In the game of craps using two dice, a person wins on the first roll if a 7 or an 11 is rolled. Find the probability of winning of the first roll. A person flipped a coin 100 times and abtained 73 heads. Can the person conclude that the coin was unbalance? Know the Die Roll Probabilities In a completely random game the chances of any given number on either die being rolled is 1 in 6. The chance of rolling any combination of numbers on the dice is 1 in 36. This “1 in 36” number can mislead you.

• Craps Dice Roll Probability Rules
• Craps Dice Roll Probability Rule
• Craps Dice Roll Probability Dice

The majority of casino games are known to offer either a low or a high house edge. Craps stands out from the crowd because it offers both. For one thing, this unique game features some of the best bets you can possibly make in a casino. For another, some of the more advanced, proposition bets in craps give the house a formidable edge over patrons that nearly reaches 17%!

Another distinctive trait of craps is that it is the only game of chance where players can actually bet something will not happen instead of backing the outcome they believe is most likely to occur, as is the case with Don’t Come and Don’t Pass bets.

These two features are what render this simple game of chance so unique and appealing in the eyes of millions of gamblers around the world.

As simple as craps seemingly is, you most definitely should take the time necessary to learn all the possible dice combinations along with their odds and probabilities before you invest any of your money in the game.

The purpose of this part of the guide is to introduce you to all dice combinations in craps and to help you make a distinction between true and casino odds. By the end, you will know whether dice control is effective in reducing the house edge and will be able to calculate your average expected losses at the craps tables.

Possible Dice Combinations in Craps

As we have previously explained in this guide, craps is a game of chance that plays with two six-sided dice, with each side having a different number of pips so as to represent numerical values 1 through 6. Each toss of the two dice can result in one of 11 possible numbers, namely numbers 2 through 12.

When two dice are in play, the number of possible dice combinations increases to 36 since each dice is practically a cube with six sides (6×6 = 36). Now, we want you to take a closer look at the table below and see whether you will be able to notice a trend.

You have probably noticed the column with the possible combinations is diamond-shaped. From this shape, it becomes apparent the number of combinations that may result in a toss of 7 is the highest, which automatically means the probability of this number being tossed is the highest.

As many as 6 combinations result in a 7 which is the main reason why the majority of seasoned craps players favor the Pass Line where a come-out roll wins when 7 or 11 are tossed, with a total of 8 possible winning combinations (six for a toss of 7 and two more for a toss of 11).

Do not be intimidated by the chart – there is one easy way for you to learn the combinations by heart. All you have to do is use 7 as a starting point and divide the remaining outcomes into groups of two. You can pair rolls of 6 and 8 since both outcomes have 5 possible combinations.

Next in line is the pair of 5 and 9, with each of these rolls having 4 possible combinations. The next pair comprises rolls of 4 and 10 with three dice combinations. Then you have 3 and 11 where the number of possible combinations drops with a unit to two.

The rolls of 2 and 12 are the easiest to memorize since there is only one possible dice combination for each of the two outcomes. Learning these combinations is essential because it helps you gain a better understanding of the odds and probabilities in craps. We tackle the subject in the next section.

Craps Dice Roll Probability Rules

Understanding the Odds and Probabilities of Craps Bets

This is the part most gamblers struggle with. Craps, like all other games of chance, is based on independent trials, which is to say the odds of the dice rolls remain constant and are not influenced by previous outcomes. This is easily the most important aspect of the game all craps players must understand. Yet, there are people who would invest their action in a given dice outcome simply because they have not seen it occur in a while.

Here it is important to draw a distinction between odds and probabilities. While interrelated, the two terms do not denote one and the same thing as many gamblers mistakenly presume. Probability is nothing more than the likelihood of an independent outcome occurring and is usually expressed either as a proper fraction or as a percentage.

To calculate the probability of rolling a certain value, you must know the number of dice combinations that result in it as well as the number of all possible dice combinations. For instance, the probability of tossing a 7 on the come-out roll is the highest at 16.67%. We can also say the probability is 6 in 36 because there are 6 winning combinations out of 36. This is called theoretical probability.

Odds, on the other hand, show you the ratio between winning and losing outcomes and are usually expressed as fractions. So for example, if you decide to make a Snake Eyes bet, which is a wager on a single roll of 2, the odds of winning will be 1 to 35 because there is only one winning combination for this roll while the remaining 35 combinations result in a loss. This corresponds to a theoretical probability of 2.78% because 1/36 = 0.02777 x 100 = 2.78%. Consult the table below to see the probabilities of rolling all two-dice totals in craps.

