Dice probability calculator
How likely is a sum of n dice? Exact odds, the full table, and keep-the-best options.
How do I find the probability of a dice sum?
Count the rolls that give the sum and divide by all possible rolls, sⁿ for n dice with s sides. With two dice, a sum of 7 happens in 6 of 36 rolls (1 + 6, 2 + 5, 3 + 4 and their reverses), so the chance is 6/36 = 1/6 ≈ 16.67%. With three dice, a sum of 10 happens in 27 of 216 rolls, or 12.5%.
How do I find the chance of rolling at least (or at most) a number?
Add up the ways for every sum from that number up (or down). For three six-sided dice, a sum of 10 or more has 135 ways out of 216, exactly 62.5%. Choose “at least”, “at most” or “between” and the table below the answer highlights the sums that are counted.
What are the average and spread of n dice?
Each die has mean (s + 1) ÷ 2 and variance (s² − 1) ÷ 12, and independent dice add, so n dice have mean n(s + 1) ÷ 2 and variance n(s² − 1) ÷ 12. For 3d6 that is a mean of 10.5 and a standard deviation of √8.75 = 2.958.
What is the chance of rolling a particular number at least once?
1 − ((s − 1) ÷ s)ⁿ: one minus the chance that none of the n dice shows it. Rolling at least one 6 with four dice is 1 − (5/6)⁴ = 671/1,296 ≈ 51.77%.
What does “4d6 drop the lowest” mean, and what are its odds?
Roll four six-sided dice and add the highest three — in this calculator, keep the 3 highest of 4d6. The lowest possible total is 3 (all four dice show 1, 1 roll in 1,296) and the highest is 18 (21 rolls in 1,296, about 1.62%); the average is 15,869/1,296 ≈ 12.2446, against 10.5 for 3d6. “Advantage” in some games is the highest of 2d20, averaging 13.825. Every number here comes from counting all the rolls exactly.
How is this different from the odds, probability and lottery calculators?
The odds calculator converts between American, decimal and fractional odds, the probability calculator combines two events, and the lottery odds calculator covers draw games. This one works out the full distribution of the sum of n dice, the chance of any total or range, and what changes when you keep only the best or worst dice.
What are the limits?
Up to 100 dice with up to 100 sides each. The keep-highest or keep-lowest option works up to 20 dice with 20 sides, because it has to count each arrangement of the dice exactly. Probabilities are shown to four significant digits; the fractions are exact.
Sources: Wikipedia, “Dice notation” (NdS notation and the 4d6 drop-lowest odds of 1/1,296 for 3 and 21/1,296 for 18) and “Discrete uniform distribution” (mean and variance of one die). Related: probability calculator, binomial probability calculator, expected value calculator and odds calculator.