Probability of All Rolls With 2 Dice #

This page enumerates the probability of hundreds of events possible when rolling two dice.

The total number of possible rolls #

With two standard 6-sided die, there are 6 * 6 = 36 possible rolls. All thirty-six possible rolls are equally likely.

All possible rolls with two dice #

The possible rolls with two dice are depicted in the table below:

die 1 \ die 2	1	2	3	4	5	6
1	1,1	1,2	1,3	1,4	1,5	1,6
2	2,1	2,2	2,3	2,4	2,5	2,6
3	3,1	3,2	3,3	3,4	3,5	3,6
4	4,1	4,2	4,3	4,4	4,5	4,6
5	5,1	5,2	5,3	5,4	5,5	5,6
6	6,1	6,2	6,3	6,4	6,5	6,6

So the probability of rolling a 6 on the first die and a 1 on the second die is 1/36 = 2.778%.

Two distinct numbers in any order #

The table above shows that there are two ways to roll a 1 and a 2. You could roll a 1 and then a 2 or you could roll a 2 and then a 1. Or if your dice are two different colors, say red and blue, then you could roll a 1 with the red die and a 2 with the blue die or you could roll a 2 with the red die and a 1 with the blue die. The point being that in the above table, the two dice are in some way distinguishable from each other (either in order or color or some other trait). The consequence of this distinction is that a roll of [1,2] is considered different than a roll of [2,1]. But what if the two dice are not distintuishable from each other. We might be interested in the number of ways that a 1 and a 2 may be rolled in any order (or on any die, ignoring which die the number came from). Looking at the above table, it is apparent that there are 2/36 ways for each. Each of the possible two number sets is listed below.

The probability of rolling two dice that are a 1 and a 2 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 1 and a 3 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 1 and a 4 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 1 and a 5 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 1 and a 6 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 2 and a 3 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 2 and a 4 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 2 and a 5 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 2 and a 6 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 3 and a 4 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 3 and a 5 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 3 and a 6 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 4 and a 5 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 4 and a 6 (in any order) is 2/36 = 5.556%
The probability of rolling two dice that are a 5 and a 6 (in any order) is 2/36 = 5.556%

Sums, rolls added #

Probability of rolling two dice that sum to a number #

The probability of rolling two dice that sum to 2 is 1/36 = 2.778%
The probability of rolling two dice that sum to 3 is 2/36 = 5.556%
The probability of rolling two dice that sum to 4 is 3/36 = 8.333%
The probability of rolling two dice that sum to 5 is 4/36 = 11.11%
The probability of rolling two dice that sum to 6 is 5/36 = 13.89%
The probability of rolling two dice that sum to 7 is 6/36 = 16.67%
The probability of rolling two dice that sum to 8 is 5/36 = 13.89%
The probability of rolling two dice that sum to 9 is 4/36 = 11.11%
The probability of rolling two dice that sum to 10 is 3/36 = 8.333%
The probability of rolling two dice that sum to 11 is 2/36 = 5.556%
The probability of rolling two dice that sum to 12 is 1/36 = 2.778%

Probability of rolling two dice that sum to less than a number #

The probability of rolling two dice that sum to 2 or less is 1/36 = 2.778%
The probability of rolling two dice that sum to 3 or less is 3/36 = 8.333%
The probability of rolling two dice that sum to 4 or less is 6/36 = 16.67%
The probability of rolling two dice that sum to 5 or less is 10/36 = 27.78%
The probability of rolling two dice that sum to 6 or less is 15/36 = 41.67%
The probability of rolling two dice that sum to 7 or less is 21/36 = 58.33%
The probability of rolling two dice that sum to 8 or less is 26/36 = 72.22%
The probability of rolling two dice that sum to 9 or less is 30/36 = 83.33%
The probability of rolling two dice that sum to 10 or less is 33/36 = 91.67%
The probability of rolling two dice that sum to 11 or less is 35/36 = 97.22%
The probability of rolling two dice that sum to 12 or less is 36/36 = 100%

