UtilibyWritten by the Utiliby editorial team · Last updated October 23, 2026 · Rules and formulas reviewed October 2026
What a dice probability calculator does
Dice feel random, but the odds are exact. A dice probability calculator turns a roll such as 3d6, d20 with advantage or 4d6 drop lowest into the precise chance of each result, so you can decide whether to attack, pick a build or design an encounter with real numbers instead of hunches.
This free online tool computes the full probability distribution for sums of dice with a modifier, keep-highest and keep-lowest rolls, advantage, disadvantage and best-of-three rolls. It also handles success-counting dice pools and a roll-versus-roll comparison. You get the chance of hitting a target, the odds as "1 in N", the average, standard deviation, median, the most likely result and a chart. Everything runs in your browser.
How to use this calculator
- Choose a tab. Dice roll for one roll, Dice pool for success counting, Compare rolls for contests.
- Enter the dice. Number of dice, sides per die and a modifier. Use a preset such as d20 advantage or 4d6 drop lowest to start.
- Pick how the roll works. Add the dice, keep the highest or lowest, advantage, disadvantage or best of three.
- Set your target. Choose at least, more than, at most, less than or exactly, then enter the number.
- Read the chance. The green bars in the chart show which results meet your target. Open the full table for every result.
The maths behind dice odds
| Roll | How it is calculated |
|---|---|
| Sum of dice | Convolution of the single-die distributions, counting every combination exactly |
| Keep highest k of n | Dynamic programming over face values, counting how many dice show each face |
| Advantage | P(max ≤ x) = P(single roll ≤ x)² |
| Disadvantage | P(min ≥ x) = P(single roll ≥ x)² |
| Dice pool | Binomial distribution with success chance (sides − target + 1) ÷ sides |
| Roll against roll | Sum over every pair of results of the product of their probabilities |
The tool uses exact calculation, not random simulation, so the results are the same every time and do not carry sampling error. Values are shown to two decimal places of a percent for ordinary chances and more for very rare ones.
Average roll of NdM: tables for d4 to d100
The average of NdM (N dice with M sides each) is N × (M + 1) ÷ 2. A single d6 averages 3.5, so 8d6 averages 8 × 3.5 = 28. A d8 averages 4.5, so 12d8 averages 54. A d4 averages 2.5, so 4d4 averages 10. Add any flat modifier to the result. The table gives the average for common dice counts.
| Dice | d4 | d6 | d8 | d10 | d12 | d20 | d100 |
|---|---|---|---|---|---|---|---|
| 1 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 10.5 | 50.5 |
| 2 | 5 | 7 | 9 | 11 | 13 | 21 | 101 |
| 3 | 7.5 | 10.5 | 13.5 | 16.5 | 19.5 | 31.5 | 151.5 |
| 4 | 10 | 14 | 18 | 22 | 26 | 42 | 202 |
| 5 | 12.5 | 17.5 | 22.5 | 27.5 | 32.5 | 52.5 | 252.5 |
| 6 | 15 | 21 | 27 | 33 | 39 | 63 | 303 |
| 7 | 17.5 | 24.5 | 31.5 | 38.5 | 45.5 | 73.5 | 353.5 |
| 8 | 20 | 28 | 36 | 44 | 52 | 84 | 404 |
| 9 | 22.5 | 31.5 | 40.5 | 49.5 | 58.5 | 94.5 | 454.5 |
| 10 | 25 | 35 | 45 | 55 | 65 | 105 | 505 |
| 12 | 30 | 42 | 54 | 66 | 78 | 126 | 606 |
| 15 | 37.5 | 52.5 | 67.5 | 82.5 | 97.5 | 157.5 | 757.5 |
| 20 | 50 | 70 | 90 | 110 | 130 | 210 | 1,010 |
The average is only the centre. How far results spread depends on the standard deviation, which is √(N × (M² − 1) ÷ 12). The table below adds the minimum, maximum and spread for popular rolls, plus the chance of the single most likely total. All values are exact and match what the calculator shows.
