Craps 7 Basic Craps Player's Guide Menu
Craps Math
This site will not show the math behind all the bets in craps at this time. There are a few good sites that have that information.
CrapsMath.com
Simple 3 pages of craps math. Shows all the math formulas. A little hard to follow if you can not understand
and follow the math symbols.
The Wizard of Odds.
A great site for the math behind all casino games.
| Outcome | Combinations Of Dice | Probability | Odds Against | % of rolls |
|---|---|---|---|---|
| #2 | 1-1 | 1/36 | 35 to 1 | 2.78 |
| #3 | 1-2, 2-1 | 2/36 | 17 to 1 | 5.56 |
| #4 | 1-3, 2-2, 3-1 | 3/36 | 11 to 1 | 8.83 |
| #5 | 1-4, 2-3, 3-2, 4-1 | 4/36 | 8 to 1 | 11.11 |
| #6 | 1-5, 2-4, 3-3, 4-2, 5-1 | 5/36 | 31 to 5 | 13.89 |
| #7 | 1-6, 2-5, 3-4, 4-3, 5-2, 6-1 | 6/36 | 5 to 1 | 16.67 |
| #8 | 2-6, 3-5, 4-4, 5-3, 6-2 | 5/36 | 31 to 5 | 13.89 |
| #9 | 3-6, 4-5, 5-4, 6-3 | 4/36 | 8 to 1 | 11.11 |
| #10 | 4-6, 5-5, 6-4 | 3/36 | 11 to 1 | 8.83 |
| #11 | 5-6, 6-5 | 2/36 | 17 to 1 | 5.56 |
| #12 | 6-6 | 1/36 | 35 to 1 | 2.78 |
| # | exactly | Probability | against | or less | or more | # | |||
|---|---|---|---|---|---|---|---|---|---|
| 2 | 1 | 37.31% | 62.69% | 0 | 36.27% | 2 | 26.42% | 100.00% | 2 |
| 3 | 2 | 27.85% | 72.15% | 1 | 39.83% | 3 | 32.32% | 100.00% | 3 |
| 4 | 3 | 23.40% | 76.60% | 2 | 41.34% | 4 | 35.26% | 100.00% | 4 |
| 5 | 4 | 20.72% | 79.28% | 3 | 42.19% | 5 | 37.09% | 100.00% | 5 |
| 6 | 5 | 18.90% | 81.10% | 4 | 42.73% | 6 | 38.37% | 100.00% | 6 |
| 7 | 6 | 17.59% | 82.41% | 5 | 43.09% | 7 | 39.33% | 100.00% | 7 |
| 8 | 5 | 18.90% | 81.10% | 4 | 42.73% | 6 | 38.37% | 100.00% | 8 |
| 9 | 4 | 20.72% | 79.28% | 3 | 42.19% | 5 | 37.09% | 100.00% | 9 |
| 10 | 3 | 23.40% | 76.60% | 2 | 41.34% | 4 | 35.26% | 100.00% | 10 |
| 11 | 2 | 27.85% | 72.15% | 1 | 39.83% | 3 | 32.32% | 100.00% | 11 |
| 12 | 1 | 37.31% | 62.69% | 0 | 36.27% | 2 | 26.42% | 100.00% | 12 |
| # | exactly each | Probability | against | or less | or more | # | |||
|---|---|---|---|---|---|---|---|---|---|
| 2,12 | 1 | 37.31% | 62.69% | 0 | 36.27% | 2 | 26.42% | 100.00% | 2,12 |
| 3,11 | 2 | 27.85% | 72.15% | 1 | 39.83% | 3 | 32.32% | 100.00% | 3,11 |
| 4,10 | 3 | 23.40% | 76.60% | 2 | 41.34% | 4 | 35.26% | 100.00% | 4,10 |
| 5,9 | 4 | 20.72% | 79.28% | 3 | 42.19% | 5 | 37.09% | 100.00% | 5,9 |
| 6,8 | 5 | 18.90% | 81.10% | 4 | 42.73% | 6 | 38.37% | 100.00% | 6,8 |
| 7 | 6 | 17.59% | 82.41% | 5 | 43.09% | 7 | 39.33% | 100.00% | 7 |
Craps Principle
A nice formula to use when wanting to know the probability of one event happening before another event.
The Craps principle formula. P = x / (x + y)
P = probability of one event happening before another.
x = probability of an event
y = probability of another event
- Example 1: probability of a 4 rolling before a 7.
- x = chances of a 4 rolling are 3/36
- y = chances of a 7 rolling are 6/36
- (3/36) / [(3/36)+(6/36)]
- = 3/9 or 1/3 or 33.33% chance of a #4 rolling before a #7
- Example 2: probability of a 7 rolling before any of the 6 place bets.
- x = chances of a 7 rolling are 6/36
- y = chances of place bet 4, 5, 6, 8, 9 and 10 rolling are 24/36
- (6/36) / [(6/36)+(24/36)]
- = 6/30 or 1/5 or a 20.00% chance of a #7 rolling before any of the 6 place bets.
- For all place bettors, the most common outcome of going across with place bets is you will win 0, zero place bets. The second most common outcome is you will win only 1, that is right, ONE place bet before the 7 rolls.
What craps math means and how to apply the math to the most poplar bets in craps. Knowing the math will not hurt you. This site applies the frequency of bets to many betting strategies.
Example: Half of all sevens show within 4 rolls of the last seven that rolled. Also half of all shooters will seven out, lose control of the dice, within 6 rolls or less. Of course the math means nothing unless you know how to apply it to actual betting situations.
