Dice probability calculator
Sums of dice follow a bell-shaped distribution because there are more ways to make a middling total than an extreme one.
Chance of exactly that total, two dice
16.6667%
- Expected total
- 7.000
- Standard deviation
- 2.4152
- Possible outcomes
- 36
- Minimum roll
- 2
- Maximum roll
- 12
- Chance of at least one maximum face
- 30.5556 %
- Chance of all maximum faces
- 2.777 78 %
- Z-score of the target
- -0.2070
Show the working
- Number of dice = 2
- Sides per die = 6
- Target total = 7
- Modifier = 0
- Possible outcomes = 6^2 = 36
- Minimum roll = 2 = 2
- Maximum roll = 2 * 6 = 12
- Expected total = 2 * (6 + 1) / 2 + 0 = 7
- variance = 2 * (6^2 - 1) / 12 = 5.833 33
- Standard deviation = sqrt(5.833 33) = 2.415 23
- Z-score of the target = 2.415 23 > 0 ? (7 - 0 - 0.5 - 2*(6+1)/2) / 2.415 23 : 0 = -0.207 02
- single_prob = 6 > 0 ? 1/6*100 : 0 = 16.6667
- Chance of exactly that total, two dice = 2 == 2 ? (7 - 0 <= 6 + 1 ? max(7 - 0 - 1, 0) : max(2*6 + 1 - (7 - 0), 0)) / 36 * 100 : 0 = 16.6667
- Chance of at least one maximum face = (1 - ((6 - 1)/6)^2) * 100 = 30.5556
- Chance of all maximum faces = (1/6)^2 * 100 = 2.777 78
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- outcomes = sides^dice
- mean of n dice = n(s+1)/2, variance = n(s^2-1)/12
- for two dice the count of ways to make t is t-1 up to s+1 and 2s+1-t beyond it
Above about five dice the total is very close to normal, which is why the z-score row is useful.