What this calculator does
Find the exact chance of rolling a total, or of a face like a six showing up, with any number of dice. When fair dice are rolled, every combination of faces is equally likely. So the chance of something happening is the number of combinations that make it, divided by the number of all combinations.
With two ordinary dice there are 6 × 6 = 36 combinations. Six of them add up to 7. So the chance of rolling a 7 is 6/36, or one in six.
Counting by hand gets slow beyond two or three dice. This calculator counts exactly, for up to 50 dice with anywhere from 2 to 100 sides each. It gives the chance of rolling exactly a total, at least a total or at most a total. You see it as a percentage, a reduced fraction and odds, with a table of every possible total. It can also find the chance that a chosen face, such as a six, shows on exactly, at least or at most a set number of the dice.
Where it's used
Why this math matters outside the classroom:
- Board game strategy. Players building near a 6 or 8 in a dice-based board game know each total comes up 5 times in 36, far more than a 2 or 12.
- Tabletop role-play. A game master knows a d20 roll of at least 15 succeeds 30% of the time, so the challenge is hard but fair.
- Math class sample spaces. Students list all 36 outcomes of two dice in a 6 × 6 grid and find 6 of them sum to 7, giving a probability of 1/6.
- Game design balance. A designer replaces 1d12 with 2d6 so the middle totals become more common and the results less swingy.
- Classroom experiments. A class rolls two dice 360 times and compares its count of 7s with the expected 60 from a probability of 1/6.
Grade level and standards: Grade 7; Common Core State Standards for Mathematics 7.SP.C.8.a, 7.SP.C.8.b, 7.SP.C.7.a. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose whether you care about the total of the dice or about how many dice show a particular face.
- Enter the number of dice and the sides per die (6 for ordinary dice; 4, 8, 10, 12 or 20 for game dice).
- Choose exactly, at least or at most. Then enter the target total, or the face value and how many dice should show it.
- Read the probability at the top. The fraction, the odds and the full table of totals or counts are below.
What the inputs mean
- Sides per die
- 6 for ordinary dice; 4, 8, 10, 12 and 20 for tabletop game dice.
- Target total
- Totals outside the possible range are allowed. With two dice, a total of 13 has a probability of 0, and at least 1 has a probability of 100%.
- Number of dice showing that face
- Use 1 with 'at least' for the common question 'what is the chance of at least one six?'.
Formula and method
P(event) = favourable outcomes ÷ total outcomes, where total outcomes = sⁿ Ways to roll a total T with n s-sided dice = Σ (−1)ᵏ × C(n, k) × C(T − k·s − 1, n − 1), for k = 0 to ⌊(T − n) ÷ s⌋ Ways for exactly j dice to show a chosen face = C(n, j) × (s − 1)^(n − j) P(at least one) = 1 − ((s − 1) ÷ s)ⁿ Average total = n(s + 1) ÷ 2
P means probability, the chance of something happening. n is the number of dice, s is the number of sides on each die and T is the target total. sⁿ means s multiplied by itself n times. C(n, k) is the number of ways to choose k things from n, ignoring order. Σ means "add up" the terms, and ⌊ ⌋ means round down to a whole number.
Every die lands on one of its s faces, no matter what the others do. So n dice give sⁿ equally likely outcomes. Any probability is then a count of the outcomes that fit, divided by sⁿ.
To count the ways to make a total, picture handing out T dots to n dice, with at least one each. There are C(T − 1, n − 1) ways to do that. Some of those give a die more dots than it has faces. The plus-and-minus sum, counting k from 0 upward, removes exactly those. (Mathematicians call this inclusion–exclusion.)
Behind the scenes, the calculator builds the whole set of totals one die at a time. That gives the same counts for every total at once.
For a particular face, first choose which j of the n dice show it. There are C(n, j) ways to do that. Each of the other dice can show any of the remaining s − 1 faces. 'At least one' is quickest as one minus the chance of none.
'At least' and 'at most' add up the ways for every total that counts before dividing.
Worked example
Two ordinary dice: what is the chance of a 7? With three dice, what about a total of 10? And with four dice, at least one six?
- Two dice have 6² = 36 outcomes. The pairs (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1) make 7, so P = 6/36 = 1/6 ≈ 16.6667%.
- Three dice have 6³ = 216 outcomes. Ways to make 10: C(9, 2) − 3 × C(3, 2) = 36 − 9 = 27.
- So P(total 10 with three dice) = 27/216 = 1/8 = 12.5%.
- Four dice: the chance of no six is (5/6)⁴ = 625/1,296, so at least one six is 671/1,296 ≈ 51.7747%.
Result: A 7 with two dice: 1/6 ≈ 16.6667%. A 10 with three dice: 12.5%. At least one six with four dice: about 51.7747%, just better than even.
Understanding the result
A probability of 16.6667% means that over many rolls, the event happens about one time in six. It says nothing about the next roll. Dice have no memory, so a 7 is exactly as likely after five 7s in a row as after none.
Totals in the middle of the range are the most likely, because they can be made in the most ways. With more dice, the totals bunch more tightly around the average and take on a bell shape.
The 'About 1 in' figure is one divided by the probability. Odds of 1 in 6 don't promise one success in every six rolls. They describe the average over many rolls.
Assumptions and limitations
- Every die is fair, so each face is equally likely. The dice don't affect each other.
- Faces are numbered 1 to s, as on standard dice. For counting, the dice are treated as told apart (say, a red one and a blue one). That's what makes all sⁿ outcomes equally likely.
- Loaded or unusually numbered dice (such as dice with a 0 face, or Fudge dice) aren't modelled.
- Rules that change the roll aren't included. That covers rerolls, dropping the lowest die, exploding dice (rolling again on a top face) and modifiers added to the total. To add a fixed modifier, adjust the target yourself.
- The table of totals is shown only when there are 200 or fewer possible totals.
Frequently asked questions
What is the most likely total with two dice?
Seven. Six of the 36 combinations make 7, more than any other total, so it comes up about one roll in six. Totals of 2 and 12 are the least likely, with one combination each (1 in 36).
What are the odds of rolling at least one six with two dice?
11 in 36, or about 30.56%. The chance of no six is (5/6)² = 25/36, so the chance of at least one is 1 − 25/36 = 11/36. With four dice it rises to about 51.77%.
Why is the probability of doubles 1/6?
There are six doubles, (1, 1) through (6, 6), out of 36 combinations, and 6/36 = 1/6. Here's another way to see it. Whatever the first die shows, the second has a 1 in 6 chance of matching it.
How do I find the chance of rolling at least a 15 on a d20?
A d20 is a 20-sided die. Totals 15 through 20 are six faces out of twenty, so the chance is 6/20 = 30%. Choose one die with 20 sides, at least, and a target of 15.
Why do 3 dice make 10 more often than 9?
Both totals have six combinations if you ignore order, but they don't count equally. With dice you can tell apart, 10 can be rolled 27 ways and 9 only 25 ways out of 216. That's because combinations with repeated numbers can be arranged fewer ways.
How this calculator is checked
Method: Classical probability for fair dice: exact counts of sums by dynamic programming (cross-checked by the inclusion–exclusion formula), and binomial counts C(n, j)(s − 1)^(n − j) for a chosen face. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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