or about 1 in 6. Asking instead for "at least 10" adds the ways to roll 10, 11 and 12 (3 + 2 + 1 = 6 ways), again 16.67%. A d20 attack with +5 against armour class 15 needs a natural 10 or better: 11 of 20 faces, 55%. With advantage it becomes 1 − 0.45² = 79.75%, and with disadvantage 0.55² = 30.25%. Rolling 4d6 and dropping the lowest gives an average ability score of 12.24, and an 18 comes up 1.62% of the time (21 of the 1,296 rolls).

Sources: Dice — probability distributions (Wikipedia)

Frequently asked questions