How the Math Behind Craps Works
Published on June 15, 2017
Glance through any primer in craps and you are sure to find a discussion of probability formulas complete with tuples like (1,3), (2,2), etc. One of the problems with math education is that the way it is taught makes it hard for the average person to intuitively understand it. Worse, math teachers are taught to just keep repeating everything until the students begin to repeat what they hear by rote. Somewhere along the way you are supposed to start understanding what it all means.
A good math teacher tries to relate all the numbers and symbols to natural everyday experiences. In probability theory, the standard rationalization exercise is to have students imagine tossing a coin a bajillion times. That example doesn’t relate well to the complexity of craps. One problem with calculating probabilities in craps is that you don’t have a unique set of numbers to work with. Another problem is that the math you need to work with isn’t always obvious.
Let’s begin our study in math by looking at some of the hidden mathematical aspects of craps.
We play craps with two cube-shaped six-sided dice. The cubic shape is important to determining how dice roll. Did you know that? Further, cube-shaped dice are important for many games because of the way they are designed.
If you look at the faces of your cube-shaped six-sided dice you should notice that the numbers on opposing faces always add up to 7. This arrangement is why 7 is the most common result you get when you roll two six-sided dice. If the numbers were arranged differently you would find a different number comes up most often. An interesting mathematical problem would be to find a configuration for six-sided dice numbers that does not produce a most common number when two dice are rolled together.
The values assigned to the various die faces are only one way that dice set the probabilities for the game. The construction of the dice also matters. Do you know why casinos use transparent dice? It is so the dealers can see any flaws in the dice. Of course, the dice are carefully weighed and measured to ensure they are perfectly balanced before they are used. Role playing gamers have learned the hard way that opaque dice can be manufactured to such low standards they have a bias. The bias is caused by inconsistent filler distribution within the dice.
Another way that dice affect the probabilities of your game is through their edges. Worn edges interact with physical surfaces differently from pristine edges. Casinos don’t change their dice frequently just to ensure that you are not slipping weighted dice into the game; they are also making sure you always play with dice that have near-perfect edges. A rounded corner on a die creates bias.
The perfect casino die is used to create a mathematical fiction. The dice are forced to lie to us about how the universe works. The casino wants the thrown dice to always come up in a random order. The universe wants the dice to seek equilibrium with their environment. If the universe had its way your casino’s dice would always land a certain way. Hence, casino dice are as unnatural as dice can be, and yet we use them to produce “natural” random numbers.
There is some debate among gamblers about whether you can practice throwing the dice to become a skilled shooter. Some players swear they can control the dice. In fact, there are players who buy dice to practice with. They also buy felt-covered tables built close to casino standards so they can practice throwing dice as if they were in a casino. There are a few flaws in this logic.
First, where do you get your practice dice? Are they manufactured to the same standards as casino dice?
Second, do you replace your dice as often as the casino replaces their dice? If not then your dice are probably experiencing more wear and tear than typical casino dice. You are practicing with biased dice.
Third, do you replicate the atmosphere of the casino? Literally, do you drink alcohol, smoke cigars, cough and sneeze, and do everything else that contributes to the quality of the air inside a casino? If not then your practice sessions with the dice are missing something. If you don’t sweat as much as the next guy, or if you sweat more than he does you are not coating the dice the same way casino dice may be coated by natural oils.
All these things can subtly alter the surface texture of the dice and the table. Changing the surface textures alters the mathematical bias of the dice when they are thrown. It’s not as easy to throw the dice the same way every time as some well-practiced shooters want to believe.
Believe it or not, everything we have discussed so far is analyzed by mathematicians and engineers in a thousand different environments. The mathematics that are at play in a “simple” game of craps are just mind-boggling. The same principles that are used to ensure that rocket ships don’t explode on the launch pad are used to ensure that casino dice produce as nearly random a result as can be achieved.
Are you not glad you don’t have to learn all those formulas and theorems? Really, all you need to do is throw the dice when your turn comes and decide when and how you should bet on the game as it is played.
Gamblers love their probability tables. It does help you to understand why craps rules are set the way they are, so let’s take a moment to examine how probabilities are calculated.
Given a pair of dice numbered 1 to 6, there are a total of 11 possible values for the dice. Only the number 1 cannot be represented by two dice. As noted above the number 7 occurs most often. Since there are three ways to denote the number 7 with a pair of six-sided dice (1 and 6, 2 and 5, and 3 and 4) you must allow for a total of 6 possible permutations. That is because you must allow for 1 on both dice and 6 on both dice, etc.
