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Dice Probability Calculator

Calculate the probability of rolling a given total with several dice. Shows the exact probability of a target sum, the at-least and at-most probabilities, expected value, variance, and a bar chart of the distribution.

Input

Enter the number of dice, the number of faces, and a target sum to find the probability that the dice add up to that value.

dice
faces
sum

The probability that the dice add up to this value.

Result

Probability of a total of 7

16.67percent

6 ways out of 36

At least 7

58.33 percent

At most 7

58.33 percent

Expected value

7

Variance

5.833

Standard deviation

2.415

Total outcomes

36

Distribution of the total

Sum 2: 2.78 percent2Sum 3: 5.56 percent3Sum 4: 8.33 percent4Sum 5: 11.11 percent5Sum 6: 13.89 percent6Sum 7: 16.67 percent16.677Sum 8: 13.89 percent8Sum 9: 11.11 percent9Sum 10: 8.33 percent10Sum 11: 5.56 percent11Sum 12: 2.78 percent12

How it works

  • The distribution of the sum of n dice with m faces (each face from 1 to m equally likely) is computed exactly by convolution.
  • The total can range from n, when every die shows 1, up to n times m, when every die shows m.
  • The probability of a target sum equals the number of ways to reach that sum divided by the total number of outcomes, m to the power n.
  • The at-least probability adds the outcomes for every sum greater than or equal to the target, and the at-most probability adds those less than or equal to it.
  • Each single die has expected value (m plus 1) divided by 2 and variance (m squared minus 1) divided by 12, so the total for n dice is n times these values.

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