Hard Eight Craps
- Eight and Six are the second most frequently rolled totals with two dice. In order to increase the payout for these number combinations, a bet can be made that requires a pair of threes for a 'Hard Six' or a pair of fours for a 'Hard Eight'. Early in the film Philip Seymour Hoffman loses while playing for a 'Hard Eight'.
- In a nutshell, there are four numbers, 4, 6, 8, and 10 and each of them has its own spot on the table. In order to roll a hardway combination, you must roll the exact pair of doubles. Once the player has made a hardway bet and a point is established, they win if a hard number is rolled before an easy one or a 7.
A gambling game played with two dice in which a first throw of 7 or 11 wins, a first throw of 2, 3, or 12 loses the bet, and a first throw of any.
The hard eight is a very popular bet in the casino game of craps. Jan 3, 2014 - Explore Jessica Rionero's board 'Hard Eight' on Pinterest. See more ideas about hard eight, eight, hard.
Introduction
Introduction
Welcome to the craps appendix. This is where I derive the player's edge for all the major bets in craps. Outside of this appendix I usually speak about the house edge, which is just the product of the player's edge and -1. To avoid multiplying by -1 for every bet I shall speak of everything in term's of the player's edge, which you can expect to be negative since the house ultimately has the edge on all bets except the free odds. Please stay a while and work through some of the bets yourself. Not only will this give you a deeper understanding of the odds but hopefully motivate you to refresh or improve your math skills.
Before going on you must have an understanding of the probability of throwing each total in one roll. This is explained in depth in my dice probability basics page. If you didn't know or can't figure out that the probability rolling a 6 is 5/36 then a visit to that page is a prerequisite for this page.
The general formula for the expected return of a bet is:
∑ (probability of event i) × (return of event i) over all possible outcomes.
The player's edge is the expected return divided by the initial bet. For example when betting against the line on a sporting event you have to bet $11 to win $10. Assuming a 50% chance of winning the expected return would be 0.5×(10) + 0.5×(-11) = -0.5 . The player's edge would be -0.5/11 = -1/22 ≈ -4.545%.
An exception to the house edge rule is when a tie is possible. In general ties are ignored in house edge calculations. To adjust for this, when a tie is possible, divide the expected return by the average bet resolved. The 'average bet resolved' is the product of the initial wager and the probability that the bet was resolved. In craps the only bets with a tie are the don't pass and the don't come.
Many of the bets in craps win if one particular event happens before another. These bets can take several rolls or more to resolve. If a wager wins with probability p, loses with probability q, and stays active with probability 1-p-q then the probability of winning eventually is:
∑ p×(1-p-q)i (for i=0 to infinity) =
p × (1/(1-(1-p-q))) = p × (1/(p+q)) = p/(p+q).
Throughout this page you will see a lot of expressions of the form p/(p+q). To save space I do not derive the expression each time since it is worked out above.
Pass/Come
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 probability of losing is obviously 1-(244/495) = 251/495
The player's edge is thus (244/495)×(+1) + (251/495)×(-1) = -7/495 ≈ -1.414%.
Don't Pass/Don't Come
The probability of winning on the come out roll is pr(2)+pr(3) = 1/36 + 2/36 = 3/36.
The probability of pushing on the come out roll is pr(12) = 1/36.
The probability of establishing a point and then winning is pr(4)×pr(7 before 4) + pr(5)×pr(7 before 5) + pr(6)×pr(7 before 6) + pr(8)×pr(7 before 8) + pr(9)×pr(7 before 9) + pr(10)×pr(7 before 10) =
(3/36)×(6/9) + (4/36)×(6/10) + (5/36)×(6/11) + (5/36)×(6/11) + (4/36)×(6/10) + (3/36)×(6/9) =
(2/36) × (18/9 + 24/10 + 30/11) =
(2/36) × (1980/990 + 2376/990 + 2700/990) =
(2/36) × (7056/990) = 14112/35640
The total probability of winning is 3/36 + 14112/35640 = 17082/35640 = 2847/5940
The probability of losing is 1-(2847/5940 + 1/36) = 1-(3012/5940) = 2928/5940
The expected return is 2847/5940×(+1) + 2928/5940×(-1) = -81/5940 = -3/220 ≈ 1.364%
Most other sources on craps will claim that the house edge on the don't pass bet is 1.403%. The source of the discrepancy lies is whether or not to count ties. I prefer to count ties as money bet and others do not. I'm not saying that one side is right or wrong, just that I prefer counting them. If you don't count ties as money bet then you should divide by figure above by the probability that the bet will be resolved in a win or loss (35/36). So 1.364%/(35/36) ≈ -1.403%. This is the house edge assuming that the player never rolls a 12 on the come out roll.