It’s no accident of number theory that there is only 1 configuration each for 2 (1 and 1) and 12 (6 and 6) on the dice. The design of the dice ensures that. If one of the dice were numbered from 2 to 7 things would be different.
What we really need to know is that the closer a number is to 7 the more likely it will be rolled up with the two dice. Hence, the game assumes that 4, 5, 6, 7, 8, and 9 will occur more frequently than 2, 3, 10, 11, and 12
The frequency of 7 therefore forces the game rules to assign a special role to 7. If you roll a 7 on your first throw you win with a natural. But 7 won’t come up often enough to make the game boring. Instead, it’s eliminated from the list of numbers that can make point. If you don’t roll a 7 (or 11) on your first throw the 7 becomes a bad number for you. 11 is just an odd man out; it can help you win a natural but after the first throw it doesn’t mean anything.
By the same token, the numbers 2, 3, and 12 are craps on your first throw but after that they are meaningless.
So why are the numbers 4, 5, 6, 8, 9, and 10 fair to use for the point? These 6 numbers, comprising just over half of the 11 available numbers, represent almost 78% of the probable distribution of unbiased die throws. There are 36 possible combinations of die faces; 28 of those possible combinations are represented by the 6 numbers allowed for the point. Your chances of making point would be less if 2, 3, 11, or 12 were used instead of one of the regular point numbers.
This is an interesting question. A lot of experiments have been designed to choose the best point numbers. If you add up all the permutations for each of the eligible point numbers, you’ll see there are four ways to roll a 4, 5, 9, or 10 and six ways to roll a 6 or 8.
The choice you make is influenced by the odds that are paid per point. The 4 and 10 are paid 2-to-1 odds and the 6 and 8 are paid 6-to-5 odds. The 5 and 9 are paid mediocre odds of 3-to-2.
You’ll find opinions swing toward 4 and 10 or toward 6 and 8 most often. The arguments are sound. As some expert guides point out, the game of craps strives to be extremely fair to players. You have a lot of different ways to win. It comes down to which betting style you enjoy most.
Aggressive bettors should favor the 4/10 point strategy. Conservative bettors should favor the 6/8 strategy. With 4/10 you get better odds but your numbers are less likely to come up. With 6/8 you get the worst odds but your numbers are more likely to come up.
If you use a conservative betting strategy (make small bets) then your money lasts longer and you can make more (losing) bets. Hence, you can be patient and allow your occasional winnings to add up.
If you use an aggressive betting strategy (make large bets) you don’t need to hit quite as often and so you want bigger payoffs per bet. Either way you are supposed to stay in the game about as long.
The real magic for Odds bets happens where casinos allow you to make large bets. The more you bet the more you risk but the Odds bet is craps’ trademark money maker. The rule of thumb among experienced gamblers is to bet the table minimum on the Pass / Don’t Pass lines and the table maximum on the Odds bet.
If the player keeps rolling that point before hitting a 7 your Odds bet could do well. But if the table maximum is not very high the casino is hedging its own bet. Although players like to calculate a House Edge of 0% on the Odds bet, there’s just one little problem with that maxim: you lose when the shooter rolls a 7.
Some experts suggest you should not put more money on the table if the shooter’s point is anything other than 6 or 8. And as an alternative strategy, if the shooter fails to make 6 or 8, you can call a Place bet on 6 or 8, setting your own point.
While anything can happen once the dice start rolling, we know that 6, 7, and 8 are probably going to be rolled more often than any other three numbers. At the beginning of your craps career it will be easier for you to decide when to bet aggressively by focusing on those three numbers. After the come out roll you want 6 or 8, not 7, and it’s good to know you can set your own points if you’re not the shooter.
When you are just learning to play craps it’s better to learn the rules than to memorize probability tables. The probability tables help you understand the rules. While it’s good to know why the 7 is such a special number in craps, you need a simpler guide for deciding how to play the game.
Using simple logic allows you to focus on the game and you don’t worry so much about whether you made the right decision. That means not making every possible bet you can. It means limiting yourself to the safest bets until you have an opportunity to go for the money.
There are no guarantees in life or craps, but if you know your numbers, you don’t have to learn to think like a mathematician.
That’s the most important lesson in craps.