Odds In Craps

• Craps Betting Strategy
• Best Odds In Craps
• Best Way To Play Craps And Win

The don't pass line bet used with maximum don't pass odds bet is the absolute best bet you can make in craps or anywhere else in the casino! This has slightly better odds than even the regular pass line bet used with pass odds, so it gives you the best mathematical odds possible. Here is an overview of craps bets with medium odds that you can make. Have a 16.7% chance of winning an Any 7 bet, with a payout of 4/1. This gives the house a substantial edge of 16.9%. This is quite high compared to some of the bets we've seen above, yet still a solid choice if you're chasing more slightly more daring bets. True odds for the 7 against the 4 are 2:1, so divide the $200 bet by 2 = $100, and then multiply the $100 by 1 = $100. The vig is based on the $100 win amount, so $100 x 5% = $100 x 0.05 = $5, which is a whole dollar amount, so it isn't rounded.

Since craps is a game of chance, you need to understand why you have a greater or lesser chance of rolling different numbers. Because you're rolling two dice, your chances of rolling a specific number in craps are determined by the number of die combinations that can add up to that number. For example, 2 can only be rolled with two 1s, but 4 can be rolled with either a 1 and a 3 or two 2s. That means you have twice the chance of rolling a 3 as you do a 2. Because the 7 has the greatest number of combinations (six), it is the number that has the potential to come up most often, which is why 7 is the magic number in craps.

There are 36 possible number combinations in craps. Here is a chart showing the possible combinations for each number using two die.

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From the chart, you can see that the most likely number you'll roll is a 7, followed by the 6 and the 8, then 5 and 9, then 4 and 10, then 3 and 11, and finally (and least likely) the 2 and the 12. This means you'll roll a 7 once out of every six rolls, a 6 or an 8 once out of every 7 to 8 rolls, and so on.

Odds for Each Number

By looking at the possible combinations, the 'true odds' for each number can be established. Knowing the odds in craps is good so you have a feel for the likelihood of one number being rolled before another one (e.g., is the 4 going to be rolled before the 7?).

House Edge

Now, true odds are not what the casino pays you unless you're also betting 'free odds' on top of your main bet. Free odds, which is an additional wager you place with your original line bet, pay true odds so the casino's edge is reduced. (We'll talk more about free odds in Strategies the Winners Use.)

To better explain how the casino edge works, let's take the example of flipping a coin. You have a 50/50 chance of the coin landing on heads, and a 50/50 chance of it landing on tails. If that were a bet on which you were being paid true odds, you would be paid even money. The casino, however, has to have an edge in order to make a profit on the game. So, the payoff for any given bet is less than what true mathematical odds would dictate. For example, on a bet that had true odds of 1:1, you would think that if you bet $1 and win, you would be paid $1 in winnings. But in a casino, depending on the bet, you might only be paid $.96. The difference between the true odds and what they pay you is how they make money -- it's called casino odds.

Another way to better understand casino odds versus true odds is to look at the definition of the casino (or house) edge. WizardOfOdds.com defines it as, 'The ratio of the average loss to the initial bet,' going on to explain that it's based on the original wager rather than the average wager so that players can have an idea of how much they are going to lose when they place a bet. For example, by knowing that the casino has a 1.41 percent edge in craps, you can know that you'll be losing 14.1 cents for every $10 bet.

For charts of odds for all types of bets, visit the Wizard of Odds.com.

On the next page, we'll talk about types of craps bets, their odds and what the casino pays for each.

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

Craps Betting Strategy

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

Best Odds In 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) =
How to leave guild wow classic. 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.

Best Way To Play Craps And Win

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

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.