| Roll | Minimum | Average | Maximum | Standard deviation | Chance of the most common total |
|---|---|---|---|---|---|
| 1d20 | 1 | 10.5 | 20 | 5.77 | 5.0% |
| 2d6 | 2 | 7 | 12 | 2.42 | 16.7% |
| 3d6 | 3 | 10.5 | 18 | 2.96 | 12.5% |
| 4d4 | 4 | 10 | 16 | 2.24 | 17.2% |
| 5d6 | 5 | 17.5 | 30 | 3.82 | 10.0% |
| 5d8 | 5 | 22.5 | 40 | 5.12 | 7.5% |
| 7d8 | 7 | 31.5 | 56 | 6.06 | 6.4% |
| 8d6 | 8 | 28 | 48 | 4.83 | 8.1% |
| 8d8 | 8 | 36 | 64 | 6.48 | 6.0% |
| 10d6 | 10 | 35 | 60 | 5.40 | 7.3% |
| 10d8 | 10 | 45 | 80 | 7.25 | 5.4% |
| 12d8 | 12 | 54 | 96 | 7.94 | 5.0% |
| 4d6 drop lowest | 3 | 12.24 | 18 | 2.85 | 13.3% |
Roughly two thirds of rolls land within one standard deviation of the average. For 8d6 that is about 23 to 33. The more dice you add, the more tightly results cluster around the average relative to the full range, which is why big damage rolls feel predictable while a single d20 is not.
Spell damage averages
These are the averages for some well-known damage rolls from the 5e System Reference Document. Use the calculator for the chance of beating a target number, for example the chance a Fireball deals at least 30.
| Spell (5e SRD) | Damage dice | Average | Range |
|---|---|---|---|
| Fireball | 8d6 | 28 | 8 to 48 |
| Lightning Bolt | 8d6 | 28 | 8 to 48 |
| Cone of Cold | 8d8 | 36 | 8 to 64 |
| Chain Lightning | 10d8 | 45 | 10 to 80 |
| Finger of Death | 7d8 + 30 | 61.5 | 37 to 86 |
| Disintegrate | 10d6 + 40 | 75 | 50 to 100 |
Advantage and disadvantage, in numbers
Advantage on a d20 is worth a lot, but its value depends on the number you need. The closer your chance is to 50%, the bigger the swing. The table gives the chance of reaching a target on a single d20.
| Need | Normal | Advantage | Disadvantage |
|---|---|---|---|
| 5 or higher | 80% | 96% | 64% |
| 11 or higher | 50% | 75% | 25% |
| 15 or higher | 30% | 51% | 9% |
| 20 | 5% | 9.75% | 0.25% |
Best of three rolls (some features and spells let you roll three times) lifts the chance of a natural 20 to about 14.3%. Notice that advantage helps little when you already need a very low number, and helps most when you need a middling or high one.
Ability scores: 3d6, 4d6 drop lowest and more
Rolling for ability scores is a classic use of keep-highest rolls. With 3d6 the average is 10.5 and an 18 appears 1 time in 216. With 4d6 drop lowest the average climbs to about 12.24 and the chance of at least 15 for a single score is about 23.1%, while at least 18 is about 1.6%. This is why 4d6 drop lowest feels generous: it removes most bad rolls.
Use the keep-highest option to test your own variants, such as 5d6 keep the highest three, or 2d20 keep the highest one for a quick comparison with advantage.
Dice pools and success counting
In many systems you roll a handful of dice and count how many beat a threshold. The chance of one success is (sides − target + 1) ÷ sides, and the number of successes follows a binomial distribution. Six six-sided dice with success on 5 or 6 give each die a 1 in 3 chance, an expected 2.0 successes and about a 64.9% chance of at least two. Use the pool tab to find how adding one more die changes the odds of reaching the number of successes you need.