Pass Line House Edge
| Roll | Outcome | Probability | Roll | Outcome | Probability | Product |
|---|---|---|---|---|---|---|
| 2 | -$1 <lose> | 1/36 | --- | --- | --- | -$1/36 |
| 3 | -$1 <lose> | 2/36 | --- | --- | --- | -$2/36 |
| 4 | established point | 3/36 | 4 | +$1 <win> | 3/9 | +$9/324 |
| 7 | -$1 <lose> | 6/9 | -$18/324 | |||
| 5 | established point | 4/36 | 5 | +$1 <win> | 4/10 | +$16/360 |
| 7 | -$1 <lose> | 6/10 | -$24/360 | |||
| 6 | established point | 5/36 | 6 | +$1 <win> | 5/11 | +$25/396 |
| 7 | -$1 <lose> | 6/11 | -$30/396 | |||
| 7 | +$1 <win> | 6/36 | --- | --- | --- | +$6/36 |
| 8 | established point | 5/36 | 8 | +$1 <win> | 5/11 | +$25/396 |
| 7 | -$1 <lose> | 6/11 | -$30/396 | |||
| 9 | established point | 4/36 | 9 | +$1 <win> | 4/10 | +$16/360 |
| 7 | -$1 <lose> | 6/10 | -$24/360 | |||
| 10 | established point | 3/36 | 10 | +$1 <win> | 3/9 | +$9/324 |
| 7 | -$1 <lose> | 6/9 | -$18/324 | |||
| 11 | +$1 <win> | 2/36 | --- | --- | --- | +$2/36 |
| 12 | -$1 <lose> | 1/36 | --- | --- | --- | -$1/36 |
| Total | 36/36 | (-7/495) -0.0141 |
House Edge on the pass line = -7/495 or 1.4141%
| Roll | Outcome | Probability | Outcome | Probability | Product | % |
|---|---|---|---|---|---|---|
| 2 | lose | 1/36 | ||||
| 3 | lose | 2/36 | ||||
| 4 | establish point | 3/36 | win | 3/9 | 9/324 | 2.7778% |
| 5 | establish point | 4/36 | win | 4/10 | 16/360 | 4.4444% |
| 6 | establish point | 5/36 | win | 5/11 | 25/396 | 6.3131% |
| 7 | win | 6/36 | - | - | 6/36 | 16.6667% |
| 8 | establish point | 5/36 | win | 5/11 | 25/396 | 6.3131% |
| 9 | establish point | 4/36 | win | 4/10 | 16/360 | 4.4444% |
| 10 | establish point | 3/36 | win | 3/9 | 9/324 | 2.7778% |
| 11 | win | 2/36 | - | - | 4/288 | 1.3889% |
| 12 | lose | 1/36 | ||||
| Total | 244/495 | 49.2929 |
Calculate Pass Line Winning Probability using a Tree
From the come out roll box just follow the branches and multiply the fractions in each blue box.
=(8/36)+(2*(((24/36)*(3/24)*(3/9))+((24/36)*(4/24)*(4/10))+((24/36)*(5/24)*(5/11))))
=(8/36)+(2*((216/7776)+(384/8640)+(600/9504))) now we must find a common denominator so we can add all the fractions which is 855,360.
=(8/36)+(2*(115776/855360))
=(8/36)+(231552/855360)
=(190080/855360)+(231552/855360)
=(421632/855360) This is the exact pass line probability written as a fraction using the probability tree. NOW we can round down to find the commonly known pass line probability.
=244/495 or 49.2929%
Winning on come out roll = 8/36 EXACTLY. Not 2/9. When working with many fractions it is
recommended to NOT round fractions down or find the lowest common denominator until ALL calculations have been completed. One error
will screw up everything!
There are EXACTLY 8 ways to win out of
the 36 possible come out
roll outcomes. A fraction shows the exact probability of an event. When working with PERCENTAGES or DECIMALS do not round down until ALL calculations have been completed.
Pass Line Probabilities
Pass line winning probability on the come out roll is 8/36 or 22.22%.
Pass line losing probability is 251/495 or 50.71% and the losing probability on the come out roll is 4/36 or 11.11%
The percentage of a pass line wins that happen on the come out roll is: (8/36)/(244/495)
= 8/36 * 495/244
= 3960/8784. Now we can round down
= 55/122 or 45.08% of pass line wins happen on the come out roll. That is slightly more than 4 out of 9.
54.92% of pass line wins happen after the come out roll while a point is established on a number. That is slightly less than 5 out of 9.
The percentage of a pass line losses that happen on the come out roll is: (4/36)/(251/495)
=
4/36 * 495/251
= 1980/9036. Now we can round down
= 55/251 or
21.91% of pass line losses happen on the come out roll. That is slightly less than 2 out of
9.
78.09% of pass line losses happen after the come out roll while a point is established on a number. That is slightly
more than 7 out of 9.
- When the pass line wins, it wins slightly more than 4 out of 9 times on the come out roll
- the pass line wins slightly less than 5 out of 9 times while a point is established.
- When the pass line loses, it loses slightly less than 2 out of 9 times on the come out roll
- the pass line loses slightly more than 7 out of 9 times while a point is established.
Expected number of wins is 244. House edge can be seen at 1.41%
The 69.88% on the right, highlighted in green, represents 1 standard deviation or exactly 23 possible outcomes out of the 496 total possible outcomes. A player has a slightly better than a 2 out of 3 chance of falling somewhere between a $75 win and a $145 loss.
Player only has a 37.65% chance of being ahead. Average win, when the player wins, is $77.24 or $75 and average loss, when the player loses, is $102.76 or $105.