Place Bets to Win
Place bet on 6 or 8: [(5/11)×7 + (6/11)×(-6)]/6 = (-1/11)/6 = -1/66 ≈ -1.515%
Place bet on 5 or 9: [(4/10)×7 + (6/10)×(-5)]/5 = (-2/10)/5 = -1/25 = -4.000%
Place bet on 4 or 10: [(3/9)×9 + (6/9)×(-5)]/5 = (-3/9)/5 = -1/15 ≈ -6.667%
Place Bets to Lose
Place bet to lose on 6 or 8: [(6/11)×4 + (5/11)×(-5)]/5 = (-1/11)/5 = -1/55 ≈ -1.818%
Place bet to lose on 5 or 9: [(6/10)×5 + (4/10)×(-8)]/8 = (-2/10)/8 = -1/40 = -2.500%
Place bet to lose on 4 or 10: [(6/9)×5 + (3/9)×(-11)]/11 = (-3/9)/11 = -1/33 ≈ -3.030%
Note: These bets are not allowed in land casinos. They can only be found in some Internet casinos.
Buy
Buy bet on 6 or 8: [(5/11)×23 + (6/11)×(-21)]/21 = (-11/11)/21 = -1/21 ≈ -4.762%
Buy bet on 5 or 9: [(4/10)×29 + (6/10)×(-21)]/21 = (-10/10)/21 = -1/21 = -4.762%
Buy bet on 4 or 10: [(3/9)×39 + (6/9)×(-21)]/21 = (-9/9)/21 = -1/21 ≈ -4.762%
Lay
Lay bet to lose on 6 or 8: [(6/11)×19 + (5/11)×(-25)]/25 = (-11/11)/25 = -1/25 ≈ -4.000%
Lay bet to lose on 5 or 9: [(6/10)×19 + (4/10)×(-31)]/31 = (-10/10)/31 = -1/31 = -3.226%
Lay bet to lose on 4 or 10: [(6/9)×19 + (3/9)×(-41)]/41 = (-9/9)/41 = -1/41 ≈ -2.439%
Big 6/Big 8
[(5/11)×1 + (6/11)×(-1)]/1 = -1/11 ≈ 9.091%
Hard 4/Hard 10
Note: The hard 4 and hard 10 pay 7 to 1, or 8for 1. In craps the odds on the cloth are listed on a for 1 basis, including the graphic above.
The probability of a hard 4 on any given roll is 1/36.
The probability of a 7 on any given roll is 6/36.
The probability of a soft 4 on any given roll is 2/36 (1+3 and 3+1).
The probability of winning on any given roll is 1/36.
The probability of losing on any given roll is 6/36 + 2/36 = 8/36.
The probability of winning the bet is p/(p+q) (see above) = (1/36)/(9/36) = 1/9
The expected return is (1/9)×7 + (8/9)×(-1) = -1/9 ≈ 11.111%.
The player's edge is also -1/9 since the bet is 1 unit.
The odds are the same for a hard 10.
Hard 6/Hard 8
Note: The hard 4 and hard 10 pay 9 to 1, or 10for 1. In craps the odds on the cloth are listed on a for 1 basis, including the graphic above.
The probability of a hard 6 on any given roll is 1/36.
The probability of a 7 on any given roll is 6/36.
The probability of a soft 6 on any given roll is 4/36 (1+5, 2+3, 3+2, and 5+1).
The probability of winning on any given roll is 1/36.
The probability of losing on any given roll is 6/36 + 4/36 = 10/36.
The probability of winning the bet is p/(p+q) (see above) = (1/36)/(11/36) = 1/11
The expected return is (1/11)×9 + (10/11)×(-1) = -1/11 ≈ 9.091%.