Differences, rolls subtracted #

The probability of rolling two dice that subtract (first die minus second die) to 0 is 6/36 = 0.1667, or 16.67%.
The probability of rolling two dice that subtract (first die minus second die) to 1 is 5/36 = 0.1389, or 13.89%.
The probability of rolling two dice that subtract (first die minus second die) to -1 is 5/36 = 0.1389, or 13.89%.
The probability of rolling two dice that subtract (first die minus second die) to 2 is 4/36 = 0.1111, or 11.11%.
The probability of rolling two dice that subtract (first die minus second die) to -2 is 4/36 = 0.1111, or 11.11%.
The probability of rolling two dice that subtract (first die minus second die) to 3 is 3/36 = 0.0833, or 8.33%.
The probability of rolling two dice that subtract (first die minus second die) to -3 is 3/36 = 0.0833, or 8.33%.
The probability of rolling two dice that subtract (first die minus second die) to 4 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that subtract (first die minus second die) to -4 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that subtract (first die minus second die) to 5 is 1/36 = 0.0278, or 2.78%.
The probability of rolling two dice that subtract (first die minus second die) to -5 is 1/36 = 0.0278, or 2.78%.

Explanation of probability calculation for differences of two dice

How to find probability of a particular difference between two dice #

The figure below depicts all possible rolls with two dice, each of which is equally likely. The color of each cell and the number in the cell shows the difference between die 1 and die 2. There are 6 ways for there to be a difference of 0 (a pair is rolled), 5 ways for the difference to be 1, 4 ways for the difference to be -2, and so forth. Here are the 5 ways for the difference to be 1 (die 1 value, die 2 value): (2,1); (3,2); (4,3); (5,4); (6,5). The figure concisely shows all 36 differences possible between two dice. The probability of any difference is equal to the number of ways that difference can occur divided by the total number of rolls possible (36).

All Possible Differences Between Two Dice

Absolute differences, larger valued die minus smaller valued die #

The probability of rolling two dice that have an absolute difference of 1 is 10/36 = 0.2778, or 27.78%.
The probability of rolling two dice that have an absolute difference of 2 is 8/36 = 0.2222, or 22.22%.
The probability of rolling two dice that have an absolute difference of 3 is 6/36 = 0.1667, or 16.67%.
The probability of rolling two dice that have an absolute difference of 0 is 6/36 = 0.1667, or 16.67%.
The probability of rolling two dice that have an absolute difference of 4 is 4/36 = 0.1111, or 11.11%.
The probability of rolling two dice that have an absolute difference of 5 is 2/36 = 0.0556, or 5.56%.

Explanation of probability calculation for absolute differences of two dice

How to find probability of a particular absolute difference between two dice #

All Possible Absolute Differences Between Two Dice

The absolute difference between two dice is greater than or equal to #

The probability of rolling two dice that have an absolute difference greater than or equal to 5 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that have an absolute difference greater than or equal to 4 is 6/36 = 0.1667, or 16.67%.
The probability of rolling two dice that have an absolute difference greater than or equal to 3 is 12/36 = 0.3333, or 33.33%.
The probability of rolling two dice that have an absolute difference greater than or equal to 2 is 20/36 = 0.5556, or 55.56%.
The probability of rolling two dice that have an absolute difference greater than or equal to 1 is 30/36 = 0.8333, or 83.33%.
The probability of rolling two dice that have an absolute difference greater than or equal to 0 is 36/36 = 1.0, or 100.0%.

The absolute difference between two dice is less than or equal to #

The probability of rolling two dice that have an absolute difference less than or equal to 0 is 6/36 = 0.1667, or 16.67%.
The probability of rolling two dice that have an absolute difference less than or equal to 1 is 16/36 = 0.4444, or 44.44%.
The probability of rolling two dice that have an absolute difference less than or equal to 2 is 24/36 = 0.6667, or 66.67%.
The probability of rolling two dice that have an absolute difference less than or equal to 3 is 30/36 = 0.8333, or 83.33%.
The probability of rolling two dice that have an absolute difference less than or equal to 4 is 34/36 = 0.9444, or 94.44%.
The probability of rolling two dice that have an absolute difference less than or equal to 5 is 36/36 = 1.0, or 100.0%.

