The dice odds behind craps
Every payout in craps is set from one small table: the 36 ways two dice can land. If you know that table, you can work out why the game pays what it pays.
Thirty-six outcomes
Each die has six faces, so two dice make 6 × 6 = 36 equally likely combinations. A total of 7 can be made six ways (1-6, 2-5, 3-4 and their reverses), while a total of 2 can only be made one way (1-1). Counting every total gives:
| Total | Ways | Chance | In other words |
|---|---|---|---|
| 2 | 1 | 2.78% | 1 in 36, about 1 roll in 36 |
| 3 | 2 | 5.56% | 2 in 36, about 1 roll in 18 |
| 4 | 3 | 8.33% | 3 in 36, about 1 roll in 12 |
| 5 | 4 | 11.11% | 4 in 36, about 1 roll in 9 |
| 6 | 5 | 13.89% | 5 in 36, about 1 roll in 7.2 |
| 7 | 6 | 16.67% | 6 in 36, about 1 roll in 6 |
| 8 | 5 | 13.89% | 5 in 36, about 1 roll in 7.2 |
| 9 | 4 | 11.11% | 4 in 36, about 1 roll in 9 |
| 10 | 3 | 8.33% | 3 in 36, about 1 roll in 12 |
| 11 | 2 | 5.56% | 2 in 36, about 1 roll in 18 |
| 12 | 1 | 2.78% | 1 in 36, about 1 roll in 36 |
Why 7 rules the table
A 7 comes up 1 time in 6, more than any other total. That is why the come-out roll favours the Pass Line (8 of 36 rolls win at once, 4 lose at once), and why a point is dangerous: with a 7 so likely, the shooter must repeat the point before it appears.
Will the point repeat before a 7?
Once a point is set, every roll is either the point, a 7 or nothing. Only the point and the 7 matter, so the chance the point comes first is its ways divided by its ways plus the six ways of 7:
| Point | Ways | Chance it repeats before a 7 | True odds against |
|---|---|---|---|
| 4 or 10 | 3 vs 6 | 33.33% | 2 to 1 |
| 5 or 9 | 4 vs 6 | 40.00% | 3 to 2 |
| 6 or 8 | 5 vs 6 | 45.45% | 6 to 5 |
These "true odds" are exactly what free odds pay, and why they cost the player nothing in the long run. When a bet pays less than the true odds, the difference is the house edge.
How often does the Pass Line win?
Combining the come-out and every possible point gives a Pass Line win chance of 244 in 495, about 49.29%. The Don't Pass bet wins 949 times in 1,980 (about 47.93%) and ties, on a come-out 12, 1 time in 36 (about 2.78%); it loses the rest of the time. Pass Line wins fewer than half of the time, but only just, and that small gap is the house edge. It is why a 1.41% bet loses slowly while a 16.67% bet loses fast.
Dice have no memory
You can check this against your own play: the analytics page counts every total you have seen at the table and compares the number of 7s with the 1 in 6 the odds predict. A roll of 7 does not make another 7 less likely, and a long run without a 7 does not make one "due". Each roll is independent, which is also why betting systems cannot change the edge, as the next page explains. In Smart Craps the dice are random for every roll, and at a table where the dice are read by a camera, the dice are the physical ones.