The Difference between True Odds and Casino Odds

It is of essential importance for you to understand there is a distinction between true odds and casino odds. It is the discrepancy between the two that gives the house the edge that eventually grinds the bankrolls of imprudent gamblers down to nothing.

If you hang around a casino for an hour or so, you are more than likely to hear (losing) players complaining about the games being rigged as they exit the venue with empty pockets. These players are, in fact, correct to a certain extent. The games are indeed rigged against them but not because the casino resorts to cheating. Why cheat your patrons when there is a perfectly legal way of extracting their money?

Another Example

The main reason gamblers lose in the long term is the above-mentioned discrepancy between true odds and the odds the house pays you at when you win. The following example demonstrates how the house edge in craps works. Let’s presume you decide to make 36 consecutive Snake Eyes bets wagering a dollar on each. The mathematically correct odds for this bet are 35 to 1, which is to say 35 of all possible combinations result in a loss whereas only 1 combination (1-1) can lead to a win.

Now take a quick glance at the layout of the craps table. Can you see what it says? The payout for a one-roll Snake Eyes bet on 2 is 30 to 1 instead of being 35 to 1, as it should be. The same goes for a roll of 12 where again there is a single winning combination (6-6). This applies to all payouts in craps – they have all been reduced, making it possible for the house to extract consistent profits from its tables in the long term.

So you wager a dollar on Snake Eyes a total of 36 times in a row and the results are mathematically perfect, meaning that one of those 36 trials was indeed a winning one. Now, this is unlikely to happen in the short term but generally, the more you play, the closer your results get to the mathematical expectations for the game.

Players can easily figure out what the house edge is for any available bet in craps as long as they know its true odds, its payout, and its probability. Let’s demonstrate how this works for the Any 7 bet. This wager wins on a roll of 7, regardless of which one of its six combinations occurs. The payout for a winning Any 7 bet is 4 to 1. The probability is 6 in 36 possible dice combinations.

The calculations will run in the following manner: (6/36) x 4 – (30/36) = (0.166 x 4) – 0.833 = 0.666 – 0.833 = (-0.166) x 100 = -16.67. The figures in the first brackets correspond to the probability of winning with an Any 7 bet, which is then multiplied by the casino’s payout of 4 units per unit wagered. You then subtract the probability of losing (the figures in the second brackets stand for 30 ways to lose out of 36 possible combinations) from the result and get a house edge of 16.67% for the Any 7 bet.

As you can see, you will lose $16.67 (hence the “-”) per every $100 wagered on the Any 7 bet. In other words, your expected return with this bet will be in the negative at -$16.67. If you are still struggling to understand this, we suggest you go back to the Craps House Edge article of this guide where you will be able to find further explanations on the matter along with the true and casino odds of all available craps bets.

Distinguishing between “To” and “For” Odds

The trouble with most casual gamblers is that they are so engulfed in the action, they rarely pay any attention to anything else, including what’s in front of their eyes, right there on the table layout. If you take the time to carefully inspect craps layouts, you will undoubtedly notice there is something weird about the proposition bets section at some craps tables.

In some casinos, the layout states that you get 5 units “for” 1 unit wagered on Any 7 instead of the usual payout of 4 “to” 1. The same goes for proposition bets like the Hard 6 and the Hard 10 which pay 10 unit for each unit wagered instead of the usual 9 to 1. Many undiscerning gamblers are misled by this phrasing (that was the purpose in the first place) and are quick to assume their winning proposition bets return at enhanced odds. There is nothing of the kind, though.

The “for” payouts are basically the same since they include your initial stake. In contrast, the phrasing “to” distinguishes your net profits from your original bet. So in the case of the Any 7 bet, you get 4 units in net profits plus your original stake of 1 unit for a total of 5 units.

The wording of these payouts is no coincidence. Quite the contrary, the house uses this sly approach for the purpose of leading inexperienced craps players into believing they get higher payouts on the wagers with the steepest house edge when in reality, they are paid at standard casino odds. No matter what phrasing is used, we would like to remind you not to waste your money on proposition bets. The monstrous house edge you combat with these wagers is not worth it even if the payouts were indeed “enhanced”.

Figuring Out Your Expected Loss in Craps

This section is somewhat a continuation of the True Odds one. You are probably scratching your head wondering why on earth would we teach you how to calculate your expected losses. What you want to know is how to determine your hourly expected profits, right? We hate to break this to you but craps is a negative expectation game which is to say your expected value in the long run will always be in the negative due to the house edge, i.e. you will inevitably lose money to the casino in the long term.

