Two fair six-sided dice have 36 ordered outcomes. They do not produce the eleven possible totals with equal frequency. Seven can appear six ways, while 2 and 12 can appear only one way each. That single fact explains most of the basic mathematics in craps: why 7 dominates the point cycle, why odds payouts differ by point, and why a large-looking proposition payout may still be poor value.
The word ordered matters. A roll of 2–5 and a roll of 5–2 both total 7, but they are separate outcomes in the 36-outcome sample space.
The complete 6 × 6 map
Read the first die down the left side and the second die across the top. Each cell is one equally likely ordered result.
| First die ↓ / Second die → | 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 |
Counting the matching totals gives the familiar probability pyramid:
| Total | Ordered combinations | Exact probability | Percentage |
|---|---|---|---|
| 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 combination counts add to 36. The probabilities add to 1, or 100%. Those two checks are useful whenever you build or verify a dice table.
Why the totals form a pyramid
The smallest total, 2, requires 1–1. Increasing the total to 3 creates two routes: 1–2 and 2–1. Every step toward 7 adds another available pairing. After 7, the pattern reverses because a die cannot show more than 6.
This symmetry is why:
- 2 and 12 have the same probability;
- 3 and 11 have the same probability;
- 4 and 10 match;
- 5 and 9 match;
- 6 and 8 match.
It does not mean bets on those totals must have the same rules or payouts. Probability describes the dice. The paytable describes the wager.
The basic probability formula
For any one-roll event:
[ P(E)=\frac{F}{36} ]
where:
- (P(E)) is the probability of event (E);
- (F) is the number of favorable ordered combinations;
- 36 is the total number of ordered outcomes for two fair six-sided dice.
For a total of 9, the favorable combinations are 3–6, 4–5, 5–4, and 6–3. Therefore:
[ P(9)=\frac{4}{36}=\frac{1}{9}\approx11.1111% ]
This formula applies cleanly to a wager settled by the next roll. Multi-roll bets require an additional step because irrelevant totals can occur without deciding the bet.
Point versus 7: remove the rolls that do not decide anything
Suppose the point is 6. Five combinations make 6, and six combinations make 7. The other 25 combinations do not resolve the Pass Line decision; the dice are rolled again.
Among the 11 deciding combinations, the point wins five and 7 wins six:
[ P(6\text{ before }7)=\frac{5}{5+6}=\frac{5}{11} ]
[ P(7\text{ before }6)=\frac{6}{11} ]
The true odds against making the 6 are therefore 6 to 5. The same reasoning produces every standard odds payout:
| Point | Point combinations | Seven combinations | Probability point wins first | True odds payout |
|---|---|---|---|---|
| 4 | 3 | 6 | 3/9 = 1/3 | 2 to 1 |
| 5 | 4 | 6 | 4/10 = 2/5 | 3 to 2 |
| 6 | 5 | 6 | 5/11 | 6 to 5 |
| 8 | 5 | 6 | 5/11 | 6 to 5 |
| 9 | 4 | 6 | 4/10 = 2/5 | 3 to 2 |
| 10 | 3 | 6 | 3/9 = 1/3 | 2 to 1 |
This is the mathematical foundation of the craps odds bet. The payouts vary because the point numbers do not have the same number of ways to appear.
Worked point example
The point is 5 and a player has $30 in odds behind the Pass Line.
- Four combinations make 5.
- Six combinations make 7.
- True odds against the point are 6:4, reduced to 3:2.
- A $30 odds wager therefore wins $45 profit when 5 appears first.
The original Pass Line wager still pays even money. The odds portion follows the 3-to-2 combination ratio.
Combination count is not the same as payout value
A bet can be easy to count and still be badly priced. Consider Any Seven:
- winning combinations: 6;
- losing combinations: 30;
- true odds against winning: 30:6 = 5:1;
- common casino payout: 4:1.
For a one-unit wager, expected value is:
[ EV=\frac{6}{36}(4)+\frac{30}{36}(-1)=-\frac{1}{6} ]
The player loses an average of one-sixth of a unit per bet, a house edge of 16.6667%. Seven is the most common individual total, but the payout is still too short for its probability.
Now compare a free-odds wager behind the Pass Line. It is paid at the exact point-versus-7 ratio shown above, so that portion has no built-in house edge. The casino retains its advantage through the required line bet, not by shortening the odds payout.
The craps house-edge guide shows how this payout gap differs across the layout.
“Ways” can mean different things at the table
Players often use the word ways loosely. Three distinctions prevent errors:
Ordered combinations. 1–6 and 6–1 are two outcomes. These are used for probability calculations.
Dice totals. Both outcomes above belong to the total 7. A total is not itself a single combination.
Hard and easy forms. A hard 8 is specifically 4–4. The four easy forms are 2–6, 3–5, 5–3, and 6–2. A hardway wager is not simply a wager that an 8 will appear; it is a race between the pair, the easy forms, and 7.
That distinction is why the hard 8 has one winning combination and ten losing combinations while it remains active: four easy 8 combinations plus six sevens. A 9-to-1 payout is shorter than the fair 10-to-1 price.
What the crew uses the combinations for
Dealers normally work from memorized payout procedures rather than rebuilding the grid during play. The underlying counts still explain why the procedures look the way they do.
A base dealer must know that odds on 4 and 10 pay 2 to 1, 5 and 9 pay 3 to 2, and 6 and 8 pay 6 to 5. Place-bet payouts use different prices because they are not paid at true odds. Proposition bets require another set of payout multiples.
The official Massachusetts craps rules, for example, define the major wagers and publish standard payout odds, including 7 to 1 for Any Craps and 15 to 1 for an 11-in-one-roll wager. The rules also specify that wagers should be made before the dice are thrown, with limited confirmation procedures for late calls. See the Massachusetts rules for craps and mini-craps.
From an operating perspective, combination knowledge supports three controls:
- Payout accuracy. The crew can recognize whether a quoted price fits the event.
- Dispute review. Surveillance can match the dice result, booked wager, and payout.
- Game explanation. A floor supervisor can show why two point numbers receive different odds without relying on “that is just the rule.”
Mistakes the 36-outcome map prevents
“Every total has a 1-in-11 chance.” There are eleven possible totals, but they do not have equal numbers of combinations.
“Seven has six combinations, so it should appear every sixth roll.” One-sixth is a long-run probability, not a schedule. Seven can repeat or disappear for many rolls without violating fair-dice mathematics.
“The last ten rolls contained no 12, so 12 is due.” Each properly randomized roll begins with the same 1/36 probability of 12. Past misses do not increase the next-roll chance.
“A bet covering four totals must be broad.” The useful count is combinations, not labels. The four Horn totals—2, 3, 11, and 12—contain only six combinations altogether.
“A high payout means a fair payout.” The fair comparison is losing combinations divided by winning combinations, followed by the actual bet rules.
A compact method for checking a craps bet
For a one-roll wager:
- list every ordered combination that wins;
- count the remaining combinations that lose;
- reduce losing-to-winning combinations to true odds;
- compare true odds with the posted payout;
- calculate expected value if the payout is shorter.
For a multi-roll wager, first identify which totals decide the bet and which merely continue it. Then compare the deciding combinations.
The 36-outcome map cannot predict the next throw. Its value is more practical: it tells you what is common, what is rare, what a fair payout would be, and exactly where the casino’s price enters the game. Continue with craps probability basics for decision-based calculations or use the craps odds calculator to test individual wagers.