Products, rolls multiplied #

The probability of rolling two dice that multiply to 12 is 4/36 = 0.1111, or 11.11%.
The probability of rolling two dice that multiply to 6 is 4/36 = 0.1111, or 11.11%.
The probability of rolling two dice that multiply to 4 is 3/36 = 0.0833, or 8.33%.
The probability of rolling two dice that multiply to 10 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 5 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 2 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 24 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 20 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 18 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 3 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 15 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 30 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 8 is 2/36 = 0.0556, or 5.56%.
The probability of rolling two dice that multiply to 36 is 1/36 = 0.0278, or 2.78%.
The probability of rolling two dice that multiply to 9 is 1/36 = 0.0278, or 2.78%.
The probability of rolling two dice that multiply to 16 is 1/36 = 0.0278, or 2.78%.
The probability of rolling two dice that multiply to 25 is 1/36 = 0.0278, or 2.78%.
The probability of rolling two dice that multiply to 1 is 1/36 = 0.0278, or 2.78%.

Probability of possible products with two dice

Quotiets, rolls divided #

Either die is a particular number #

When rolling two dice, the probability that at least one of the two dice is a 1 is 11/36 = 0.3056, or 30.56%.
When rolling two dice, the probability that at least one of the two dice is a 2 is 11/36 = 0.3056, or 30.56%.
When rolling two dice, the probability that at least one of the two dice is a 3 is 11/36 = 0.3056, or 30.56%.
When rolling two dice, the probability that at least one of the two dice is a 4 is 11/36 = 0.3056, or 30.56%.
When rolling two dice, the probability that at least one of the two dice is a 5 is 11/36 = 0.3056, or 30.56%.
When rolling two dice, the probability that at least one of the two dice is a 6 is 11/36 = 0.3056, or 30.56%.

Neither of two dice is a particular number #

When rolling two dice, the probability that neither of the two dice is a 1 is 25/36 = 0.6944, or 69.44%.
When rolling two dice, the probability that neither of the two dice is a 2 is 25/36 = 0.6944, or 69.44%.
When rolling two dice, the probability that neither of the two dice is a 3 is 25/36 = 0.6944, or 69.44%.
When rolling two dice, the probability that neither of the two dice is a 4 is 25/36 = 0.6944, or 69.44%.
When rolling two dice, the probability that neither of the two dice is a 5 is 25/36 = 0.6944, or 69.44%.
When rolling two dice, the probability that neither of the two dice is a 6 is 25/36 = 0.6944, or 69.44%.

Both dice are the same number, a pair #

When rolling two dice, the probability that the two dice are a pair (both show the same number) is 6/36 = 0.1667, or 16.67%.
When rolling two dice, the probability that two dice are both 1, a pair of ones, is 1/36 = 0.0278, or 2.78%.
When rolling two dice, the probability that two dice are both 2, a pair of twos, is 1/36 = 0.0278, or 2.78%.
When rolling two dice, the probability that two dice are both 3, a pair of threes, is 1/36 = 0.0278, or 2.78%.
When rolling two dice, the probability that two dice are both 4, a pair of fours, is 1/36 = 0.0278, or 2.78%.
When rolling two dice, the probability that two dice are both 5, a pair of fives, is 1/36 = 0.0278, or 2.78%.
When rolling two dice, the probability that two dice are both 6, a pair of sixes, is 1/36 = 0.0278, or 2.78%.

Only one of the dice is a particular number #

When rolling two dice, the probability that one die is a 1 and the other die is not a 1 is 5/36 = 0.1389, or 13.89%.
When rolling two dice, the probability that one die is a 2 and the other die is not a 2 is 5/36 = 0.1389, or 13.89%.
When rolling two dice, the probability that one die is a 3 and the other die is not a 3 is 5/36 = 0.1389, or 13.89%.
When rolling two dice, the probability that one die is a 4 and the other die is not a 4 is 5/36 = 0.1389, or 13.89%.
When rolling two dice, the probability that one die is a 5 and the other die is not a 5 is 5/36 = 0.1389, or 13.89%.
When rolling two dice, the probability that one die is a 6 and the other die is not a 6 is 5/36 = 0.1389, or 13.89%.