Thirty-six outcomes, eleven totals
Craps is played with two six-sided dice. Each die has six faces, so there are 6 x 6 = 36 equally likely ordered outcomes. This is the sample space for every craps bet. A roll of a 3 and a 4 is a different outcome from a 4 and a 3, which is why the totals are not equally likely.
Counting the ways to reach each total gives:
| Total | Ways | Probability |
|---|---|---|
| 2 | 1 | 1/36 (2.78%) |
| 3 | 2 | 2/36 (5.56%) |
| 4 | 3 | 3/36 (8.33%) |
| 5 | 4 | 4/36 (11.11%) |
| 6 | 5 | 5/36 (13.89%) |
| 7 | 6 | 6/36 (16.67%) |
| 8 | 5 | 5/36 (13.89%) |
| 9 | 4 | 4/36 (11.11%) |
| 10 | 3 | 3/36 (8.33%) |
| 11 | 2 | 2/36 (5.56%) |
| 12 | 1 | 1/36 (2.78%) |
The ways add up to 36, as they must. The distribution is symmetrical around 7, which is the single most likely total. This is the same pattern that matters in many board games, and the general idea is covered in the article on dice odds.
The come-out roll
A round of craps begins with the come-out roll, and the pass line bet is the most common wager. The rules for that roll are:
- A 7 or 11 wins for the pass line. That is 6 + 2 = 8 ways out of 36, or 22.2%.
- A 2, 3 or 12 (craps) loses. That is 1 + 2 + 1 = 4 ways, or 11.1%.
- Any other total (4, 5, 6, 8, 9, 10) becomes the point. That is 24 ways, or 66.7%.
Check: 8 + 4 + 24 = 36.
Resolving the point
Once a point is set, the shooter keeps rolling. The pass line wins if the point repeats before a 7 appears, and loses if a 7 appears first. All other totals are ignored. This makes the question simple: of the rolls that matter, how many are the point and how many are a 7?
The chance the point is made before a 7 is (ways to make the point) divided by (ways to make the point + 6 ways to make a 7):
- Point 4 or 10: 3 / (3 + 6) = 1/3
- Point 5 or 9: 4 / (4 + 6) = 2/5
- Point 6 or 8: 5 / (5 + 6) = 5/11
A 7 always has six ways, so it beats any point number in a head-to-head race.
Adding it up: the pass line probability
To get the overall chance of winning a pass line bet, combine the come-out and point stages. Work in thirty-sixths:
- Immediate win: 8/36.
- Point 4 or 10 (probability 6/36 combined) times 1/3: 2/36.
- Point 5 or 9 (8/36 combined) times 2/5: 3.2/36.
- Point 6 or 8 (10/36 combined) times 5/11: about 4.545/36.
Sum: (8 + 2 + 3.2 + 4.545)/36 = 17.745/36, about 0.4929. In exact fractions it is 244/495.
A pass line bet pays even money, so the expected value per unit is about 0.4929 - 0.5071 = -0.0142 with rounded figures, or -0.0141 using the exact fraction 244/495. The house edge on the pass line is therefore roughly 1.41%. The don’t pass bet, which takes the other side, has a very similar edge of roughly 1.36% because a come-out 12 is treated as a push rather than a win for the bettor.
This is a good example of how the multiplication rule combines the chance of reaching a stage with the chance of finishing it.
The odds bet and why it is different
After a point is set, a player can add an odds bet behind the pass line. It pays according to the true odds of the point beating the 7:
- Point 4 or 10: pays 2 to 1 (true odds, since the chance is 1/3).
- Point 5 or 9: pays 3 to 2 (chance 2/5).
- Point 6 or 8: pays 6 to 5 (chance 5/11).
Since the payout equals the fair price, the odds bet by itself has an expected value of zero. This is the only place in a standard casino where a bet has no built-in margin. It can only be made alongside a pass line bet, which carries its 1.41% edge, so the combined edge is lower as a percentage of total money wagered, but the player is also risking more money in total. The expected loss in units is the same as the pass line bet alone, since the odds bet adds nothing to it. A zero-edge bet does not remove the long-run loss from the pass line; it only dilutes the percentage.
Why the table shapes the other bets
Most side bets on a craps layout are single-roll wagers on a total, such as a bet that the next roll is 12. If the true chance of a 12 is 1/36, a fair payout would be 35 to 1. Payouts below that, such as 30 to 1, lead to an edge of 5/36 = 13.9% on a unit bet (36/36 chances: expected = (1/36 x 30) - (35/36) = -5/36). Bets that sound exciting often carry larger margins, so the shape of the distribution is a good tool for checking any offered payout.
Remember that each roll is independent of the last. Past results do not change the table above, a belief explored in the article on the gamblers fallacy. Outcomes in craps are uncertain and the figures here describe long-run averages only.