Introduction
This document is meant to serve as a source for playing strategy for the game of Craps. Betting strategies along with mathematical odds are presented here. Basic rules for playing Craps can be found at:
http://www.DotComCasinoGuide.com/info/rules/craps.html
One good thing about the game of Craps is that the rules tend to be the same at every online and land casino. While some sites may not offer some of the less popular bets, like the big 6 or 8 which are awful to place anyway, I’ve yet to seen a site try to tinker with the general rules for the pass and don’t pass bets to give the house more of an edge. One variable which is adjusted from site to site is the amount the house will let the user place for an odds bet. Some sites offer 2X odds while others offer 3X odds for example. We will go into more detail below on how the greater the odds bet a house will allow the more beneficial it is for the player. If you do find a site which appears to “tinker” with the general rules, stay away from it!
Odds
Craps is all about the outcome of two dice thrown. Let’s start by looking at a chart showing the combinations which can be rolled.
You can see from this that there are 36 possible combinations of the dice. 7 is the most popular number here, with 6 possible combinations. Things get worse as you get away from 7, with 2 and 12 having only one combo on the board. And from this is where the bets for craps have come to be decided upon. Let’s continue our analysis by looking at the pass bet. From our basic instructions we know that the pass bet is won when the shooters rolls a 7 or 11 on the come-out roll, and it is lost if a 2,3, or 12 is rolled. 7 is the most common combination so this is a good bet for the shooter to win on their initial roll, right? Not so fast my gambling friend. With the 6 possible seven combinations and the 2 eleven combinations, the probability of the player winning on the come-out roll is 8 out of 36, or 22.22%. (8/36=0.2222) The 2,3, or 12 give 4 combinations out of 36, or a probability of 11.11% of the player losing on his come-out roll. (4/36=0.1111) While you do have twice the probability of winning on your come-out bet compared to losing, you still have a 66.67% chance that the outcome of your bet will be decided upon by whether or not you make the point that you established from your come-out roll. And this is where the house regains their edge, for you will now lose your bet if you roll one of the common 7 combinations before you roll your point. No matter what your point was, be it 4, 5, 6, 8, 9, or 10, there is still a better chance that you’ll now be rolling a 7 and losing. When all the math is said and done, you have a probability of 49.29% that you will win a pass bet, compared to a probability of 50.71% that you will lose. That gives a house edge of 1.42%. Let’s derive this below.
Bets
Pass Bet
P(W) = Probability of Winning; P(co) = Prob. of winning on your Togel Singapore come-out roll; P(p) = Prob. you will have won by establishing and making a point p.
P(W) = P(co) + P(p)
P(W) = P(7) + P(11) + P(4p) + P(5p) + P(6p) + P(8p) + P(9p) + P(10p)
The point probabilities are calculated by multiplying the probability that you will establish that point, i.e. there are 3 chances out of 36 that your point will be a 4 or 8.33%, by the probability that you will roll that point before a 7, i.e. for a point of 4 you have 3 possibilities to get a 4 compared to 6 to get a 7, or a 3 in 9 chance to now win.
P(W) = (6/36) + (2/36) + (3/36)(3/9) + (4/36)(4/10) + (5/36)(5/11) + (5/36)(5/11) + (4/36)(4/10) + (3/36)(3/9)
P(W) = 16.67% + 5.56% + 2.78% + 4.44% + 6.31% + 6.31% + 4.44% + 2.78%
P(W) = 49.29%
If there is a 49.29% chance that you will win, then there is a 50.71% chance that you will lose. 50.71% – 49.29% = 1.42% (Actually because of rounding errors the math here works out to 1.42, but if you take the numbers to more decimal places you get 1.4141414141%, so that’s what we’ll use for further discussion) And that is the house edge that you are up against for this bet. The 1.41% is not as bad as Roulette, but it is also not as good as a lot of Video Poker machines that can be found. Attempting to adjust your betting by increasing your bets after you lose in anticipation of a hot steak will gain you no more advantage than what was analyzed in our Roulette strategy guide:
Derivation of the don’t pass bet works out to be slightly better, at 1.36% house edge. I’ll post the math below. Rolling a 12 on the initial roll is considered a stand-off, and that is how the house keeps their edge instead of simply reversing the pass bet rules.
