# Probability of Rolling Sums on a Pair of D20 (Icosahedral) Dice

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D20 dice are icosahedrons (20-sided polyhedra).

An icosahedron is a 20-sided polyhedron composed of equilateral triangular faces. Five equilateral triangles meet at each vertex of an icosahedron. This solid geometric figure is commonly used to create 20-sided dice, also known as D20 dice. For a pair of such dice whose 20 faces are labeled with the numbers 1 through 20, the smallest possible sum is 2 (getting a 1 on each die) and the largest possible sum is 40 (getting a 20 on each die). The distribution of possible sums from 2 to 40 is not uniform as certain sums are more likely to come up than others.

The number of ways to obtain each sum and the corresponding probabilities are given in the list below.

## List of Probabilities

There are 20x20 = 400 possible outcomes on a pair of 20-sided dice. Rolling a 2 on the first die and a 7 on the second die is considered distinct from rolling a 7 on the first and a 2 on the second. For each sum, if there are "r" different ways to roll that sum, then the probability "p" is r/400. Here is the full list for the sums 2 through 40.

• 2, r = 1, p = 1/400 = 0.0025
• 3, r = 2, p = 2/400 = 0.005
• 4, r = 3, p = 3/400 = 0.0075
• 5, r = 4, p = 4/400 = 0.01
• 6, r = 5, p = 5//400 = 0.0125

The probability of rolling a sum of 17 on a pair of D20 dice is 16/400 = 0.04.
• 7, r = 6, p = 6/400 = 0.015
• 8, r = 7, p = 7/400 = 0.0175
• 9, r = 8, p = 8/400 = 0.02
• 10, r = 9, p = 9/400 = 0.0225
• 11, r = 10, p = 10/400 = 0.025
• 12, r = 11, p = 11/400 = 0.0275
• 13, r = 12, p = 12/400 = 0.03
• 14, r = 13, p = 13/400 = 0.0325
• 15, r = 14, p = 14/400 = 0.035
• 16, r = 15, p = 15/400 = 0.0375
• 17, r = 16, p = 16/400 = 0.04
• 18, r = 17, p = 17/400 = 0.0425
• 19, r = 18, p = 18/400 = 0.045
• 20, r = 19, p = 19/400 = 0.0475
• 21, r = 20, p = 20/400 = 0.05
• 22, r = 19, p = 19/400 = 0.0475
• 23, r = 18, p = 18/400 = 0.045
• 24, r = 17, p = 17/400 = 0.0425
• 25, r = 16, p = 16/400 = 0.04
• 26, r = 15, p = 15/400 = 0.0375
• 27, r = 14, p = 14/400 = 0.035
• 28, r = 13, p = 13/400 = 0.0325
• 29, r = 12, p = 12/400 = 0.03
• 30, r = 11, p = 11/400 = 0.0275
• 31, r = 10, p = 10/400 = 0.025
• 32, r = 9, p = 9/400 = 0.0225
• 33, r = 8, p = 8/400 = 0.02
• 34, r = 7, p = 7/400 = 0.0175
• 35, r = 6, p = 6/400 = 0.015
• 36, r = 5, p = 5/400 = 0.0125
• 37, r = 4, p = 4/400 = 0.01
• 38, r = 3, p = 3/400 = 0.0075
• 39, r = 2, p = 2/400 = 0.005
• 40, r = 1, p = 1/400 = 0.0025

This is an example of a discrete triangular distribution. Its median, mean, and mode are 21, the min is 2, and the max is 40. The variance of the distribution is 66.5 and its standard deviation is about 8.155. Another name for this distribution is the discrete uniform sum distribution.

You can simulate throws of a pair of D20 dice using this online discrete uniform sum generator. To use the tool, enter a = 1 (the minimum number on a die), b = 20 (the maximum number on a die), k = 2 (the number of dice you are rolling simultaneously), and N = 100 (to simulate 100 rolls). You can change N to any number you want.

Below is a chart showing the triangular distribution of sums on a pair of D20 dice.

## More Probabilities

The chance of throwing a sum that is a multiple of 3 is 134/400 = 0.335, which is very close to one third.

The probability of rolling a multiple of 4 as a sum is 100/400 = 1/4 = 0.25, exactly one fourth.

The probability of the faces summing to a multiple of 5 is 80/400 = 1/5 = 0.2, exactly one fifth.

The most likely sums are clustered in around the average, 21. The probability of rolling a sum between 17 and 25 is 160/400 = 0.4. The likelihood of throwing a sum between 16 and 26 is 190/400 = 0.475. And finally, the changes of the faces adding up to a number between 15 and 27 is 218/400 = 0.545.

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