Comparing rolls and contests
Opposed rolls and contested checks compare two totals. The compare tab shows the exact chance that A wins, ties or loses. A flat bonus matters less than it seems when the dice are wide: a +3 on 2d6 against a d20 is a coin flip, with A winning about 45% and B about 50%, and ties of about 5%. If ties favour the attacker, add the tie chance to the win chance.
Tips for using dice odds
- Think in terms of the chance of the result you need, not the average.
- Remember that probabilities describe the long run. A 90% roll still fails one time in ten.
- Use the standard deviation to see how swingy a roll is: fewer, larger dice are swingier.
- For game design, compare two options by chance of success and average result together.
- Check the rules of your game for rerolls and special cases that change the maths.
Common dice probability mistakes
- Gambler's fallacy. Past rolls do not change future ones for fair dice.
- Adding probabilities of independent rolls. Two d6 do not have a 2 in 6 chance of a six; it is 11 in 36, about 30.6%.
- Assuming averages equal likely. The average of 2d6 is 7, but 7 only occurs 1 time in 6.
- Mixing up "at least" and "more than". A difference of one on a d20 is 5 percentage points.
Dice in game design: d12 or 2d6?
Two dice rolls can have a similar average and feel completely different. A d12 averages 6.5 and a 2d6 averages 7, but the d12 has a standard deviation of about 3.45 while 2d6 has about 2.42. The d12 is flat and swingy, with every value from 1 to 12 equally likely. The 2d6 clusters around 7, with 6, 7 and 8 together making up 44% of rolls. Designers choose flat distributions for excitement and bell curves for reliability. Use the compare tab and the distribution chart to see the difference for any pair.
Modifiers shift the whole distribution but do not change its shape. On a d20 each +1 modifier is worth 5 percentage points of chance until you hit the 5% minimum or 95% maximum. Advantage at a 50% base chance is worth about +5; at 30% it is worth about +4.2; and it is worth less at both extremes.
Critical hits and natural 20s
A natural 20 has a 5% chance on one d20. Over two attacks in a turn the chance of at least one is 1 − 0.95² = 9.75%, the same as one attack with advantage. Over three attacks it is 14.3%. Over ten attacks in a fight it is about 40%. These numbers explain why extra attacks matter so much to damage output: the more rolls you make, the more often the rare high result shows up. Use the keep-highest option or the dice pool tab to explore similar questions for your own game.
Privacy
The calculator runs in your browser. The numbers you enter are not sent to a server and are not stored. Reloading the page returns the example. Downloaded files are created on your device.
Related free tools
Use these free online tools together with this one:
- Work out how many pulls a gacha banner needs with the gacha pity calculator.
- Balance a fight for your party with the D&D encounter calculator.
- Rank players or teams with the Elo rating calculator.
- Turn chances into percent changes with the percentage calculator.
- Pick someone at random for a classroom game with the random student picker.
- Split players into balanced teams with the group generator.
Frequently asked questions
How do I calculate dice probability?
Count the ways to get the result and divide by the total number of equally likely outcomes. For two six-sided dice there are 36 outcomes, and 6 of them sum to 7, so the chance of a 7 is 6/36 = 16.7%. This calculator does the counting exactly for you.
What is the probability of rolling a certain number or higher?
Add the probabilities of every result at or above the number. Choose "at least" in the dice roll tab and enter your target, and the tool shows the chance and the odds as 1 in N.
How does advantage work in probability terms?
You roll twice and keep the higher result. The chance of getting at least a target T on a d20 becomes 1 − (chance of missing)². At least 11 goes from 50% to 75%. At least 15 goes from 30% to 51%.
What is disadvantage in D&D?
You roll twice and keep the lower result, so both rolls must be high. At least 11 drops from 50% to 25%, and at least 15 drops from 30% to 9%.
How much does advantage increase the average roll on a d20?
A normal d20 averages 10.5. With advantage it averages 13.825, an increase of about 3.3 points. With disadvantage it averages 7.175.
What is 4d6 drop lowest?
It is a common method for generating ability scores: roll four six-sided dice and add the highest three. The average is about 12.24, compared with 10.5 for 3d6. Use the keep-highest option with 4 dice and keep 3.