The player's edge is also -1/11 since the bet is 1 unit.
The odds are the same for a hard 8.
Craps 2/Craps 12
[(1/36)×30 + (35/36)×(-1)]/1 = -5/36 ≈ -13.889%
Craps 3/Craps 11
[(2/36)×15 + (34/36)×(-1)]/1 = -4/36 ≈ -11.111%
Any Craps
[(4/36)×7 + (32/36)×(-1)]/1 = -4/36 ≈ -11.111%
Any 7
[(6/36)×4 + (30/36)×(-1)]/1 = -6/36 ≈ -16.667%
Horn
The probability of rolling either a 2 or 12 is 1/36 + 1/36 = 2/36.
The probability of rolling either a 3 or 11 is 2/36 + 2/36 = 4/36.
The probability of roling anything else is 1-2/36-4/36 = 30/36.
Remember that the horn bet is like all four craps bets in one. Even if one wins the other three still lose. The house edge is:
[(2/36)×27 + (4/36)×12 + (30/36)×(-4)]/4 = (-18/36)/4 = 12.500%
Field
When the 12 pays 2:1 the expected return is:
2×(pr(2)+pr(12)) + 1×(pr(3)+pr(4)+pr(5)+pr(10)+pr(11)) + -1×(pr(6)+pr(7)+pr(8)+pr(9)) =
2×(1/36 + 1/36) + 1×(2/36 + 3/36+ 4/36 + 3/36 + 2/36) + -1×(5/36 + 6/36 + 5/36+ 4/36) =
2×(2/36) + 1×(14/36) + -1×(20/36) = -2/36 = -1/18 ≈ 5.556%.
When the 12 pays 3:1 the expected return is:
3×pr(2) + 2×pr(12)) + 1×(pr(3)+pr(4)+pr(5)+pr(10)+pr(11)) + -1×(pr(6)+pr(7)+pr(8)+pr(9)) =
3×(1/36) + 2×(1/36) + 1×(2/36 + 3/36+ 4/36 + 3/36 + 2/36) + -1×(5/36 + 6/36 + 5/36+ 4/36) =
3×(1/36) + 2×(1/36) + 1×(14/36) + -1×(20/36) = -1/36 ≈ 2.778%.
Buying Odds
4 and 10: [(3/9)×2 + (6/9)×(-1)]/1 = 0.000%
5 and 9: [(4/10)×3 + (6/10)×(-2)]/2 = 0.000%
6 and 8: [(5/11)×6 + (6/11)×(-5)]/5 = 0.000%
Laying Odds
4 and 10: [(6/9)×1 + (3/9)×(-2)]/1 = 0.000%
5 and 9: [(6/10)×2 + (4/10)×(-3)]/2 = 0.000%
6 and 8: [(6/11)×5 + (5/11)×(-6)]/5 = 0.000%
Combined Pass and Buying Odds
The player edge on the combined pass and buying odds is the average player gain divided by the average player bet. The gain on the pass line is always -7/495 and the gain on the odds is always 0. The expected bet depends on what multiple of odds you are allowed. Lets assume full double odds, or that the pass line bet is $2, the odds bet on a 4, 5, 9, and 10 is $4, and the odds on a 6 or 8 is $5.
The average gain is -2×(7/495) = -14/495.
The average bet is 2 + (3/36)×4 + (4/36)×4 + (5/36)×5 + (5/36)×5 + (4/36)×4 + (3/36)×4] =
2 + 106/36 = 178/36
Hard Eight Craps Scene
The player edge is (-14/495)/(178/36) = -0.572%.
The general formula if you can take x times odds on the 6 and 8, y times on the 5 and 9, and z times on the 4 and 10 is (-7 / 495) / [ 1 + ((5x + 4y + 3z) / 18) ]
Combined Don't Pass and Laying Odds
The player edge on the combined don't pass and laying odds is the average player gain divided by the average player bet. The gain on the don't pass is always -3/220 and the gain on the odds is always 0. The expected bet depends on what multiple of odds you are allowed. Lets assume double odds and a don't pass bet of $10. Then the player can lay odds of $40 for a win of $20 on the 4 and 10, $30 for a win of $20 on the 5 and 9, and $24 on the 6 and 8 for a win of $20. The average gain is -10×(3/220) = -30/220.
