Counting the ways to make seven
Roll two fair six-sided dice and add them. The totals run from 2 to 12, but they are not equally likely. Seven is the most common, and the reason is a matter of counting.
Treat the dice as different, for example one red and one blue. There are 6 x 6 = 36 equally likely ordered results. The pairs that add to seven are:
- 1 and 6
- 2 and 5
- 3 and 4
- 4 and 3
- 5 and 2
- 6 and 1
That is six pairs, so the chance of a seven is 6/36 = 1/6, about 16.7 percent.
Why seven gets the most pairs
For any first die, there is exactly one result on the second die that makes the total seven. If the first die shows 1, you need a 6. If it shows 2, you need a 5. Whatever the first die shows, a matching face exists on the second die, because the needed value, 7 minus the first, is always between 1 and 6. So all six faces of the first die work, giving six pairs.
No other total has this property. For a total of 8, a first die showing 1 would need a 7, which does not exist, so that face cannot lead to an 8. Only first-die values from 2 to 6 work, giving five pairs. For 6, a first die of 6 would need a 0, so only values 1 to 5 work, again five pairs. The further a total is from seven, the more first-die values are ruled out.
The full distribution
Counting the pairs for every total gives:
| Total | Ways | Chance |
|---|---|---|
| 2 | 1 | 1/36 (2.8%) |
| 3 | 2 | 2/36 (5.6%) |
| 4 | 3 | 3/36 (8.3%) |
| 5 | 4 | 4/36 (11.1%) |
| 6 | 5 | 5/36 (13.9%) |
| 7 | 6 | 6/36 (16.7%) |
| 8 | 5 | 5/36 (13.9%) |
| 9 | 4 | 4/36 (11.1%) |
| 10 | 3 | 3/36 (8.3%) |
| 11 | 2 | 2/36 (5.6%) |
| 12 | 1 | 1/36 (2.8%) |
The ways add up as 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36, so the table covers every outcome. Plotted as a bar chart, this is a triangle rising to a peak at seven and falling symmetrically. The shape is symmetric because swapping each die face x for 7 - x turns a total of s into a total of 14 - s.
Summarising a few handy facts from the table:
- Seven is 6 times as likely as a 2 or a 12.
- Totals of 6, 7 or 8 together have 5 + 6 + 5 = 16 ways, which is 16/36, about 44 percent.
- Totals of 2, 3 or 12 together have 1 + 2 + 1 = 4 ways, which is 4/36, about 11 percent.
A common confusion
People sometimes reason that there are eleven possible totals, so each should have a chance of 1/11. That ignores that totals are made in different numbers of ways. A similar confusion explains why a 1 and 2 (two ways) is twice as likely as a double 1 (one way). The ordered pairs are the equally likely outcomes; the totals are groupings of them. This basic idea of listing equally likely cases first underlies dice probability generally.
Average and spread
The expected value of one die is 3.5, so for two dice it is 7. The most likely total and the average coincide here because the distribution is symmetric. Over many throws, the average total settles near 7, an example of the law of large numbers. In any single throw, though, totals from 2 to 12 can all appear, and long runs without a seven are normal. The chance of not rolling a seven in a throw is 30/36 = 5/6, so the chance of five throws in a row without a seven is (5/6)^5, about 40 percent. Streaks of this kind are not evidence of anything unusual.
Where it shows up in games
Many board games use two dice with rules attached to specific totals. Since the distribution is uneven, a rule triggered by a seven happens more often than one triggered by a ten or a four. A designer can use this to tune how frequently events happen. Adding more dice makes the shape smoother and more bell-shaped, while using one die gives a flat spread.
Cautions
These are the true chances for fair dice, but they are not a route to profit. In dice games run for money, payouts are set so that the operator keeps a long-run advantage, and knowing which total is common does not remove this. Past rolls do not alter the next throw. Gambling can be harmful, and local support services exist for people who need help.