What is the chance of rolling an 18 with 4d6 drop lowest?
About 1.62%. With plain 3d6 it is 0.46%, which is 1 in 216.
What does keep highest or keep lowest mean?
You roll a number of dice but only add some of them. 2d20 keep highest 1 is the same as advantage. The calculator supports keeping the top or bottom dice for up to 12 dice.
What is a dice pool?
A dice pool is a handful of dice where each die is a success or a failure, usually by meeting a target number. You count the successes. Many tabletop games use this, for example 6d6 with a success on 5 or 6.
How do I find the chance of at least 2 successes in a dice pool?
Use the dice pool tab. Enter the number of dice, sides, the success number and the successes needed. For 6d6 with success on 5+, the chance of at least 2 successes is about 64.9%.
How do I compare two different rolls?
Use the compare tab. It computes the exact chance that roll A beats, ties or loses to roll B. For example 2d6+3 against a d20 wins 45.0%, ties 5.0% and loses 50.0%.
What is standard deviation for dice?
It measures how spread out the results are. Rolling more dice makes results cluster around the average, so the standard deviation grows slower than the total.
Why do dice sums form a bell curve?
There are more ways to make middle totals than extremes. With 2d6 there is one way to roll 2 and six ways to roll 7. The more dice you add, the closer the shape gets to a normal bell curve.
Is a d100 the same as 2d10?
A d100 is uniform: each value from 1 to 100 has a 1% chance. Two d10s read as tens and units are also uniform. Adding two d10s is not the same, because the sum has a triangular shape.
Does this calculator handle exploding dice or rerolls?
No. It models fair dice with the options listed. Exploding dice and reroll rules vary by game and are not included.
Can I use it for board games and RPGs?
Yes, for any game that uses sums, keep-highest rolls, advantage or success counting. It is useful for Dungeons & Dragons style games, board games and war games.
Can I export or print the results?
Yes. Print / save as PDF prints only the active tab's results, CSV downloads them and copy summary puts them on your clipboard.
Is my data private?
Yes. Everything runs in your browser. Nothing is sent to a server or stored.
What is the average of 8d6?
The average of 8d6 is 28, because each d6 averages 3.5 and 8 × 3.5 = 28. Results range from 8 to 48 with a standard deviation of 4.83, and the chance of rolling 28 or more is about 54.0%.
How do I calculate the average of NdM?
Multiply the number of dice by the average of one die: N × (M + 1) ÷ 2. For 12d8 that is 12 × 4.5 = 54. For 5d6 it is 5 × 3.5 = 17.5. Add any flat modifier afterwards.
What is the average of 4d4?
The average of 4d4 is 10, since a d4 averages 2.5. Results range from 4 to 16 with a standard deviation of 2.24, and 10 is the most likely total at about 17.2%.
What is the average roll of a d4, d6, d8, d10, d12 and d20?
A d4 averages 2.5, a d6 3.5, a d8 4.5, a d10 5.5, a d12 6.5 and a d20 10.5. The rule is (sides + 1) ÷ 2. A d100 averages 50.5.
What is the average of 10d8 and 12d8?
10d8 averages 45 and 12d8 averages 54. 10d8 ranges from 10 to 80 with a standard deviation of 7.25; 12d8 ranges from 12 to 96 with 7.94.
How far from the average can a dice roll land?
Use the standard deviation, √(N × (M² − 1) ÷ 12). About two thirds of rolls fall within one standard deviation of the average. For 8d6 the spread is about 4.83, so most rolls land between 23 and 33.
Sources and further reading
- Standard probability theory for discrete uniform distributions, convolution of independent variables and the binomial distribution.
- The publicly available System Reference Document 5.1 (Wizards of the Coast, Creative Commons licence) for the definitions of advantage and disadvantage.
- Combinatorics references on counting ordered outcomes and order statistics of independent random variables.
Sources were consulted in October 2026. Standards and rules can change, so always check the official page for your situation.