The average bet is 10 + 2×[(3/36)×40 + (4/36)×30 + (5/36)×24] = 30.
The player edge is (-30/220)/30 = -0.455%.
The general formula if you can buy x times odds then the house edge on the combined don't pass and laying odds is (3/220)/(1+x).
Net Gain/Loss per Session
The chart below shows the net gain or loss you can expect over 100 trials, or come out rolls. For purposes of creating the chart the player would bet $1 on the pass line and take full double odds.
Here are some actual numbers that show the probability of falling into various intervals.
Session Win/Loss
| Interval | Probability |
|---|---|
| loss of over $100 | 0.0422% |
| loss of $76-$100 | 0.6499% |
| loss of $51-$75 | 4.6414% |
| loss of $26-$50 | 16.3560% |
| loss of $1-$25 | 30.0583% |
| break even | 0.6743% |
| win of $1-$25 | 28.6368% |
| win of $26-$50 | 14.4257% |
| win of $51-$75 | 3.9097% |
| win of $76-$100 | 0.5639% |
| win of over $100 | 0.0418% |
The graph and table were created by simulating 1,000,000 sessions of 100 trials, or come out rolls, and tabulating the results of each session.
Internal Links
- How the house edge for each bet is derived, in brief.
- The house edge of all the major bets on both a per-bet made and per-roll basis
- Dice Control Experiments. The results of two experiments on skillful dice throwing.
- Dice Control Advantage. The player advantage, assuming he can influence the dice.
- Craps variants. Alternative rules and bets such as the Fire Bet, Crapless Craps, and Card Craps.
- California craps. How craps is played in California using playing cards.
- Play Craps. Craps game using cards at the Viejas casino in San Diego.
- Number of Rolls Table. Probability of a shooter lasting 1 to 200 rolls before a seven-out.
- Ask the Wizard. See craps questions I've answered about:
- Simple Craps game. My simple Java craps game.
Written by: Michael Shackleford
Craps Bets: Payouts, True Odds, and House Edges
This is a basic guide for new players of craps with regards to learning the different craps bets and payouts of the game. A brief explanation on each craps bet is also included.
Seven - A payout of 4:1 is often given by casinos on a seven even if the true odds are 5:1. The house edge for this craps bet is 16.66% thus making it a poor bet in craps.
Eleven - Even if the true odds of an eleven are 17:1, the casinos often give the payout of 14:1. The house has full advantage over this bet in craps.
Place Bets - Hitting a four or a ten has a payout of 9:5, but the true odds of this craps bet is 10:5 making the house edge at 6.66%. Hitting a five or a nine has a payout of 7:5, but the true odds of this craps bet is 7.5:5 making the house edge at 4%. Hitting a six or an eight has a payout of 7:6, but the true odds of this craps bet is 6:5 making the house edge at 1.51%.
Buy Bets - The craps players will have to pay a 5% commission to get the true odds of all the numbers. But the worthwhile bets in buy bets are when a four or a ten is rolled because they decrease the house edge from being 6.66% down to 4.76%.
Hard Ways - This craps bet can only be done on the hard four, hard six, hard eight, and hard ten. The payouts are 9:1 for the hard six or hard eight and 7:1 for the hard four and hard ten. And the house edges of these craps bets are 9.09% and 11.11% each, which makes both of them not good betting options.
Come and Don't Come Bets - These are even money craps bets that have low house edges of 1.414% and 1.402%, respectively.
Pass Line - The house edge of these craps bets is the same as that of the come bets.
Don't Pass - The craps bets that have similar house edge with don't come bets, thus making them a good bet.
Best Way To Throw Dice In Craps
The Field - This is a one-roll craps bet that has a payout of 2:1 on a two or 12 with a house edge of 5.55%.
Big Six and Eight - Even if the true odds of these craps bets are 6:5, the casinos will pay you even money and the house edge is 9.09%.
How To Shoot Craps Dice
Now that you know the payouts, true odds, and house edges of the various craps bets, make the wise decision in where to place your bets and take advantage on those bets that have lower house edges for the chance of winning.