Don’t Pass Bet
P(dp) = Probability that you win by not rolling the point establish before you roll a 7.
P(W) = P(co) + P(dp)
P(W) = P(2) + P(3) + P(4dp) + P(5dp) + P(6dp) + P(8dp) + P(9dp) + P(10dp)
P(W) = (1/36) + (2/36)+ (3/36)(6/9) + (4/36)(6/10) + (5/36)(6/11) + (5/36)(6/11) + (4/36)(6/10) + (3/36)(6/9)
P(W) = 2.78% + 5.56% + 5.56% + 6.67% + 7.58% + 7.58% + 6.67% + 5.56%
P(W) = 47.93%
We must keep in mind the 12 is a standoff, so we do not have a 52.07% chance of losing, it is somewhat better, derived below.
P(L) = Probability of losing; P(ldp) = Probability you’ll lose off of that point
P(L) = P(co) + P(dp)
P(L) = P(7) + P(11) + P(5ldp) + P(6ldp) + P(8ldp) + P(9ldp) + P(10ldp)
P(L) = (6/36) + (2/36) + (3/36)(3/9) + (4/36)(4/10) + (5/36)(5/11) + (5/36)(5/11) + (4/36)(4/10) + (3/36)(3/9)
P(L) = 16.67% + 5.56% + 2.78% + 4.44% + 6.31% + 6.31% + 4.44% + 2.78%
P(L) = 49.29%
Therefore, for the don’t pass bet, the house edge is 49.29%-47.93%=1.36%. Slightly better, but about the same.
Odds Bet
There is some hope, however, and that comes from the odds bet which the user can place after establishing a point. These bets pay at true odds, and help to even up the game for the player. This bet pays at true odds, and from this bet is where we gain our strategy methods to play the game of Craps. Most guides will tell you that a lot of casinos do not publicize the odds bet since the casino has no edge, but I’ve found that most online casinos will list the odds bet in their instructions as far as where to click your mouse to make the bet. Most either have an “odds” placement point on the betting table on the screen, or use the traditional method of allowing the user to place the bet by clicking directly behind their chips on the pass line spot. When you make the odds bet after a point has been established from your pass bet, you are betting the additional amount that you will roll your point before you roll a 7 and lose. For a 6 or an 8 point, the bet pays at 6:5. For a 5 or a 9 point, the bet pays at 3:2. For a 4 or 10 point, the bet pays at 2:1. You can see here that these are the true odds for the game. For example, with 6 possible 7 combinations on the board, and 5 possible 6 or 8 combinations, the true odds are as simple as that: 6 versus 5. If you’ve placed a don’t pass bet, you can lay an odds bet as well, but you’re betting that the 7 will be rolled before your point, so the true odds bay less than 1:1, 5:6 for example, but they are still true odds. The math works out almost identical, so I won’t go into details. Now that we’ve proved the casino is paying at true odds, let’s go through the math to see exactly how this works to our advantage.
We know from the math we’ve done on the pass bet that the house edge here is 1.41%. What the odds bet does is help reduce this number closer to a 0% house edge. No matter how great of an odds bet a casino will offer, the house will still have an edge greater than 0% overall. I’ve seen land based casinos advertise on billboards outrageous odds bet allowances, but the greatest we’ve reviewed here at .com Casino Guide has offered 3X odds. I’ll use the 3X allowance for my examples. What the 3X odds bet means is that the casino will allow you to place an odds bet at 3 times the amount of your initial pass bet. So for example, if you bet $10 on a pass bet you will be allowed to wager $30 on your odds bet. Therefore you now have $40 in total wagers. The good news here is that $30 out of that $40 is being played at true odds! Only the $10 pass bet is subject to the 1.41% house edge. This calculates to an overall house edge of less than 1.41%. Let’s derive the house edge we are up against now. The main thing to keep in mind here is that 33% of the time your bet is decided on the come-out roll so you do not get to always place your odds bet. (one 2, two 3s, six 7s, two 11s, and one 12) (12 / 36 = 33%)
x = Amount being wagered on pass bet
3x = Amount being wagered on 3X odds bet
Let HE = House edge
HE=(1.41%)(x) / ((33%)(x)+(67%)(3x+x))=0.471%
Intuitively, this is derived from the fact that your expected loss, no matter if you place the odds bet or not, is 1.41% of your initial pass bet. However, 67% of the time you are able to wager an additional 3X that amount, or 4X for a total wager. 33% of the time you are only able to wager X amount. We can quickly calculate the overall house edge for any N(X) factor.
