Hard 8 In Craps

Introduction

1. How To Play Craps
2. Hard 8 In Blackjack

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.

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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

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 To Play Craps

• 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

Hard 8 In Blackjack

The Hard 8 Why do I gamble? It’s not for entertainment - though often it is entertaining. It's not so I can tip the dealers and show what a good sport I am. I play to win. It’s not just a goal. It’s the only goal. The rest of it -- the comps, the excitement, the roller-coaster thrill -- is all secondary. I’m in the casino for one reason. I want their money - and lots of it. I look at gambling as the best possible part time job for me. But as in any profession, there are certain basic tools you must have to ply your trade. I call these tools the Hard Eight. They’re really quite easy, but players around the world struggle with them every day. Let’s open the tool box and take a brief look at them. 1. Knowledge of the game. One of the silliest questions I hear people ask at the craps table is “How do you play this game?” Standing at the table with your hard-earned cash at risk is not the time to learn. Before placing your first bet you should have mastered all of the basic rules of the game, understand the terminology, have a working knowledge of the correct odds and pay-off, and be comfortable with both the pass and don’t pass sides of the game. To that end, you should read some of the top books on the game - including Scarne on Dice, Sam Grafstein's The Dice Doctor, and John Patrick's Advanced Craps. Explore the game on the internet through forums like this and through 'play for fun' sites and downloads such as WinCraps. Once you feel you have a good command of the game - head to the casion the try it out. But be sure you have a good command of the other seven tools of the trade. 2. Conservative Strategy. A friend of mine loves to play marathon craps session, but long sessions in a negative expectation game will eventually grind you down. The only way to survive these sessions over the long haul is to adopt an extremely conservative betting philosophy. In this gentleman's case, that consists of a single Don’t Pass bet on each shooter. If he wins a bet his next wager is a single Don't Pass bet with single odds. On any loss he reverts to his original single unit bet. And while he will never win a huge amount of money, he will rarely lose a large amount either. He is patient and plays his strategy flawlessly, and it achieves exactly what he wants. You must approach the table with the same type of logically conceived, conservative game plan. That doesn’t mean you have to limit yourself to the pass line or don’t pass - or limit yourself to a single bet. It simply means you step up to the table with a plan. A strategy that allows you to adjust the size of your bets based on your bankroll, minimize your losses, and maximize your wins. How? By positioning yourself to take advantage of the gambler’s friend - the almighty streak. 3. Sufficient Bankroll. I like to think of my casino chips as bullets. On the battlefield, if you run out of bullets you’re as good as dead. That’s why it is important to build a sufficient bankroll before stepping into the casino. You will always have limited funds to play with when your bankroll is compared to the casinos. The Pit Boss can roll out the reserves anytime by calling the cage and having the security boys bring out a chip refill. There are lots of ways you can use your “bullets” against the casino. You can use them like artillary, playing those long shot bets, or a machine gunner raining chips all over the table. Or, if you’re smart, you’ll launch a sniper attack, making every shot count. But to stand any chance at all in these bankroll battles, you must have enough capital to start. 4. Money Management. I can only guess at how many times a player standing next to me at the table has pointed to a stack of chips on the layout and asked, “Is that my money?” I’ve heard it hundreds of times through the years. But money management is more than watching your bets. As I mentioned before, you have to adjust the size of your bets in relation to the size of your bankroll. Money management goes beyond that, as well. It requires you to set specific win objectives and loss limits based on your total bankroll. It means knowing exactly how much you will bet in every conceivable win/loss situation you encounter. And it means having the self-discipline to execute those bets flawlessly. 5. Self Discipline. Most gamblers don’t have it. Simply put, self-discipline is how you control your emotions while gambling. A few months back I bought in at a table and -- by using good money management and discipline -- doubled my money in about forty-five minutes. At the same time, the player next to me lost a little over $14,000. The more he lost the more he relied on crazy, scared-money wagers - $100 hop bets on two or three numbers every roll, and placing the hardways for $500 each. He was pitting his bankroll bullets against the casino’s. The house had him out-manned and out-gunned. Everyone at the table could see he was destined to lose it all -- everyone except him. As the old saying goes, you gotta know when to walk away -- and know when to run. 6. Trends and Streaks. Craps is a game of streaks. Streaks of Point - Seven Out. Streaks of Point - Pass. And, most often, choppy streaks where there is no dominant trend. Since craps is a game of independent trials -- what happened on the last roll of the dice has no influence on what happens this roll of the dice. Predicting a trend is impossible. However, every forty-five minute monster hand starts out with a dozen tosses of the dice. It moves on to the five-minute mark, then ten, fifteen, and twenty. At some point virtually every player at the table recognizes what is happening and the layout fills up with chips. They have spotted the almighty streak. You can spot it too. 7. Precision Shooting. Have you ever noticed that some shooters at the craps table seem to have consistently longer hands than other players? While random trends and streaks do occur in this game, some players prefer to create their own. They do this, consciously or subconsciously, by influencing the outcome of the roll. Generally they take great care in pre-setting the dice to a particular arrangement. Then they affect a consistent, soft toss to a particular point on the table. Casino personnel often crank up the heat on these individuals in an attempt to break their rhythm. Often, though, these shooters appear unflappable as they throw the dice with zen-like precision, banging out number after number after number. Casinos fear them - and for good reason. The best of them have the ability to significantly alter the odds of the game in their favor - all within the confines of the house rules. Can you master this technique? Absolutely. 8. Winning Attitude. Let’s face it. The only one who likes a loser is the casino. And yet you hear people reinforcing a losing attitude at every turn. The science of neuro-linguist-programming - NLP for short - teaches that you can actually program your mind for success through positive affirmations. Yet so often we program ourselves for failure instead. On a recent Vegas outing I walked into the casino and turned a quick $360 profit at the craps table. Recognizing that the streak at that table was over, I colored up and strolled over to the Wheel-of-Fortune carousel to see if my companion was ready to go to dinner. “Just a minute,” she said, shaking her coin cup. “I just want to lose these last few dollars, then we can go.” Think about that. How many times have you stood in the casino and heard someone say something similar? They don’t expect to win - so they don’t. They just stand there until they throw their last chip in. Well, there you have them - the Hard Eight. Are they really that hard? Not really. Can you master them and become a consistent winner? I believe you can. Back To Heavy On...