It is important for you to learn to calculate your expected loss so that you know at what average rates you will lose money per hour. This depends on your action, the types of bets you make, and their house edge. The formula is quite simple – you must multiply the number of rounds you play per hour by the amount you wager, the house edge, and the number of hours you intend to play.

So, let’s suppose you play at a slower pace and go through 160 rolls, betting $5 on Any 7 for one hour. The calculations will look like this: 160 x $5 x 0.166 x 1 = $133 on average. This is the average rate at which you lose money with Any 7 wagers but you arrive at this amount after a gazillion of independent trials. In the short term, you might end up losing much more or far less within an hour. The bottom line is the longer you play, the closer you get to these expected loss figures.

Can Dice Control Influence the Probability of Rolling Certain Combinations?

Some craps players would attempt to influence the outcome of the rolls by using an advanced technique, called dice setting or dice control. The main idea here is that you can skew the odds in your favor by tossing the dice in a specific manner. To achieve this, the shooter must toss the dice at the correct angle in order to allow them to produce the desired outcome.

The toss itself should be performed with as little hand movements as possible so that the dice do not tumble as much before they hit the back wall of the craps table. The dice must be picked in a particular way as well. The shooter’s wrist remains locked during the toss, i.e. there is no twisting motion when the dice are thrown. It makes sense this technique takes ages to master but the real question is does it really help you influence the outcome of the roll?

Renowned gambling author Stanford Wong is among the proponents of the efficiency of this technique and wrote extensively about it in his book Wong on Craps. The subject was also tackled by author Christopher Pawlicki in his 2002 book Get the Edge at Craps: How to Control the Dice. Some mathematicians and gambling experts are of the opinion the technique either does not work or is impossible to execute successfully in the casino environment.

Others are willing to give dice control some credence arguing that it might possibly work, provided that the dice do not hit the back wall of the craps table. Unfortunately, most casinos require the dice to hit the back wall in order for the roll to be considered valid. The bottom line is most members of the advantage play community still distrust the concept of overcoming the house edge in craps through dice control.

Beginning gamblers often shy away from the craps table because the game looks complicated. It’s actually easy to play craps because the math keeps everyone honest. A rule of thumb to live by in any casino game is “the more they pay the less likely you are to win the bet”. Hence, there is no shame and a lot of wisdom in playing a conservative craps strategy. Here is a look at 12 secrets every craps player should learn to improve their game.

1. Why are Casino Dice Special?

Casinos use transparent dice because they hide no flaws. Opaque dice can be manufactured to varying standards and can hide balancing flaws. Unbalanced dice do not roll randomly.

And casinos replace their dice often. Casino dice have machine-tooled straight edges. These edges eventually wear down, accumulating imperfections. Imperfections add bias to rolls.

Casino dice are larger and straighter than board gaming dice because players must throw the dice so far on a craps table. The felt top and lining help the dice bounce more randomly than a smooth table top does.

So while you may be practicing your die throws at home, you’re not going to get the same action as at a casino, especially if you never replace your practice dice.

Craps Dice Roll Probability Rule

2. How the 5-Count System Works

Since 1994 craps players have debated whether the Captain’s 5-Count system is legit. This system tells you when to bet on a shooter other than yourself. Here are the 5 counts:

• Any point on the Come Out roll roll.
• Any good roll after the 1^st Count roll.
• Any good roll after the 2^nd Count roll.
• Any good roll after the 3^rd Count roll.
• The first point rolled after the 4^th Count roll.

You begin placing low bets on the shooter after he hits his 5^th Count roll. If he never gets there then you never bet on that shooter. Never bet big on another shooter.

The 5-Count method reduces the number and size of bets you place on other shooters, thus reducing your overall risk. The downside of using the 5-Count method is that you watch more than play, but betting on a drunk guy to throw dice the way you want is a pretty risky bet.

3. You Can Stop the Game for a Dispute

Sometimes the dice roll funny, or maybe you’re not sure you were paid correctly. Before the dice are thrown again, if you are certain something is wrong, you can stop the game. You can ask the dealers to recount or reconsider or, if you disagree with their decisions, ask to speak to the pit boss. This is an option of last resort when you are sure you are right. Casinos want to keep the table in play and will work to resolve disputes quickly but they’ll also ask troublesome or argumentative players to leave.

Stopping play is a mix of courtesy, privilege, and right. It’s not a gambling strategy, at least not a winning one.

4. The More Bets You Place the Worse Your Chances of Winning

This is true in any table game, but some craps players love to place multiple bets. You’re taking on more risk, not spreading the risk, when you place several bets at the same time.

