Dice Combinations and Probabilities in Craps

The majority of casino games are based solely on chance and the outcome often depends only on random events rather than players’ skill and decision-making. But it is exactly this unpredictability that makes gambling games so appealing and craps is no exception – to choose the best bets, however, players should understand the concept of probability.

Probability is a mathematical tool that measures the likelihood of an event occurring. It is used in various areas of science and statistics and basically, it shows us the chances that something will happen. Probability, often used as equivalent to odds, also helps casino players distinguish the good and bad bets in games. In craps, it is fundamental to playing correctly and increasing one’s chances to win.

The probabilities for winning and losing on each roll of the dice can be determined easily once we know all possible ways the pair of dice could land after being tossed in the air. This article focuses on the importance of odds and probabilities in craps by explaining what the dice combinations are, how to calculate the probabilities, and how all this helps players find the bets with the highest expected return.

Possible Dice Combinations

Dice Probabilities

The game of craps uses two identical, six-sided dice, which are tossed by the shooter and the outcome of the game round depends on the sum of the values of the dice. Each of the two dice may show numbers from 1 through 6, which means that if you toss one of them, the chances for hitting any number is 1 out of 6.

Now, this is quite simple but calculating the mathematical probability for a random roll of a pair of dice could be very confusing for many people. So, we should start by seeing all possible outcomes – for instance, it is possible to roll the number 2 only if both dice show 1. For a total of 3, we have two possible combinations – one dice showing 1 and the other one 2, or vice versa (1-2, 2-1). The highest number that we can roll is 12 and the only way to do that is by throwing two 6s.

Meanwhile, 11 different totals can be rolled with a pair of standard dice – these are the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. We can instantly calculate the number of possible combinations by multiplying the six sides of the first dice to the six sides of the second dice – 6×6, which is 36. Of course, certain numbers are more likely to be rolled than others and this depends on the number of dice combinations for these particular numbers:

Number	Combinations	Possible Combinations
2	1	01/01/19
3	2	1-2, 2-1
4	3	1-3, 3-1, 2-2
5	4	1-4, 4-1, 2-3, 3-2
6	5	1-5, 5-1, 2-4, 4-2, 3-3
7	6	1-6, 6-1, 2-5, 5-2, 3-4, 4-3
8	5	2-6, 6-2, 3-5, 5-3, 4-4
9	4	3-6, 6-3, 4-5, 5-4
10	3	4-6, 6-4, 5-5
11	2	5-6, 6-5
12	1	06/06/19

As we can see from the table above, the highest number of possible dice combinations can be observed for number 7. This means that the 7 is more likely to come out on each roll than any other number. For this reason, the game of craps revolves around this particular number.

The probability of rolling any particular number in craps is quite easy to calculate once we have laid out the possible dice combinations. It is the ratio of the number of ways an event can occur and the total number of outcomes. In other words, we can find the probability of a total coming out by diving the number of combinations for it by all possible combinations.

Probability = Number of possible combinations ⁄ All dice combinations

To calculate the probability for rolling a 7, for example, we simply divide 6 by 36 (6/36), which is 0.1667. The same probability can be expressed as a percentage – we need to multiply the result by 100. So, the likelihood of rolling a 7 is around 16.67%. And while this may not sound like too much, it is higher than the probability of rolling any other number. Let’s take one of the craps numbers as a second example – the probability for a 2 is 1/36, which is 0.0278 or 2.78%. The table below shows the probabilities for rolling any particular number in the game:

Number	Probability	Likelihood (in %)
2	0.0278	2.78%
3	0.0556	5.56%
4	0.0833	8.33%
5	0.1111	11.11%
6	0.1389	13.89%
7	0.1667	16.67%
8	0.1389	13.89%
9	0.1111	11.11%
10	0.0833	8.33%
11	0.0556	5.56%
12	0.0278	2.78%

These fractions and percentages may be interpreted in several ways but more importantly, they show how likely a certain bet is to win or lose. It is easy to demonstrate that with the most common wager in this game, the Pass Line bet. With it, players bet that the shooter will hit a 7 or an 11 in the come-out roll. As we can see from the table above, however, this will happen only 22.23% (16.67% + 5.56%) of the time on average.

Since this bet loses if 2, 3, or 12 are rolled, it is expected to lose around 11.12% of the time (we simply add the probability for these numbers). Clearly, the shooter is more likely to win than to lose the mandatory Pass Line bet on the come-out roll. Shooting a point number, however, is much more likely to happen – around 66.65% of the time.

Often, the terms probability and odds are used interchangeably in gambling literature but this is not accurate. Both express the likelihood of an event happening but in different ways and for a different purpose. Furthermore, some players talk of casino odds, whereas others are more interested in the so-called true odds. So, how are all these different from each other and why are they so important when analyzing the game of craps?

