There are 11 numbers that can be made with two six-sided dice, by 36 combinations. (Only 21 of these combinations can be observed with identical dice, since inverse combination pairs are indistinguishable.)
2 can be made 1 way
3 can be made 2 ways
4 can be made 3 ways
5 can be made 4 ways
6 can be made 5 ways
7 can be made 6 ways
8 can be made 5 ways
9 can be made 4 ways
10 can be made 3 ways
11 can be made 2 ways
12 can be made 1 way
Odds for Points vs. 7
4 & 10 = 6:3 (2:1)
5 & 9 = 6:4 (3:2)
6 & 8 = 6:5
| Free Odds Multiplier | Fader / House Edge (vs. Pass) | Casino Edge (vs. Don’t Pass) |
| 0x | 1.41% | 1.36% |
| 1x | 0.848% | 0.682% |
| 2x | 0.606% | 0.455% |
| Full Double Odds* | 0.572% | 0.431% |
| 3x | 0.471% | 0.341% |
| 3-4-5x** | 0.374% | 0.273% |
| 5x | 0.326% | 0.227% |
| 10x | 0.184% | 0.124% |
| 20x | 0.099% | 0.065% |
| 100x | 0.021% | 0.014% |
| 1000x | 0.002% |
*Full Double Odds means players can take 2.5x odds on a point of 6/8,and 2x on the other points.
**3-4-5X Odds means players can take 3x on the 4/10, 4x on the 5/9, and 5x on the 6/8.
Source: http://www.nextshooter.com/odds
The probability of a Pass is 244/495 = 49.3%
The probability of a Miss (Don’t Pass) is 251/495 = 50.7%
The probability of winning on the come out roll is pr(7)+pr(11) = 6/36 + 2/36 = 8/36.
The probability of establishing a point and then winning is pr(4)×pr(4 before 7) + pr(5)×pr(5 before 7) + pr(6)×pr(6 before 7) + pr(8)×pr(8 before 7) + pr(9)×pr(9 before 7) + pr(10)×pr(10 before 7) =
(3/36)×(3/9) + (4/36)×(4/10) + (5/36)×(5/11) + (5/36)×(5/11) + (4/36)×(4/10) + (3/36)×(3/9) =
(2/36) × (9/9 + 16/10 + 25/11) =
(2/36) × (990/990 + 1584/990 + 2250/990) =
(2/36) × (4824/990) = 9648/35640
The overall probability of winning is 8/36 + 9648/35640 = 17568/35640 = 244/495
.
The average game has 3.3757 rolls.
| Event | Probability | Expected Rolls | Product |
| Craps/Natural | 12/36 | 1 | 12/36 |
| Point 4 | 3/36 | 1 + 36/9 | 3/36 + 1/3 |
| Point 5 | 4/36 | 1 + 36/10 | 4/36 + 2/5 |
| Point 6 | 5/36 | 1 + 36/11 | 5/36 + 5/11 |
| Point 8 | 5/36 | 1 + 36/11 | 5/36 + 5/11 |
| Point 9 | 4/36 | 1 + 36/10 | 4/36 + 2/5 |
| Point 10 | 3/36 | 1 + 36/9 | 3/36 + 1/3 |
| Total | 1 + 392/165 = 3.3757 |
Source: https://www.casinocitytimes.com/donald-catlin/article/how-long-is-a-craps-roll-1240
The average turn has 8.5255 rolls.
The probability that any given round will end in one roll (with a 2, 3, 7, 11, or 12) is 1/3. If the player rolls a 4 or 10 on the come out roll the expected number of additional rolls is 4, because the probability of rolling a 4 or 7 is (6+3)/36 = 1/4. Likewise If the player rolls a 5 or 9 on the come out roll the expected number of additional rolls is 3.6 and for a 6 or 8 is 36/11. Assuming a point was thrown the probability of it being a 4 or 10 is 3/12, a 5 or 9 is 4/12, and a 6 or 8 is 5/12. So the expected number of throws per round is 1+(2/3)((3/12)4 + (4/12)3.6 + (5/12)(36/11)) = 3.375758. Next, the probability that the player will seven out is (2/3)((3/12)(2/3) + (4/12)(3/5) + (5/12)(6/11)) = 0.39596. The probability that the player will not seven out is 1 – 0.39596 = 0.60404.
x = 3.375758 / 0.39596 = 8.52551
Source: https://wizardofodds.com/ask-the-wizard/craps/probability/
The average turn has 2.5255 games:
2.5255 come-outs, 6 point round rolls, 1.2449 passes, and 1.2806 misses.
8.5255/3.3757 = 2.5255
8.5255 – 2.5255 = 6
2.5255 x (244/495) = 1.2449
2.5255 x (251/495) = 1.2806
The average turn has 0.5612 naturals, 0.6836 points (made), 0.2806 craps, and 1 seven-out.
The probability that the shooter makes a point is pr(point is 4 or 10) ×pr(making 4 or 10) + pr(point is 5 or 9) × pr(making 5 or 9) + pr(point is 6 or 8) ×pr(making 6 or 8) = (6/24) × (3/9) + (8/24) × (4/10) + (10/24) × (5/11) = 201/495 = 0.406061.
The expected number of points per shooter is 0.406061/(1-0.406061) = 0.683673.
1.24491770204 – 0.68367458611 = 0.56124311593
1.28063255417 – 1 = 0.28063255417
Source: https://wizardofodds.com/ask-the-wizard/craps/probability/
54.92% of passes are points, 45.08% are naturals.
0.2707070707 / 0.49292929292 = x / 1
x= 0.54918032786486
0.22222222222 / 0.49292929292 = y / 1
y = 0.45081967213514
78.09% of misses are seven-outs, 21.91% are craps.
0.39595959596 / 0.50707070707 = x / 1
x = 0.78087649402579
0.11111111111 / 0.50707070707 = y / 1
y = 0.21912350597421
Average number of rolls per point
4 and 10: 36/9 = 4
5 and 9: 36/10 = 3.6
6 and 8: 36/11 = 3.2727
Average number of rolls until a point decision (including the come out)
196/55 + 36/24 = 557/110 = 5.063636364
Source: mustangsally, https://wizardofvegas.com/forum/questions-and-answers/math/17934-how-is-average-of-rolls-calculated-in-craps/
Pass (Light Side)
Natural 7 = 6/36 = 1/6 = 16.7%
Natural 11 = 2/36 = 1/18 = 5.5%
6 & 8 = (5/36)×(5/11) = 25/396 = 6.3%
5 & 9 = (4/36)×(4/10) = 2/45 = 4.45%
4 & 10 = (3/36)×(3/9) = 1/36 = 2.8%
Don’t Pass (Dark Side)
3 Craps = 2/36 = 1/18 = 5.5%
2 & 12 Craps = 1/36 = 2.8%
7-Out
against 6 & 8 = (5/36)×(6/11) = 5/66 = 7.6%
against 5 & 9 = (4/36)×(6/10) = 1/15 = 6.7%
against 4 & 10 = (3/36)×(6/9) = 1/18 = 5.5%
= 39.6%
Litchfield County Craps League 