5. Know the Die Roll Probabilities

In a completely random game the chances of any given number on either die being rolled is 1 in 6. The chance of rolling any combination of numbers on the dice is 1 in 36. This “1 in 36” number can mislead you. There are only 11 possible values (2 through 12) that you can roll.

“7” is the most frequent die roll combination. There are 6 ways to roll a “7”. Some writers say there are three ways to roll a “7”: 1 and 6, 2 and 5, or 3 and 4. However, the math has to account for each die separately; hence, the probability of rolling a “7” in craps is 1 in 6.

In declining order of probability, the possible combinations in craps are:

• 7 (1 in 6)
• 6 or 8 (5 in 36)
• 5 or 9 (4 in 36)
• 4 or 10 (3 in 36)
• 3 or 11 (2 in 36)
• 2 or 12 (1 in 36)

6. The “Pass” Bet is More Likely to Pay on Come Out than the “Don’t Pass” Bet

Both Pass and Don’t Pass pay even money so you can bet either way. Still, when you look at the probability table above, the shooter has 8 chances in 36 of rolling 7 or 11 on the Come Out roll and 3 chances in 36 of rolling a 2 or 3. If you are just hoping to win on the Come Out roll, go with the “Pass” bet.

7. The 6 and 8 Points Pay the Most over Time

The 6:5 odds for the 6 and 8 points are the worst and the 2:1 odds for the 4 and 10 points are the best. But the probabilities are best for the 6 and 8 and worst for the 4 and 10.

The premium on a 6:5 payoff for 6 or 8 is 20% over your bet. The premium on a 3:2 payoff for a 5 or 9 is 50% of your bet. The premium on a 2:1 payoff for a 4 or 10 is 100% of your bet. In a perfect distribution of 36 die rolls your expected total premiums are:

• 5 * 20% = 100% (betting on 6 or 8)
• 4 * 50% = 200% (betting on 5 or 9)
• 3 * 100% = 300% (betting on 4 or 10)

Although the 300% ROI for 4/10 looks great there is a slight edge for 6/8 bettors. Because you are losing all those other bets, you lose the least amount of money with the 6/8 points. Note also that multiplying (bets + premiums) by expected wins across the board results in a 600% return. The distribution with the fewest losses is the way to bet.

8. The More Complicated Your Strategy the More Risk You Take

The more you have to think about where your money goes, the odds and probabilities, and when you can bet, the more likely you will make a mistake. High risk strategies pay off less often than low risk strategies. Most experts agree that the long, slow game works best in craps, especially for non-expert players. Keep your money on the Pass Line until you’re way ahead.

9. Avoid Hedge Bets

Ignore dealer calls for “any craps” bets. Your expected return declines your risk grows when you hedge bets. “Any craps” betting is a bet on a bet. This just adds conditions to your Pass Line bet. The strategic way to gamble is to minimize risk while maximizing potential return on bet. The house will drain your bankroll any way it can and hedge bets are a favored gimmick.

10. Use the Tower of Hanoi Method to Manage Your Betting

The Tower of Hanoi is a math puzzle about moving stacks of disks among three pegs. You can never place a disk on a smaller disk. The Tower of Hanoi rule assumes you are willing to lose everything in your bankroll. To conserve your money and manage risk, begin by making minimum bets. Increase your bets only when your bankroll is above its starting value.

Many craps players only risk 5% of their stakes on any bet. The 5% method works well enough but you’ll eventually run into the table minimum. The Tower of Hanoi method starts with the minimum bet as a floor, not 5%. As long as your bankroll is growing you can increase your bets toward the table maximum.

11. Never Return to Your Starting Stake

Let’s say your betting strategies have paid off enough that you have doubled your money. Once you reach that goal you should set a new floor. Walk away from the table if your stake drops to 150% of your original bankroll. This way you walk away a winner.

But there is another reason to do this. If you play any game too long you become tired, especially if you have been drinking. Your decision-making suffers when you are tired. Take “winner’s breaks” as often as possible so that you can give your brain a chance to rest.

12. The House Edge is not Determined by the Odds

Craps Dice Roll Probability Dice

Some gamblers assume the house loses more money on the basis of the odds on a given bet. It doesn’t work that way. The game is designed to pay about the same over time on any basic bet but to dilute your return with extra bets. In other words, the house edge is determined by the math behind the game. The odds are just what they are willing to pay you to maintain that edge over time.

Conclusion

Craps is a fine game for any gambler who enjoys taking risks, but you do need to understand the game. Fortunately, craps is designed for players of all experience levels. You don’t have to play all the different types of bets. And isn’t it interesting that the best strategies favor beginner-level bets anyway?