How this instrument works
Two standard six-sided dice produce 36 equally likely combinations in total (6 outcomes on the first die times 6 on the second), but those 36 combinations do not spread evenly across the 11 possible sums from 2 to 12. A sum of 2 can only happen one way — both dice showing 1 — while a sum of 7 can happen six different ways: 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1. Counting the sums this way produces a triangular pattern that peaks at 7 and tapers symmetrically toward the extremes: the number of ways to make sum s works out to 6 - |s - 7| for any s from 2 through 12.
Turning that count into a probability just means dividing by the 36 total equally likely outcomes: P(sum = s) = (6 - |s-7|) / 36. Because 7 has the most ways (6 out of 36, or 1/6 ≈ 16.67%), it is the single most likely sum on two dice — more likely than any other individual total, which is exactly why 7 plays such a central role in games like craps.
This distribution is a textbook example of how summing two independent uniform random variables produces a result that is no longer uniform itself — even though each die alone is perfectly uniform (every face 1-6 equally likely), their sum clusters toward the middle. It is the same underlying idea, in miniature, behind the central limit theorem's tendency for sums of independent variables to pile up around their average.
- Enter the sum you want the odds for into Target sum (2-12) — it opens on 7, the most common sum.
- Read Number of ways to make this sum and Probability (%) beneath it; both recalculate the instant the target sum changes.
- Values outside 2-12 are rejected, since those are the only sums two six-sided dice can produce.
- To compare sums quickly, try the extremes (2 or 12, each just 1 way) against the middle (7, six ways) to see the full spread at a glance.
Worked example — the probability of rolling a 7
Enter 7 into Target sum. Number of ways to make this sum reads 6, computed as 6 - |7-7| = 6 - 0 = 6, matching the six actual combinations that sum to 7: 1-6, 2-5, 3-4, 4-3, 5-2, and 6-1. Probability (%) reads 16.66667%, which is 6 divided by the 36 total two-dice combinations — the single highest probability of any sum on two dice, which is why 7 is the target roll (or the 'natural') in the first round of craps.
Questions
Why is 7 the most likely sum on two dice?
Because more distinct combinations of the two dice add up to 7 than to any other total: six combinations (1-6, 2-5, 3-4, 4-3, 5-2, 6-1) all sum to 7, versus just one combination each for the extreme sums of 2 or 12. No other sum from 2 to 12 has more than six combinations, so 7 sits at the peak of the distribution.
Why do 2 and 12 have the same probability?
Because they are mirror images of each other around the midpoint of 7 — a sum of 2 requires both dice to show their minimum (1 and 1), while a sum of 12 requires both to show their maximum (6 and 6), and each has exactly one way to happen. This symmetry holds for every pair of sums equidistant from 7: 3 and 11 each have 2 ways, 4 and 10 each have 3 ways, and so on.
Where does the '6 minus the distance from 7' shortcut come from?
It falls directly out of counting ordered pairs (a,b) with a,b in 1-6 and a+b equal to the target sum. For any sum s from 2 to 7, there are exactly s-1 ways; for any sum from 7 to 12, there are exactly 13-s ways. Both of those expressions simplify to the single formula 6-|s-7|, which is why the distribution forms a perfect triangle centered on 7.
Does this only work for two standard six-sided dice?
Yes — this exact triangular-distribution formula assumes two fair, independent six-sided dice. Different-sized dice, more than two dice, or loaded dice all produce a different distribution of sums and need a different calculation (for three or more dice, the distribution approaches a bell curve rather than a triangle, an early illustration of the central limit theorem).
Why do casino craps tables treat 7 and 11 differently from 2, 3, and 12?
Because of exactly this probability distribution: on the come-out roll, 7 (6 ways) and 11 (2 ways) are 'natural' wins for the shooter precisely because they are relatively likely, while 2, 3, and 12 ('craps') are automatic losses in part because they are comparatively rare (1, 2, and 1 ways respectively) — the game's payout structure is built directly on top of this same 36-combination distribution.