Odds Wager House Edge
1X 0.85%
2X 0.61%
3X 0.47%
4X 0.39%
5X 0.33%
6X 0.28%
7X 0.25%
8X 0.22%
9X 0.20%
10X 0.18%
100X 0.021%
Now let’s talk about some betting strategies to allow us to use all of this to our advantage. While there are other bets in craps besides the pass/don’t pass and odds bets, none have a lower house edge than these. I will go into the math later in this document to show why you want to stay away from them. For the following betting strategies let’s use the pass bet with 3X odds limits.
For starters, let’s say you have $1,000 to wager at a casino, and let’s say you want to place at least 200 bets in a session. We know we’ll be winning as well as losing, so we can probably get away with $20 per bet. We also must keep in mind that the odds bets will be costing us 3X each placement. At $20 per bet multiplied by 200 bets, this gives $4000 wagered. At a 1.41% house edge, this means that we expect to lose approximately $56.56 on the session. Anything less than this and we’ve beat the odds.
Let’s write out what we expect to mathematically happen during those 200 bets. Of the 200 bets placed, we expect to be able to place an odds wager around 133 times. If we are able to place more than 133 odds wagers, then this will only work in our favor. Of the 133 odds wagers we expect to be rolling for a 4 or 10 point 25% of the time. (6/24), a 5 or 9 point 33.33% of the time (8/24), and a 6 or 8 point 41.67% of the time. (10/24) The odds that we’ll win those are the same as the true odds that they pay, 2:1, 3:2, and 6:5 respectively. For the other 67 wagers, we expect to win twice as many as we lose on the come-out roll. (8 combinations of 7 or 11 compared to 4 for 2,3 or 12) Here’s a chart.
$20 bets
Come-Out Quantity Point Roll Win/Lose Money Won/Lost
2 5.55 NA -$111.11
3 11.11 NA -$222.22
4 16.67 5.55W/11.11L -$111.11
5 22.22 8.89W/13.33L -$88.89
6 27.78 12.63W/15.15L -$50.51
7 33.33 NA +$666.67
8 27.78 12.63W/15.15L -$50.51
9 22.22 8.89W/13.33L -$88.89
10 16.67 5.55W/11.11L -$111.11
11 11.11 NA +$222.22
12 5.55 NA -$111.11
Total 200 -$56.57
This agrees with our 1.41% house edge on $4000 worth of bets wagered. Since the odds bets on the point rolls pay even money, the math says that those will cancel out. You can see here that all of our + money is on the come-out rolls of 7 and 11. So what happens if we get a little luckier, say we actually roll 40 7s instead of 33.33. Well that now shifts the balance $133.33 ($20X6.67 bets) to the positive side. Our net would now be $210.10! (-$56.57 + 2 ($133.33)) So what are the odds of this happening? Well, there’s some things we’ll take for granted and assume the 6.67 winning 7 bets either replace losing come-out rolls or lost rolls on the point, but all in all it’s around 38%. (Determined from glancing at some rough Poisson probability estimates). Of course then there is also a 38% chance you’ll roll 6.67 fewer 7s, and then the rest of the probability is spread in between. An easier way to think of this is that you’ll need to get lucky on 6.67/200 bets, or 3.33% of your bets. Give yourself 200 bets in practice, in a lot of sessions you’ll get an extra 3% lucky rolls! But what does this prove? You get lucky and win, but we all know that if the dice go your way then you’ll walk away a winner, so let’s instead concentrate our efforts on forming a sound betting strategy.
