Two dice are not flat: the 36 outcomes in full
This is the single most misread thing in board-game probability, and it costs people games. One die is flat: six faces, each equally likely, nothing to think about. Two dice are not flat at all, and no amount of staring at a single die reveals why.
The method: count ordered pairs, not sums
The mistake is counting sums. There are eleven possible totals, 2 through 12, so it feels as though each should be worth about a ninth. But the totals are not what the dice produce. The dice produce a pair — a value on the first die and a value on the second — and there are 6 × 6 = 36 such pairs, each equally likely. A total is just a label shared by several of those pairs, and the labels are shared very unevenly.
A total of 2 has exactly one pair behind it: 1 and 1. A total of 7 has six: 1+6, 2+5, 3+4, 4+3, 5+2, 6+1. Those last two lines are the whole lesson — 3 on the first die with 4 on the second is a different outcome from 4 then 3, and both land on 7.
All 36 outcomes, by total
Out of 36 equally likely pairs:
- 2 — 1 pair (1+1) — 1 in 36, about 2.78%
- 3 — 2 pairs (1+2, 2+1) — 2 in 36, about 5.56%
- 4 — 3 pairs (1+3, 2+2, 3+1) — 3 in 36, about 8.33%
- 5 — 4 pairs (1+4, 2+3, 3+2, 4+1) — 4 in 36, about 11.11%
- 6 — 5 pairs (1+5, 2+4, 3+3, 4+2, 5+1) — 5 in 36, about 13.89%
- 7 — 6 pairs (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) — 6 in 36, about 16.67%
- 8 — 5 pairs (2+6, 3+5, 4+4, 5+3, 6+2) — 5 in 36, about 13.89%
- 9 — 4 pairs (3+6, 4+5, 5+4, 6+3) — 4 in 36, about 11.11%
- 10 — 3 pairs (4+6, 5+5, 6+4) — 3 in 36, about 8.33%
- 11 — 2 pairs (5+6, 6+5) — 2 in 36, about 5.56%
- 12 — 1 pair (6+6) — 1 in 36, about 2.78%
The counts add to 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36, which is the check that nothing was missed. Read down the list and the shape is a triangle: up by one each step to the peak at 7, then down by one each step.
What the shape actually means
7 is six times as likely as 2. Six pairs against one. That single ratio explains more board-game outcomes than any other fact about dice.
The middle dominates. Totals 6, 7 and 8 account for 5 + 6 + 5 = 16 of the 36 pairs, or about 44.44% — nearly half of all rolls land in a three-number band out of eleven possible totals. The four extremes, 2, 3, 11 and 12, account for 1 + 2 + 2 + 1 = 6 pairs, about 16.67%: all four of them together are exactly as likely as 7 on its own.
The symmetry is exact. Every total is as likely as its mirror about 7, so 5 and 9 are both 4 in 36. Nothing about a die favours high or low.
If you place resources on a two-dice board, this is the map. A number one step from the centre gives up a sixth of its chances against 7; a number at the edge keeps a sixth of them.
Why more dice make it sharper
The same counting applies to any number of dice, and the pile leans further toward the middle each time you add one. Three six-sided dice have 6 × 6 × 6 = 216 ordered outcomes, and the extremes still have exactly one each — 1+1+1 for a total of 3, and 6+6+6 for 18 — so each of them is 1 in 216, against 1 in 36 with two dice. Extremes get rarer faster than the middle does.
The tool
The Dice Roller rolls up to 50 dice of any side count from 2 upward, prints every die separately and then the sum, so you can see the pairs rather than only the total. Each die is drawn independently by the same seeded generator the rest of the site uses — fine for a game, not a source of secrets, and not something to bet money on. Roll two dice a few dozen times and the triangle starts showing up on its own.