Probability and Odds

True Odds or Casino Odds?

Probability and Odds

As we have mentioned above, the probability is the ratio of ways to success to all possible outcomes. The term “odds” is also an estimate of the chance that something may occur but it compares the number of ways something can occur and the number of ways it cannot occur. The concept may sound a bit too vague, but it is, in fact, quite straightforward.

Let us use the previous illustration of rolling a 7 as an example. The probability for rolling this number is 6/36 or 6 to 36. To express the odds for this happening, we compare the 6 possible combinations for the number 7 and the combinations, which will be losing. There are 6 ways to succeed in this case and 30 ways to lose or 6 to 30. The odds are usually written as fractions, so in this case, the odds for rolling a 7 will be 6/30 (also expressed as 6:30, which is 1:5).

However, when describing events that are, in general, not very likely to occur, we usually use “odds against” the event happening. For this, we simply flip the odds so the odds against rolling a 7 are 30:6 or 5:1.

True Odds or Casino Odds?

The above-mentioned odds are also known as true odds and should not be mistaken with the term casino odds. Casino odds are used to express the payout offered for a particular bet. They are commonly used in sports betting since it is impossible to calculate the true odds. If the odds that team A will win the championship, for instance, are 5:1, this means that the team has a slim chance of winning. It also means that if the bet is successful, it would bring a payout of $5 for every $1 wagered.

In craps, however, the casino odds or the payouts are based on the true odds for each bet. However, the payouts that are offered are seldom equal to the true odds, which would mean that the risk associated with a particular bet is equal to its potential reward. In reality, the risks are higher than the payouts offered by casinos. For example, the odds against rolling a 7 are 5:1 and instead of offering to pay 5:1 if the bet wins, casinos offer a smaller payout of 4:1.

There are more than 30 different types of bets in craps and although most players tend to stick to traditional ones such as the Pass Line and the Come bet, there are better wagers than them. Understanding the odds for each bet is essential to being successful on the craps table – whether you play RNG craps online or a live game in a brick-and-mortar casino.

How to Assess Craps Bets

Pass/Don’t Pass Odds

How to Assess Craps Bets

When deciding what bet to make, craps players should take several things into consideration. First and foremost, craps is a negative expectation game, which means that players are more likely to lose than to win in the long term. Casinos guarantee the long-term profits by establishing rules and payouts that are, in most cases, unfavorable to the players.

This can be easily noticed in the difference between the true odds and the casino odds of the different bets. As explained in the example above, the odds against winning for the Any 7 bet is 5 to 1 (5:1), which would suggest a payout worth 5 times the bet size if the wager wins. Still, the offered payout is only 4 times the size of the bet. Moreover, the majority of bets in this game are, overall, more likely to lose than to win.

All this is expressed in the so-called house edge, the mathematical advantage of casinos over its patrons. It can be described as the ratio of the expected loss to the initial bet and almost all wagers in craps have a house edge of at least 1.41%. This means that players can expect to lose an average of 1.41% of their total bet. The exact percentages of the house edge vary between the different bets and across casinos, which sometimes pay differently for the same type of a winning wager. Still, several betting options offer great odds and nice payouts.

Pass/Don’t Pass Odds

Players who have already placed a Pass Line or a Don’t Pass wager are allowed to take Odds after the point number has been established. This wager functions as an additional bet to the main Pass/Don’t Pass bet and it wins when the main bet wins. It also loses together with the main wager. Since it pays at true odds, however, the Odds bet on the Pass Line or the Don’t Pass comes with no built-in house advantage.

The Pass Odds bet wins when the point number rolls before the 7 and the payout depends on the point. However, the payout here is always the same as the true odds:

Point 4, 10 – The payout is 2:1, house edge 0%
Point 5, 9 – The payout is 3:2, house edge 0%
Point 6, 8 – The payout is 6:5, house edge 0%

Players who take Odds on the Don’t Pass bet will win if 7 is rolled before the point number. Once again, the payout corresponds to the true odds:

Against Point 4, 10 – The payout is 1:2, house edge 0%
Against Point 5, 9 – The payout is 2:3, house edge 0%
Against Point 6, 8 – The payout is 5:6, house edge 0%

However, players are not allowed to take Odds on the Line bets without making Pass on Don’t Pass bets first, which is why the casino always retains a certain advantage over them. Players who are in brick-and-mortar casinos should also keep in mind that betting against the shooter (the Don’t Pass bet and taking Odds on it) is not usually approved by the rest of the players around the craps table.

Dice Combinations and Probabilities in Craps