After 200 bets we expect to lose $56.57, therefore after 50 bets we expect to lose $14.14. 50 bets at $20 will go through one cycle of your $1000. Perhaps set a minimum limit for yourself to be at after 50 bets to decide whether or not you should continue your session. If you’re down more than double the expected loses at this point, say $30, then cut yourself off for the night. This will give you an out so that you can survive to gamble another session. 50 bets will average you being able to place 33.33 odds bets. At 3X, this is $666.67 additional wagered. Since this money pays at true odds, we would expect to break even. One way to secure your winnings is to pocket a certain amount of what you have won at the true odds bets. Since the 4 or 10 point comes at a 25% clip compared to the 5,6, 8 or 9, you could plan to keep the odds bets won on these points. That would leave you with around $166.67 secured per 50 bets. This is less money that you’ll wager against the house and their 1.41% edge on your next round of pass bets!
What becomes difficult with online gambling is that you can’t “pocket” chips. Somehow you have to keep track of how many bets you have made to use the above methods. A more practical method is to set some type of dollar amount you won’t let your balance dip below during your session. Even better is if you’re at an online casino that will let you “cash-out” any dollar amount at anytime during your session without automatically going through some fee withdrawal mess. Perhaps every time you are up 5% or 10% you cash out at this amount. For example, if you begin with a balance of $1000, cash out every time you find yourself up $50 and pocket that. ($1050) If your balance dips to $900, cash out every time you find yourself up $40. ($940) As your overall balance goes down keep pocketing a little bit as you hit a winning streak. This will prohibit you from gambling yourself down to $0 which is what nobody wants to do!
Other Bets
Come/Don’t Come
The come and don’t come bets in craps are identical odds wise to the pass and don’t pass bets. Their sole purpose is to have new bets constantly flowing to the Craps table while the shooter is trying to make their point.
Win Bets
Don’t let the name fool you, the house still usually wins these. These are bets that a number will be rolled before a 7. I will chart the payouts and house edges below.
Bet Payout True odds House Edge
Win 4 9:5 (1.8:1) 6:3 (2:1) 6.67%
Win 5 7:5 (1.4:1) 6:4 (3:2) 4.00%
Win 6 7:6 (1.167:1) 6:5 1.51%
Win 8 7:6 (1.167:1) 6:5 1.51%
Win 9 7:5 (1.4:1) 6:4 (3:2) 4.00%
Win 10 9:5 (1.8:1) 6:3 (2:1) 6.67%
Win 4,10 House Edge Derivation
HE = Loses – Winnings
HE = (6/9)(1) – (3/9)(1.8) = 6.67%
Win 5,9 House Edge Derivation
HE = (6/10)(1) – (4/10)(1.4) = 4.00%
Win 6,8 House Edge Derivation
HE = (6/11)(1) – (5/11)(1.167) = 1.51%
Lose Bets
They got the name right here at least, as you will lose your money if you play these. In all seriousness though, the win bet with the 1.51% house edge, and lose bet at 1.82% house edge aren’t bad I guess compared to some casino games. The lose bets are that a 7 will be rolled before a number that you have picked. They pay at odds less than 1:1, hence the house retains their edge. Note, I’ve seen variations of these payouts before at different online casinos. This is probably worse case scenario here.
Bet Payout True odds House Edge
Lose 4 2:5(0.4:1) 3:6(1:2) 6.67%
Lose 5 4:7(0.57:1) 4:6(2:3) 5.71%
Lose 6 4:5(.8:1) 5:6 1.82%
Lose 8 4:5(.8:1) 5:6 1.82%
Lose 9 4:7(0.57:1) 4:6(2:3) 5.71%
Lose 10 2:5(0.4:1) 3:6(1:2) 6.67%
Lose 4,10 House Edge Derivation
HE = (3/9)(1) – (6/9)(0.4) = 6.67%
Lose 5,9 House Edge Derivation
HE = (4/10)(1) – (6/10)(0.57) = 5.71%
Lose 6,8 House Edge Derivation
HE = (5/11)(1) – (6/11)(0.8) = 1.82%
Buy Bets/Lay Bets
Buy and Lay bets are similar to the win and lose bets, except you pay the house a 5% commission in return for the payout of these bets to be paid at true odds. The 5% commission negates all benefits of the bet paying at true odds and should be avoided. What you must watch is how online casinos round their payouts. Usually you must wager at least $20 since that is the minimum amount to get the 5% commission to be at least $1. Any bet less than this and at most casinos you will still be paying at least $1 in commission as they round up.
