The count behind every craps bet
Games with dice are games of chance, and casino versions carry a built-in house edge. Knowing the counts does not change the odds in a player’s favor, most players lose money over time, and nobody should gamble money they cannot afford to lose. The counting is still worth learning, because it explains every payout at a craps table and a good deal of everyday probability.
Why there are 36 outcomes
Each die has six faces, and each face is equally likely. Rolling two dice therefore gives 6 × 6 = 36 possible pairs. A useful trick is to imagine one die is red and the other blue. Then a red 1 with a blue 2 is a different result from a red 2 with a blue 1, even though both add to 3.
This is the key point many people miss. Treating the two dice as interchangeable would suggest that 1-2 and 2-1 are one outcome. They are two, and each has the same chance as any other pair, 1 in 36. Dice in a real game are physically distinct even when they look identical, and the math treats them as such.
Here is the complete grid of sums, with the red die down the side and the blue die across the top:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Each cell is one equally likely outcome.
How the sums are distributed
Counting each total in the grid gives the following:
| Total | Ways | Probability |
|---|---|---|
| 2 | 1 | 1/36, about 2.8% |
| 3 | 2 | 2/36, about 5.6% |
| 4 | 3 | 3/36, about 8.3% |
| 5 | 4 | 4/36, about 11.1% |
| 6 | 5 | 5/36, about 13.9% |
| 7 | 6 | 6/36, about 16.7% |
| 8 | 5 | 5/36, about 13.9% |
| 9 | 4 | 4/36, about 11.1% |
| 10 | 3 | 3/36, about 8.3% |
| 11 | 2 | 2/36, about 5.6% |
| 12 | 1 | 1/36, about 2.8% |
The ways add up to 36 (1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1). The shape is a triangle, symmetric around 7. Totals above and below 7 are mirror images: 6 and 8 each have 5 ways, 5 and 9 have 4, and so on.
Why seven is the most common
Seven is the only total that can be made with any first die. Whatever one die shows, there is exactly one face on the other die that completes the sum: a 1 needs a 6, a 2 needs a 5, and so on through a 6, which needs a 1. Every other total has some first-die results that cannot reach it. A total of 2, for instance, requires both dice to show 1, and a total of 11 needs a 5 and a 6.
The result is that a 7 comes up in 6 of 36 outcomes, or one roll in six on average. Moving to a basic probability view, the odds are 5 to 1 against a 7 on any one roll, which is the same thing expressed in a different form. Converting between 1/6, about 16.7% and 5 to 1 is routine once the counts are in hand.
Other groupings worth knowing
- Doubles (both dice alike) appear in 6 of 36 outcomes, 1 in 6.
- A total of 6 or less appears in 15 of 36 outcomes (1 + 2 + 3 + 4 + 5), and so does 8 or more. The remaining 6 are the sevens.
- Even and odd totals are equally common, with 18 outcomes each.
These groupings are the building blocks of every bet. In craps, the 7 and the six point numbers are compared against one another. A bet’s fair payout is set by the ratio of ways, and the casino pays slightly less than that ratio. The gap is the house edge.
What “common” does and does not mean
Seven being the most frequent total is a statement about long-run frequency. It says nothing certain about the next roll.
Hypothetical example: the chance that a particular six rolls in a row contain no 7 is (5/6)^6, which is about 33.5%. For twelve rolls it is (5/6)^12, about 11%. Such droughts are therefore normal, not signs that a 7 is “due.” The rolls are independent events: the dice do not remember earlier results, and the chance of a 7 on the next roll is 1 in 6 regardless of the past. Believing that a streak must end soon is the gamblers fallacy.
The reverse is also true. A run of sevens does not mean the dice are “hot.” Small samples of rolls often show patterns that disappear as the sample grows, because the law of large numbers works slowly. Frequencies settle near the expected values only over a very large number of rolls.
Putting the count to work
With the table above, any dice bet can be checked. To test a payout, count the winning ways out of 36, compute what a fair payout would be, and compare. If the payout is lower, the difference is the house edge on the bet. That check is the same expected value calculation used throughout probability.
Knowing the counts does not make a casino bet profitable. It makes the cost visible. Anyone who plays should set a limit of truly spare money, expect uncertain results, and stop when the limit is reached. A local support service can help anyone worried about their gambling.