How to use the dice probability calculator
- Type a dice expression: NdS means N dice with S sides, such as 3d6 or d20. Add a number with + or -, as in 3d6+2.
- Add kh3 after a term to keep the highest 3 dice, or kl1 to keep the lowest one. Terms can be combined, as in 1d8+1d4-1.
- Choose at least, at most or exactly and enter a target total, or leave the target blank.
- Read the chance of your target, the average, the spread, and the table with the chance of every total.
Formula
chance of a total after adding one die = the average, over the S faces, of the chance of the earlier totals
mean of NdS = N x (S + 1) / 2, variance = N x (S^2 - 1) / 12
keep highest or lowest: count every set of faces, weighted by N! / (product of the counts of each face)! / S^N
separate terms are independent, so their distributions are combined
All results are exact for fair dice. They come from counting, not from simulation, so running the page twice gives identical numbers.
Rolling four dice and keeping three
Roll 4d6, drop the lowest and add 2 (written 4d6kh3+2). The average of the kept dice is 12.2446, so the average total is 14.2446 with a standard deviation of 2.8468. The chance of a total of 15 or more is 48.765%, which matches counting the 1,296 possible rolls one by one.
For 3d6+2 the chance of 15 or more is 7 / 27 = 25.926%. Rolling two twenty-sided dice and keeping the highest (2d20kh1) gives 51% to reach 15, and keeping the lowest (2d20kl1) gives 9%. For 1d8+1d4-1, the average is 6 and the chance of a 5 is 12.5%.
Why sums are not flat
With one die every face is equally likely, but with two or more dice the middle totals can be made in more ways than the extremes. Two six-sided dice make a 7 in six ways and a 2 in one way, so 7 is six times as likely. This is why the table looks like a hill. Keeping only the highest dice pushes the hill upward, and keeping the lowest pushes it down.
For simple events like coins and cards, the Probability Calculator works with chances you enter, and the Binomial Probability Calculator finds the chance of a number of successes in repeated trials.
Limits
Keep highest and keep lowest are worked out by counting, so the dice must not be too many for the number of sides. If an expression has too many cases to count exactly, the page says so. Rules for rerolls, exploding dice, successes or ties are not modeled.
More probability tools
For other scores, ratings and odds, try the Elo Rating Calculator, Balanced Team Generator and Race Pace Chart Generator.
Assumptions and limits
- Every die is fair and every roll is independent.
- A term can have 1 to 100 dice with 2 to 1,000 sides, and 200 dice in all.
- A minus sign subtracts the whole term that follows it.
- Rerolls, exploding dice and house rules are not modeled.
Frequently asked questions
What is the chance of rolling 15 or more on 4d6 drop lowest plus 2?
It is 48.765%. The kept dice average 12.24 and the bonus of 2 moves the whole range up.
How do I calculate advantage or disadvantage on a d20?
Advantage is rolling two d20 and keeping the highest (2d20kh1), and disadvantage keeps the lowest (2d20kl1). To reach a 15, advantage gives 51% and disadvantage gives 9%, against 30% for one die.
Why is the middle total the most likely?
There are more combinations of faces that give a middle total than an extreme one, so after summing two or more dice the totals form a hill.
What does 4d6kh3 mean?
Roll four six-sided dice, keep the three highest and add them. kl keeps the lowest instead, and the number after it says how many to keep.