Hardway Bets
This is a bet that a 4, 6, 8, or 10 will be rolled as a double before a 7 is rolled or the number you picked. For example betting on a hard 4 means that you must roll a pair of twos before either rolling a 7 or a combo of 3 and 1 to total 4. None of these bets are good, for the hard 4 and 10 have a house edge of 11.11%, and the house edge for the hard 6 and 8 is 9.09%. We can chart and derive below.
Bet Payout True odds House Edge
Hard 4 7:1 8:1 11.11%
Hard 6 9:1 10:1 9.09%
Hard 8 9:1 10:1 9.09%
Hard 10 7:1 8:1 11.11%
Hard 4,10 House Edge Derivation
HE = (8/9)(1) – (1/9)(7) = 11.11%
Hard 6,8 House Edge Derivation
HE = (10/11)(1) – (1/11)(9) = 9.09%
Big 6/Big 8
These are two bets that really have no business being on the Craps table. This is a bet that a 6 or 8 will be rolled before a 7. The only different between it and the win 6/8 bet is that it pays at even money as opposed to 7:6. This amounts to a house edge of 9.09%.
Big 6,8 House Edge Derivation
HE = (6/11)(1) – (5/11)(1) = 9.09%
Field Bet
The standard for this bet is that a 2,3,4,9,10,11 or 12 will be on the next roll of the die. Pays even money except for 2 or 12 which pays at 2:1. House edge is 5.55%.
Field Bet House Edge Derivation
HE = Loses – Winnings
HE = P(5) + P(6) + P(7) + P(8) – 2 * P(2) – P(3) – P(4) – P(9) – P(10) – P(11) – 2 * P(12)
HE = 4/36 + 5/36 + 6/36 + 5/36 – 2 * (1/36) – 2/36 – 3/36 – 4/36 – 3/36 – 2/36 – 2 * (1/36)
HE = 0.1111 + 0.1389 + 0.1667 + 0.1389 – 0.0556 – 0.0556 – 0.0833 – 0.1111 – 0.0833 – 0.0556 – 0.0556
HE = 5.55%
Any Seven
The seven bet is that a seven will be rolled next. This bet pays at 4:1. House edge is 16.67%. Needless to say, this bet is stupid.
Any Seven House Edge Derivation
HE = (30/36)(1) – (6/36)(4) = 16.67%
Eleven
The eleven bet is that an eleven will be rolled next. This bet pays at 15:1. House edge is 11.11%
Eleven House Edge Derivation
HE = (34/36)(1) – (2/36)(15) = 11.11%
Any Craps
This is a bet that either a 2, 3, or 12 will be thrown on the next roll. This bet pays at 7:1. House edge is 11.11%.
Any Craps House Edge Derivation
HE = (32/36)(1) – (4/36)(7) = 11.11%
Craps 2, 3 or 12
These are bets that a 2, 3, or 12 will be the next roll. None of these bets are good.
Bet Payout True odds House Edge
Craps 2 30:1 35:1 13.89%
Craps 3 15:1 17:1 11.11%
Craps 12 30:1 35:1 13.89%
Craps 2,12 House Edge Derivation
HE = (35/36)(1) – (1/36)(30) = 13.89%
Craps 3 House Edge Derivation
HE = (34/36)(1) – (2/36)(15) = 11.11%
Horn Bet
This is a bet that your next roll will be either a 2, 3, 11, or 12. This bet pays at 27:4 on the 2,12 and 3:1(12:4) on the 3,11. House edge is 12.5%
Horn Bet House Edge Derivation
HE = ((30/36)(4) – (1/36)(27) – (2/36)(12) – (2/36)(12) – (1/36)(27))/4 = 12.5%
Summary
As you can see from the odds presented here, the best bet to make are the odds bets which pay at true odds. If you’re looking to up your wagering, save your money for these, instead of increasing your amount on the pass line bets. The edge is always in the house’s favor, but only by a small margin compared to some casino games. You can expect to see a share of winning sessions over a period of time. As long as you limit your losing sessions, you have a chance to have a lot of entertainment and win some money now and then.
