Dice Probability Calculator
Calculate the probability of rolling specific outcomes with any number and type of dice.
Dice Probability Calculator
Calculator
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Formula: P(sum=k) = count of combinations that sum to k / total outcomes (sides^dice)
Worked example — 16.67% (1 in 6)
Formula
P(sum=k) = count of combinations that sum to k / total outcomes (sides^dice)
Total outcomes = sides^numDice. The number of ways to achieve a specific sum follows a bell curve centered on the average roll.
Worked Examples
Example 1: 2d6 rolling 7
Problem:2 six-sided dice, target sum 7
Solution:6 ways out of 36 = 16.67%
Result:16.67% (1 in 6)
Frequently Asked Questions
What is the most common sum with 2d6?
7 is the most common (6 ways out of 36 = 16.67%). The probability curve is triangular for 2 dice and approaches normal for more dice.
What are probabilities for common tabletop RPG dice?
Common RPG dice probabilities for rolling a specific number: a d4 (4-sided) has a 25% chance per face; a d6 has 16.67% per face; a d8 has 12.5% per face; a d10 has 10% per face; a d12 has 8.33% per face; and a d20 has 5% per face. For advantage in D&D 5th Edition (roll 2d20 and take the higher), the probability of rolling at least 15 jumps from 30% with one die to approximately 51% with advantage.
How does the central limit theorem apply to dice rolling?
The Central Limit Theorem states that the sum of many independent random variables tends toward a normal (bell curve) distribution regardless of the shape of individual distributions. One die produces a flat uniform distribution where every outcome is equally likely. Two dice create a triangular distribution peaked at the middle sum. By six or more dice, the distribution becomes visibly bell-shaped. This explains why rolling multiple dice and adding them (common in RPGs for damage) produces results clustered near the average, making extreme highs and lows increasingly rare.
What is the gambler's fallacy in dice probability?
The gambler's fallacy is the mistaken belief that past independent events influence future probabilities. When rolling dice, each roll is completely independent — rolling a 6 on one throw does not affect the 1-in-6 probability of the next roll. Many gamblers incorrectly think that after a long streak without rolling a 6, a 6 becomes 'due.' In reality, dice have no memory. This fallacy also applies to slot machines, roulette wheels, and coin flips. Understanding independent probability is essential for making rational decisions in games of chance and avoiding costly betting mistakes based on false patterns.
How many combinations produce each sum when rolling 3d6?
With three six-sided dice there are 6^3 = 216 equally likely ordered outcomes, and the count of combinations for each sum forms a symmetric hump from 3 to 18. The extremes have one combination each, so a sum of 3 or 18 has probability 1/216 (about 0.46 percent). The peak sits at 10 and 11, which each have 27 combinations, or 27/216 = 12.5 percent. Counting is done by convolution: the ways to make a sum with n dice are built by adding one die at a time to the tally for n-1 dice.
What is the difference between the probability of exactly a sum and at least that sum?
The probability of exactly a target sum uses only the combinations that hit that value, while an at-least probability adds together every combination from the target up to the maximum. With 2d6 the chance of exactly 10 is 3/36 = 8.33 percent, but the chance of at least 10 adds the 3 ways for 10, 2 for 11 and 1 for 12, giving 6/36 = 16.67 percent. Cumulative form matters for skill checks and saving throws, where any result meeting or beating a target counts as success.
Why are snake eyes and boxcars the rarest results on two dice?
A sum of 2 can only be made one way, 1 and 1, and a sum of 12 only as 6 and 6, so each occupies a single cell of the 36-outcome grid for a probability of 1/36, about 2.78 percent. Middle sums have many more orderings because each die can compensate for the other; 7 has six orderings. The number of combinations rises linearly toward the centre and falls away symmetrically, which is why the two-dice distribution is triangular rather than flat.
How does adding a flat modifier change the dice distribution?
A modifier such as 2d6+3 shifts the entire distribution to the right without changing its shape. The mean rises by the modifier, so 2d6 averages 7 and 2d6+3 averages 10, while the spread stays identical: the variance of 2d6 is 70/12, about 5.83, giving a standard deviation near 2.42 either way. That is why bonuses are a reliable way to raise typical results, whereas adding extra dice raises the average and widens the spread at the same time.
How do you find the chance of rolling at least one specific face across several dice?
Use the complement rule. For n dice with s sides, the chance that a given face never appears is ((s-1)/s)^n, so the chance it appears at least once is 1 minus that value. Rolling four six-sided dice and hoping for at least one 6 gives 1 - (5/6)^4 = 1 - 625/1296 = 671/1296, about 51.8 percent. Adding dice raises this quickly but never reaches certainty, because each die remains independent of the others.
When should a game use a single d20 instead of 3d6 for resolution?
A single d20 is uniform: every result from 1 to 20 is equally likely and its standard deviation is about 5.77, so outcomes swing wildly and modifiers move the success rate by a flat 5 percent per point. Summing 3d6 produces a bell-shaped spread with a standard deviation near 2.96, clustering results around 10.5 and making a plus-one bonus matter far more near the middle of the range. Choose the flat die when you want dramatic upsets, and the dice sum when you want competence to dominate luck.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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