Craps probability becomes manageable once you stop treating every number as equally likely. Two fair six-sided dice create 36 ordered outcomes, but the totals are uneven: 7 has six ways to appear, while 2 and 12 have only one way each.
That 36-outcome map solves one-roll wagers directly. Multi-roll bets need a second step: identify which outcomes actually resolve the bet, then calculate the race between them.
Build the 36-outcome map
Each die can show 1 through 6. Because the first die has six possibilities for each of the second die’s six possibilities:
[ 6\times 6=36 ]
The outcomes are ordered. A 2 on the first die and 5 on the second is distinct from 5 on the first and 2 on the second, even though both total 7.
| Total | Ordered combinations | Probability on one roll |
|---|---|---|
| 2 | 1 | 1/36 = 2.7778% |
| 3 | 2 | 2/36 = 5.5556% |
| 4 | 3 | 3/36 = 8.3333% |
| 5 | 4 | 4/36 = 11.1111% |
| 6 | 5 | 5/36 = 13.8889% |
| 7 | 6 | 6/36 = 16.6667% |
| 8 | 5 | 5/36 = 13.8889% |
| 9 | 4 | 4/36 = 11.1111% |
| 10 | 3 | 3/36 = 8.3333% |
| 11 | 2 | 2/36 = 5.5556% |
| 12 | 1 | 1/36 = 2.7778% |
The pattern rises to 7 and then mirrors back down. For a visual listing of every ordered pair, use Craps Dice Combinations.
Three probability operations appear repeatedly
Addition is used for mutually exclusive outcomes. The chance of rolling 2 or 12 is:
[ rac{1}{36}+ rac{1}{36}= rac{2}{36} ]
Complement is used for “anything except.” The chance of not rolling 7 is:
[ 1- rac{6}{36}= rac{30}{36} ]
Multiplication is used for independent sequences. The chance of avoiding 7 on three consecutive rolls is:
[ \left( rac{30}{36} ight)^3pprox57.87% ]
These operations are enough to solve many basic questions, provided the bet’s rules are defined correctly.
One-roll probability is simple counting
For a one-roll event:
[ P(\text{event})=\frac{\text{winning ordered outcomes}}{36} ]
An Any Seven wager wins on six outcomes, so:
[ P(7)=\frac{6}{36}=\frac{1}{6}=16.6667% ]
A wager on the next roll being 11 wins on two outcomes, 5-6 and 6-5:
[ P(11)=\frac{2}{36}=5.5556% ]
A high probability does not automatically mean a good price. Payout must be compared with the losing outcomes. If an Any Seven wager pays 4:1, its expected value per $1 is:
[ \left(\frac{6}{36}\times 4\right)-\left(\frac{30}{36}\times 1\right)=-\frac{6}{36} ]
The house edge is 16.67%, even though 7 is the most common total.
True odds come from winning and losing combinations
Suppose a point of 6 is established. The relevant race is 6 before 7:
- 6 has five combinations;
- 7 has six combinations;
- all other totals leave the race unresolved.
Conditional on one of those two decisive totals appearing, the chance of 6 winning is:
[ P(6\text{ before }7)=\frac{5}{5+6}=\frac{5}{11} ]
The chance of 7 winning is (6/11). Fair net odds for backing the 6 are therefore 6:5. The casino’s free odds wager uses that ratio. A Place 6 wager pays less than true odds, which is where its house edge enters.
The same method gives:
| Point | Ways to make point | Ways to make 7 | Chance point wins race | Fair net payout |
|---|---|---|---|---|
| 4 or 10 | 3 | 6 | 3/9 = 1/3 | 2:1 |
| 5 or 9 | 4 | 6 | 4/10 = 2/5 | 3:2 |
| 6 or 8 | 5 | 6 | 5/11 | 6:5 |
This is conditional probability: non-deciding rolls are removed from the denominator because they do not end the point-versus-7 race.
A full Pass Line probability requires two stages
The Pass Line is not one event. It has a come-out stage and, when a point is established, a point stage.
On the come-out roll:
- 7 or 11 wins: 8 combinations;
- 2, 3, or 12 loses: 4 combinations;
- 4, 5, 6, 8, 9, or 10 establishes a point: 24 combinations.
The overall win probability is the immediate-win probability plus the probability of each point being established and then made before 7:
[ P(\text{Pass win})=\frac{8}{36} +\frac{6}{36}\left(\frac{3}{9}\right) +\frac{8}{36}\left(\frac{4}{10}\right) +\frac{10}{36}\left(\frac{5}{11}\right) ]
The grouped terms represent points 4/10, 5/9, and 6/8. The result is:
[ P(\text{Pass win})=\frac{244}{495}\approx49.2929% ]
The loss probability is:
[ P(\text{Pass loss})=\frac{251}{495}\approx50.7071% ]
With even-money settlement, expected loss per unit is:
[ \frac{251-244}{495}=\frac{7}{495}\approx1.4141% ]
That derivation shows why simply saying “7 is common” is not enough. The Pass Line uses different rules before and after a point.
Official rules define the event before math prices it
Probability work begins with the exact wager definition. Craps, crapless craps, and electronic variants can change which totals win, lose, or establish points. Side bets can add entirely different resolution conditions. The Washington State craps game description is one example of a regulated rules source showing how wagers and variants are defined.
Do not calculate a familiar name while ignoring the displayed rules. “Field,” “crapless,” and bonus bets may use paytables or conditions that differ by approved version.
Pushes must remain separate
Some wagers can win, lose, or push. A push returns the stake and contributes zero net profit to expected value. It should not be counted as a win, and it should not be silently removed unless the calculation is explicitly conditioned on a decisive outcome.
For the Don’t Pass come-out roll, 12 commonly pushes while 2 and 3 win for the bettor and 7 and 11 lose. The all-roll expected value must include the 12 as a zero-result outcome. A separate “among decisions only” percentage may remove it, but that percentage answers a different question.
This distinction also matters when comparing hit rate with return. A wager can settle as a push fairly often without giving the player a positive expectation.
Probability does not remember the last roll
For fair independent dice, the probability of the next total remains based on the same 36 outcomes. Five rolls without a 7 do not remove any 7 combination from the next roll. A long hand is possible; it does not make the next shooter or number mathematically hotter.
The chance of at least one 7 in (n) rolls is:
[ 1-\left(\frac{30}{36}\right)^n ]
For 10 rolls, that is about 83.85%. This cumulative probability describes the whole 10-roll block. After nine non-sevens, the probability of 7 on roll ten is still 6/36.
Frequent mistakes in craps probability
- Treating all totals from 2 through 12 as equally likely.
- Counting unordered labels instead of ordered dice outcomes.
- Using 36 as the denominator for a point-versus-7 race after non-deciding totals have been excluded.
- Removing pushes from one calculation but comparing the result with an all-roll house edge.
- Using payout odds as if they were probability odds.
- Adding probabilities for events that can overlap on the same roll.
- Assuming a cumulative chance makes the final roll due.
- Reporting a percentage without stating whether it is per roll, per resolved bet, or per dollar wagered.
A repeatable method for any craps wager
- Write the exact win, loss, and push conditions.
- Decide whether the bet resolves in one roll or through a multi-roll race.
- Count ordered combinations for every decisive total.
- Remove non-deciding outcomes only when using conditional probability for a race.
- Compare true probability with the posted net payout.
- Calculate expected value using net profit, not total return.
- Keep variance separate from price: a good or bad session does not alter the edge.
For payout comparisons, continue to Craps Odds and True Odds vs Casino Payouts. For individual wager costs, use Craps House Edge or the craps odds calculator.