Dice and chance
Casino games are games of chance with a built-in house edge. No strategy, system, pattern or timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose.
Craps looks busy because of its many bets, but the mathematics underneath is compact: two fair dice, 36 equally likely outcomes, and a payout for each bet that is set below what the true odds would justify. This article works through the numbers. Payouts vary by place, so the ones used below are labeled as examples.
The 36 combinations
Each die has six faces, so two dice have 6 × 6 = 36 ordered combinations. They are equally likely, and the two dice are independent. The outcome 1-then-6 is a different combination from 6-then-1, even though both total 7.
Counting the combinations for each total gives:
| Total | Combinations | Probability |
|---|---|---|
| 2 | 1 | 1/36 |
| 3 | 2 | 2/36 |
| 4 | 3 | 3/36 |
| 5 | 4 | 4/36 |
| 6 | 5 | 5/36 |
| 7 | 6 | 6/36 |
| 8 | 5 | 5/36 |
| 9 | 4 | 4/36 |
| 10 | 3 | 3/36 |
| 11 | 2 | 2/36 |
| 12 | 1 | 1/36 |
The counts add up to 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36, as they must. A total of 7 is the most common at 6/36 = 1/6, about 16.7%. Totals of 2 and 12 each have a single combination, about 2.8%. This uneven distribution is the key to every bet on the table.
A one-roll bet
Some bets are decided by the next roll alone. Take a bet that the next roll will be 7, and suppose, as an illustrative payout, it pays 4 to 1.
- Win probability: 6/36 = 1/6.
- Loss probability: 30/36 = 5/6.
- EV per $1 = (1/6 × 4) − (5/6 × 1) = 4/6 − 5/6 = −1/6, about −$0.167.
The edge on this hypothetical bet is about 16.7%, very high compared with other bets. The true odds against a 7 are 5 to 1, and a payout of 4 to 1 falls one unit short of that, which is where the edge comes from. The same gap between payout ratios and true odds underlies roulette and other games.
The pass line
The most basic bet is more involved because it can take several rolls to settle. On the first roll, called the come-out:
- A total of 7 or 11 wins. That is 6 + 2 = 8 combinations, so 8/36.
- A total of 2, 3 or 12 loses. That is 1 + 2 + 1 = 4 combinations, so 4/36.
- Any other total (4, 5, 6, 8, 9, 10) becomes the point. That is 36 − 8 − 4 = 24 combinations, so 24/36.
Once a point is set, the dice are rolled until either the point repeats (win) or a 7 appears (lose). Only those two totals matter, so the chance of making the point is the number of combinations for the point divided by the combinations for the point plus the 6 for 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.
Now combine. The probability of winning is the direct win plus each point’s chance of being set and then made:
- Direct win: 8/36 ≈ 0.2222
- Points 4 and 10: 2 × (3/36) × (1/3) = 2/36 ≈ 0.0556
- Points 5 and 9: 2 × (4/36) × (2/5) = 3.2/36 ≈ 0.0889
- Points 6 and 8: 2 × (5/36) × (5/11) = 50/396 ≈ 0.1263
The total is about 0.4929. Exactly, it is 244/495. The chance of losing is 251/495, or about 0.5071.
For a bet that pays 1 to 1, the EV per $1 is 244/495 − 251/495 = −7/495, about −0.0141. The edge is about 1.41%. That is lower than the one-roll example above, and it shows how different bets on the same table, with the same dice, can have very different costs.
Why each bet has its own edge
Every bet is a selection of dice combinations paired with a payout. Different selections cover different fractions of the 36 outcomes, and payouts are chosen separately. As a result:
- The edge is a property of the bet, not of the table.
- Bets that cover few combinations and pay a high ratio tend to have a larger edge than bets that cover many combinations.
- Making several bets at once does not average the edges out. The total expected loss is the sum of the expected losses of each bet.
There are extra bets that pay at true odds after a point is established, and those free odds bets carry no edge themselves. They require a base bet that does, so the overall cost per dollar falls but never reaches zero, and the amount at risk grows.
Independence of the dice
The dice have no memory. A run of sevens does not make a seven less likely next, and a long wait for one does not make it “due.” The sense that the table is hot or cold is a feature of how people notice runs: in any random sequence, hot hands and cold streaks appear by chance alone. Rituals such as blowing on the dice or choosing a particular throwing style do not change the combination counts. They are gambling superstitions, not influences on the dice.
Pace
Craps rounds can be quick, with many bets resolving each hour. A bet with a modest percentage can still cost real money if it is placed repeatedly at speed. Expected loss per hour is average bet × bets per hour × edge. At $10 per bet, 30 resolved bets an hour and the 1.41% pass line edge, that is 10 × 30 × 0.0141 ≈ $4.23 per hour on average. At the one-roll 7 bet’s 16.7% edge, it is 10 × 30 × 0.1667 ≈ $50 per hour. Both are averages, with wide swings around them.
Summary of the math
Counting the 36 combinations gives every probability in the game. The edge on a bet is the shortfall between its payout and the payout that would be fair. No pattern in earlier rolls and no choice of bets changes that structure. If gambling is causing worry or money trouble, a local support service can help.