BYTETOOLS

Dice Probability Calculator

Get exact probabilities for the sum of any number of dice with any number of sides, plus 'at least one' odds, a histogram and the full ways-to-roll table.

16.667%
P(sum = 7)
41.667%
P(sum < 7)
58.333%
P(sum ≤ 7)
41.667%
P(sum > 7)
58.333%
P(sum ≥ 7)

2d6 summary

P(sum = 7) = 6 / 36 = 0.16666667

66.667%
P(5 ≤ sum ≤ 9)
30.556%
P(at least one 6)
27.778%
P(exactly one 6)
2 – 12
Possible totals
7
Mean total
5.8333
Variance
2.4152
Std deviation

Totals are counted exactly by convolving the single-die distribution n times — no simulation, so the figures are the true fractions rather than an estimate. “At least one 6” is a different question from the total and uses the complement 1 − ((s−1)/s)ⁿ. Mean = n(s+1)/2 and variance = n(s²−1)/12.

Distribution of the total (7 highlighted)

2: 0.027777777783: 0.055555555564: 0.083333333335: 0.11111111116: 0.13888888897: 0.16666666678: 0.13888888899: 0.111111111110: 0.0833333333311: 0.0555555555612: 0.0277777777823456789101112

Every total

TotalWaysProbabilityCumulative
212.7778%2.7778%
325.5556%8.3333%
438.3333%16.6667%
5411.1111%27.7778%
6513.8889%41.6667%
7616.6667%58.3333%
8513.8889%72.2222%
9411.1111%83.3333%
1038.3333%91.6667%
1125.5556%97.2222%
1212.7778%100.0000%

What is the Dice Probability Calculator?

The ByteTools Dice Probability Calculator works out the exact distribution of the total when you roll several dice.

  • Exact counts by dynamic-programming convolution — no simulation
  • P(sum = t) plus all four cumulative tails and a range probability
  • 'At least one' and 'exactly one' odds for any face value
  • Any number of sides, from coins to d100s
  • Mean n(s+1)/2, variance n(s²−1)/12 and standard deviation
  • Histogram and a full ways-to-roll table you can copy

How to use the Dice Probability Calculator

  1. 1

    Set the number of dice and how many sides each die has.

  2. 2

    Enter the target total you want the probability for.

  3. 3

    Optionally set a lower and upper bound for a range probability.

  4. 4

    Pick a face value to get the 'at least one' and 'exactly one' odds.

  5. 5

    Read the probabilities, then explore the histogram and the ways-to-roll table.

About the Dice Probability Calculator

The ByteTools Dice Probability Calculator works out the exact distribution of the total when you roll several dice. Set the number of dice and how many sides each has, name a target total, and you get the probability of hitting it exactly along with every cumulative tail and the chance of landing in a range.

The totals are counted, not simulated: the tool convolves the single-die distribution once per die, so 2d6 totalling 7 comes back as exactly 6 ways out of 36, or 16.667%. It also answers the separate question of rolling at least one particular face, using the complement 1 − ((s−1)/s)ⁿ, which is what most people actually mean when they ask about the odds of "getting a six".

Alongside the numbers you get a histogram of the whole distribution and a scrollable table showing how many ways each total can be made, which is far more useful than a single probability when you are balancing a game. Everything runs in your browser with no uploads and no network calls.

Frequently asked questions

What is the probability of rolling a 7 with two dice?

There are 6 ways to make 7 out of 36 equally likely combinations, so the probability is 6/36, which is 1/6 or about 16.67%. Seven is the most likely total with two six-sided dice, which is why it sits at the peak of the histogram.

How do you calculate the odds of rolling at least one six?

Work out the probability of no sixes and subtract it from 1. With four dice that is 1 − (5/6)⁴, which comes to about 51.8%. Adding up the individual chances instead would double-count the rolls where more than one six appears.

Why is the distribution of dice totals bell-shaped?

There are many more combinations that produce a middling total than an extreme one — only one way to roll snake eyes, but six ways to roll a seven. As you add dice, the central limit theorem takes over and the shape gets closer and closer to a normal curve.

Does this calculator use a simulation?

No. It builds the exact distribution by convolving the single-die distribution once for each die, so the answers are true fractions rather than estimates. With small numbers of dice it even shows the exact count out of sⁿ.

Can I use non-standard dice like d20s?

Yes, set the sides to anything from 2 upward, so coins, d4s, d10s, d20s and d100s all work. The mean is always n(s+1)/2 and the variance n(s²−1)/12, and both update as you change the dice.

Related tools