Dice Probability Calculator
Choose your dice type and how many you are rolling, pick a probability mode, set your target, and get the exact chance instantly. The calculator handles seven standard polyhedral dice (d4, d6, d8, d10, d12, d20) plus any custom die, and covers six probability modes: exact sum, sum at least, sum at most, exactly k dice matching a value, at least k dice matching, and the single-die chance of hitting a threshold. A live distribution chart shows every possible sum and its probability, so you can see the whole picture at a glance.
Formula
Worked example
Rolling 2d6 and asking for a sum of exactly 7: there are 6^2 = 36 equally likely outcomes. Using inclusion-exclusion, 6 outcomes sum to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so P = 6/36 = 16.67%. The expected sum is 2*(6+1)/2 = 7.
How dice probability works
Every face on a fair die is equally likely, so the probability of any single outcome on one die is 1 divided by the number of sides. When you roll multiple dice, the total number of equally likely outcomes is the number of sides raised to the power of the number of dice. For two standard six-sided dice, that is 6 squared = 36 possible combinations. Finding the probability of a specific sum means counting how many of those combinations produce that sum, then dividing by 36. The distribution of sums follows a roughly bell-shaped (but discrete) curve that becomes smoother as you add more dice.
Exact sum probability: the inclusion-exclusion formula
For n dice each with s sides, the probability that all dice sum to exactly t is calculated with the inclusion-exclusion principle: P(sum = t) = (1/s^n) times the alternating sum of binomial coefficients C(n,j) and C(t-js-1, n-1) for j from 0 up to floor((t-n)/s). This formula handles any valid combination of dice and target. For example, with two d6 dice and a target sum of 7, there are exactly 6 favourable outcomes out of 36 total, giving a probability of 16.67%. This calculator applies the same formula for d4 through d20 and custom dice.
Binomial probability: counting matching dice
When you want to know the chance that a specific number of dice land on a particular face (for example, exactly two sixes when rolling five d6 dice), the binomial probability formula applies. P(X = k) = C(n,k) * p^k * (1-p)^(n-k), where n is the number of dice, k is how many must match, and p is 1 divided by the number of sides (the chance of hitting the target face on any single die). The "at least k" mode sums this probability for every value from k to n, giving the cumulative chance that k or more dice match.
Expected value and the bell curve effect
The expected (average) sum when rolling n dice with s sides is n times (s+1)/2. For two d6 dice, that is 2 * 3.5 = 7, which is why 7 is the most common total. Adding more dice pulls the distribution toward a normal (bell) curve because of the Central Limit Theorem: extreme sums become increasingly improbable while sums near the mean become dominant. The sum distribution chart below your result shows exactly this shape for your chosen dice combination.
Common dice probability quick reference
| Dice | Target sum | P (at least) | Odds against |
|---|---|---|---|
| 1d6 | 4 | 50.00% | 1:1 |
| 1d6 | 5 | 33.33% | 2:1 |
| 1d6 | 6 | 16.67% | 5:1 |
| 2d6 | 7 | 58.33% | 0.71:1 |
| 2d6 | 8 | 41.67% | 1.4:1 |
| 2d6 | 10 | 16.67% | 5:1 |
| 2d6 | 12 | 2.78% | 35:1 |
| 1d20 | 11 | 50.00% | 1:1 |
| 1d20 | 15 | 30.00% | 2.33:1 |
| 1d20 | 20 | 5.00% | 19:1 |
Probability of rolling at least the target sum with the stated number and type of dice (rounded to 2 decimal places).
Frequently asked questions
What is the probability of rolling a 7 with two standard dice?
There are 36 equally likely outcomes when rolling two six-sided dice. Six of them sum to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so the probability is 6/36 = 1/6, approximately 16.67%. Seven is the most probable sum for two d6 dice.
How do I calculate the probability of rolling at least a certain sum?
Use the "sum at least" mode. The calculator adds up P(sum = k) for every k from your target up to the maximum possible sum. For two d6 dice, the probability of rolling at least 10 is P(10) + P(11) + P(12) = 6/36 = 16.67%.
What does "odds" mean compared to probability?
Probability is the chance expressed as a fraction of 1 (or a percentage), while odds compare the chance of success to the chance of failure. A probability of 1/6 (16.67%) translates to odds of 1:5 in favour (or 5:1 against), meaning for every 1 success you expect 5 failures on average.
How do I calculate the chance that at least k dice show a specific face?
Select "at least k dice show a value" from the mode dropdown. The calculator sums the binomial probability P(X = i) for every i from k up to n (the number of dice). For example, the probability that at least one 6 appears when rolling three d6 dice is 1 - (5/6)^3, approximately 42.13%.
Why does adding more dice make the distribution bell-shaped?
This is the Central Limit Theorem in action. Each die contributes a uniform random variable. When you add independent random variables together, their combined distribution approaches a normal (bell-shaped) curve no matter the shape of the individual distributions. With two dice the distribution is already triangular; with five or more dice it looks convincingly bell-shaped. Extreme sums require every die to be high or every die to be low, which becomes exponentially less likely.
Does a previous roll affect the next one?
No. Dice have no memory; each roll is statistically independent. Rolling six dice and getting six sixes does not make the next roll any more or less likely to produce six sixes. This is known as the independence of events in probability theory.
What is the difference between d10 and d12 dice?
A d10 has ten numbered faces (typically 0-9 or 1-10) and is common in role-playing games for percentile rolls. A d12 has twelve faces and is used in games like Dungeons and Dragons for certain weapons. Selecting the correct die type in the calculator matters because the number of sides determines both the probability of each face and the range of possible sums.