Dice Probability Calculator

How dice probability works

Each fair die has equally likely faces, and separate dice are independent events – one die's result does not affect another's. The sample space is every possible combination: 6 outcomes for one d6, 6 × 6 = 36 for two, 6³ = 216 for three, and in general sidesⁿ for n dice. The dice roll probability formula is simply favourable outcomes ÷ total outcomes.

For one d6, the chance of any particular face is 1/6 ≈ 16.7%. The chance of rolling a 5 or 6 is 2/6 = 33.3%. For dice with other numbers of sides, the chance of any single face is 1 ÷ sides: 25% on a d4, 12.5% on a d8, 10% on a d10, 8.3% on a d12 and 5% on a d20.

Two dice probability chart (2d6)

Seven is the most likely sum because six combinations make it (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). The probability of rolling a sum of 7 with two dice is 6/36 = 1/6. Doubles also have a 1-in-6 chance, and so does any total of 4 or less.

Three dice probability chart (3d6)

With three dice, results cluster even more tightly around the middle: 10 and 11 each have a 12.5% chance, while 3 and 18 each come up only once in 216 rolls (0.46%).

Rolling multiple dice: at least one, exactly k and all the same

At least one

The easiest way to find the chance of at least one six is to work out the chance of no sixes and subtract it from 1. For five dice: (5/6)⁵ = 40.2% chance of no six, so 1 − 0.402 = 59.8% chance of at least one.

Exactly k successes (binomial probability)

To find the chance of exactly k dice showing a six, use the binomial formula C(n, k) × (1/6)ᵏ × (5/6)ⁿ⁻ᵏ, where C(n, k) counts the ways to choose which dice are sixes. For five dice:

All dice the same

For n dice with s sides, the chance that all match is s ÷ sⁿ = 1 ÷ sⁿ⁻¹. Two d6 match 1 in 6 times; five d6 match (a Yahtzee in one roll) 1 in 1,296 times, or 0.077%.

D20 probability for D&D

To succeed on a d20 roll you need the die plus your modifier to meet a target number. If you need a natural t or higher, the chance is (21 − t) ÷ 20. Advantage (roll two, keep the higher) gives 1 − (1 − p)²; disadvantage (keep the lower) gives p².

Advantage helps most when you need around an 11: it lifts the chance from 50% to 75%. A natural 20 goes from 5% to 9.75% with advantage, and to about 14.3% if you roll three dice and keep the best (the Elven Accuracy feat).

Dice odds in popular games

• Craps: on the come-out roll, 7 or 11 wins immediately – 8 of 36 combinations, or 22.2%; 2, 3 or 12 loses – 4 of 36, or 11.1%.
• Catan: 6 and 8 are each rolled 5 times in 36 (13.9%), while 2 and 12 come up only once in 36 – which is why settlements on 6 and 8 are valuable.
• Monopoly: doubles come up 1 time in 6; three doubles in a row (go to jail) is 1 in 216.
• Yahtzee: a Yahtzee in the first roll is 1 in 1,296; with up to three rolls and good strategy the chance per turn rises to about 4.6%.

Expected rolls and the gambler's fallacy

If an event has probability p on each roll, you need 1 ÷ p rolls on average to see it: six rolls for a particular face on a d6, 20 rolls for a natural 20. But dice have no memory. After ten rolls without a six, the chance of a six on the next roll is still 1/6 – believing it is 'due' is the gambler's fallacy. Streaks of bad luck are normal: there is a 16% chance of going ten rolls without a single six (5/6 to the 10th power).

Order matters: sequences vs combinations

Rolling a 1 then a 2 with two dice is one specific outcome out of 36 (2.8%). Rolling 'a 1 and a 2 in any order' covers two outcomes (1-2 and 2-1), so it is twice as likely (5.6%). Many dice puzzles hinge on this difference. When the question says 'in that order', count one arrangement; when it does not, count every arrangement – the number of arrangements of n different values is n! (for example, 3! = 6 ways to roll 1, 2 and 3 on three dice).

Dice pools: counting successes

Some games roll a handful of dice and count how many beat a target. In a pool of five d10 where 8 or higher is a success, each die succeeds with probability 0.3. The chance of at least one success is 1 − 0.7⁵ = 83.2%, and the average number of successes is 5 × 0.3 = 1.5. The 'exactly k' binomial formula above gives the full spread, and the calculator's dice-pool mode does this for any pool size and target.

Opposed rolls

When two players each roll a d20 and the higher roll wins, each player wins 47.5% of the time and they tie 5% of the time (20 ties out of 400 combinations). If one side adds a +2 bonus, the odds shift noticeably – which is why small modifiers matter more in opposed contests than they first appear.

How many rolls to see every face?

On average it takes about 14.7 rolls of a d6 to see all six faces at least once (6 × (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6)). The first new face takes one roll, but the last missing face takes six rolls on average by itself – a classic example called the coupon collector's problem.

Is my die fair?

To test a die, roll it many times – at least 300–600 for a d6 – and count each face. A fair d6 should show each face about one-sixth of the time, but random variation is large: in 600 rolls, counts between roughly 80 and 120 for each face are normal. A chi-square goodness-of-fit test tells you whether the differences are bigger than chance would explain. Cheap dice with uneven drilling or rounded edges can be slightly biased, but a short run of 'bad luck' is almost never proof.

Odds, percentages and '1 in N'

Probability can be written several ways. A 1/6 chance is 16.7%, '1 in 6', or odds of 5 to 1 against (five ways to fail for every way to succeed). Betting odds are usually quoted 'against', so a payout of 5 to 1 on a single number of a d6 would be exactly fair; casinos pay less than fair odds, which is where the house edge comes from. The calculator shows all three formats so you can compare game rules or betting offers directly.

Probability of a sum with many dice

For larger pools, counting combinations by hand becomes impractical – 10d6 has more than 60 million outcomes. The calculator builds the distribution by adding one die at a time (a convolution), which is exact. As the number of dice grows, the shape approaches a normal (bell) curve with mean n × (s + 1)/2 and variance n × (s² − 1)/12, which is a handy approximation for quick estimates.