Why the Sum of Two Dice Is Not Equally Likely
An ordered dice outcome records both dice, so (2,5) and (5,2) are separate equally likely outcomes even though both add to 7.
A worked example with numbers
List the 36 outcomes as a 6 by 6 table: the first die has six rows and the second die has six columns. A total of 2 appears once: 1+1. A total of 7 appears as 1+6, 2+5, 3+4, 4+3, 5+2, and 6+1, giving six cells. A total of 12 appears once: 6+6. Dividing the count by 36 gives 1/36, 6/36, and 1/36 respectively.
How to set the rule before the result
Use the two-dice roller to see two faces and their total, then count combinations when a game rewards a particular total. For a target such as total at least 10, add the counts for 10, 11, and 12: 3 + 2 + 1 = 6 outcomes, so the chance is 6/36 = 1/6. The dice pool roller is a different rule because it usually counts individual die successes rather than their sum.
Common mistakes that change the odds or the process
Do not count only the possible totals 2 through 12 and conclude that each has 1/11 chance. That ignores how many face pairs produce each total. Do not merge (2,5) and (5,2) when the dice are distinguishable during the count; merging requires keeping its multiplicity of two. Do not use the standard table for loaded dice, different-sided dice, or rules with rerolls and modifiers.
Where this method stops being appropriate
The triangular distribution applies to two independent, fair d6 results that are added without special rules. It is useful for board games and probability exercises, not as a claim about a physical die’s condition or a gambling prediction. Check the actual game: a system may keep the highest die, explode a maximum face, discard dice, or use a target number, each of which changes the event.
How the random source fits into the rule
Wolfram MathWorld’s dice reference presents probability distributions for totals from standard dice. The 36-outcome count follows multiplication: 6 faces on die 1 times 6 faces on die 2. A browser roller can draw each face using the strong integer source described by MDN’s Crypto.getRandomValues reference, but the distribution of the sum comes from addition, not from the animation.
Read the distribution as a table
| Sum | Ordered d6 pairs | Probability |
|---|---|---|
| 2 | 1 | 1/36 = 2.78% |
| 6 | 5 | 5/36 = 13.89% |
| 7 | 6 | 6/36 = 16.67% |
| 8 | 5 | 5/36 = 13.89% |
| 12 | 1 | 1/36 = 2.78% |
The count rises from 2 to 7 because more pairs can add to the middle, then falls symmetrically toward 12.
Use the complement for broad events
To find P(sum at least 9), count 9 through 12: 4 + 3 + 2 + 1 = 10 pairs. The result is 10/36 = 5/18, about 27.78%. The complement gives the same answer: P(sum at most 8) has 26 pairs, so 1 − 26/36 = 10/36. Counting named pairs is safer than relying on an impression that high totals are merely “hard.”
Separate a total from individual faces
A game may pay differently for doubles and for sums. The outcomes (3,3) and (1,5) both total 6 but are not the same event if doubles have a special rule. Start with the 36 face pairs, tag the event the game actually describes, and only then collapse to totals if the rule permits it. This prevents a sum distribution from being applied to an individual-face condition.
Expect frequencies only over many trials
In 36 fair rolls, the expected count of sum 7 is 6, but the observed count need not equal 6. “Expected” means the long-run average over repeated sets, not a quota that a short game owes. A run with no 7 values is possible, and a reroll to force the table closer to its theoretical percentages changes the game rather than testing fairness.
Apply the count to a game choice
If a game rewards sum 7 but penalises sum 2, its chance of the reward is six times the chance of the penalty under fair d6 dice: 6/36 versus 1/36. That comparison comes from combinations, not from the numerical gap between 7 and 2. A target of 8 is also not “almost as likely” by intuition; it has 5/36 because five ordered pairs reach it.
Write a small row of combination counts beside the game rule before choosing a strategy. The table makes clear when a high payout is paired with a rare total and when a middle total is common.
Draw the diagonal for each total
On a 6 by 6 outcome grid, cells with the same sum form a diagonal. The diagonal for 7 crosses six cells; the diagonals for 2 and 12 touch only one corner each. This geometry is a quick check on the combination count and shows why middle sums occur more often without treating the values as special.