One of my philosophies of life is that if you want to be good at anything I believe you should try to understand all facets of it, even parts you don’t necessarily have direct contact with. If you want to be a great plumber, for example, you should know how copper pipe is made, how solder is made, the mechanics of heat and glue bonding, the geometry of tube routing, the physics of water and atmospheric pressure, the history of plumbing, the bylaws and building codes, on and on and on. It’s no different with the game of Craps. Before we just go out and play the game somewhere we should look into understanding some basic concepts and aspects of it first. This would help us play it more intelligently and at least give us a realistic expectation of the outcome of things. Let's start with the 'tools of the game' --- Dice. Two of them are called ‘dice’. A single one is called a ‘die’. A modern die has a specific design. It’s a cube marked on its respective faces with 1, 2, 3, 4, 5 and 6 dots so that the dots on opposite faces always total 7. If the 2 side is vertical and facing you, with 4 on the top, 1 should be at your right and 6 on your left. In gambling places the dice provided there have been manufactured to be perfectly balanced at all faces and to be geometrically perfect cubes with sharp crisp corners and edges. There are unique probability characteristics with dice, that I avoided getting into in one of my other little essays (about the 6/49 lotto), but of which I'll talk about now. In most games (Craps included) two dice are rolled and the two numbers are added together. The result of the roll can be 2 (rolling a pair of 1’s) or anything up to and including 12 (rolling a pair of 6’s). Is any combination of a roll as equally probable as any other? There are a total of 36 different pair-combinations.
Possible combinations of rolling 2 dice: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
One of my philosophies of life is that if you want to be good at anything I believe you should try to understand all facets of it, even parts you don’t necessarily have direct contact with. If you want to be a great plumber, for example, you should know how copper pipe is made, how solder is made, the mechanics of heat and glue bonding, the geometry of tube routing, the physics of water and atmospheric pressure, the history of plumbing, the bylaws and building codes, on and on and on.
It’s no different with the game of Craps. Before we just go out and play the game somewhere we should look into understanding some basic concepts and aspects of it first. This would help us play it more intelligently and at least give us a realistic expectation of the outcome of things.
Let's start with the 'tools of the game' --- Dice.
Two of them are called ‘dice’. A single one is called a ‘die’.
A modern die has a specific design. It’s a cube marked on its respective faces with 1, 2, 3, 4, 5 and 6 dots so that the dots on opposite faces always total 7. If the 2 side is vertical and facing you, with 4 on the top, 1 should be at your right and 6 on your left. In gambling places the dice provided there have been manufactured to be perfectly balanced at all faces and to be geometrically perfect cubes with sharp crisp corners and edges.
There are unique probability characteristics with dice, that I avoided getting into in one of my other little essays (about the 6/49 lotto), but of which I'll talk about now.
In most games (Craps included) two dice are rolled and the two numbers are added together. The result of the roll can be 2 (rolling a pair of 1’s) or anything up to and including 12 (rolling a pair of 6’s).
Is any combination of a roll as equally probable as any other?
There are a total of 36 different pair-combinations.
Possible combinations of rolling 2 dice:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
This little grid shows all 36 combinations possible by rolling a pair of dice.
If you examine the grid you’ll notice that only 1 possible combination gives you a sum of 2, that's (1,1). So this means that the odds of getting 2 on a roll of the dice is just 1 chance in 36 or 1/36 or (if you divide 36 into 1) a 2.78% chance. On the other hand, look at all the combinations that add up to 7. They are: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). You can see that there are 6 ways to roll a 7 so this means the odds are 6 chances in 36 or 6/36 or a 16.7% chance you'll get a 7 on a typical roll.
By looking at just these two sums, 2 and 7, we already realize that different sums have different odds of showing up on a roll of a pair of dice. Let’s skip going through each remaining dice combination. Allow me to simply list all the probabilities and combinations for you:
Do you notice the patterns in this chart?
Look at the top line first. To throw a 2 (column 1) there is only 1 possible combination (column 2) of the dice that will give it to you, that is (1,1). At the other end of the scale (the bottom line of the chart), 12 also has only 1 combination and that’s (6,6). Both of their ‘probabilities of occurring’ and ‘% of chance’ are equal (column 3 & 4). Next we look at 3 and 11 (column 1). Each have 2 possible dice combinations (1,2) (2,1) & (5,6) (6,5) and both have identical ‘probabilities of occurring’ (2/36), etc. And so it goes with all the possible dice combinations as you discover the pattern of an upper number matching the parameters of a lower number in the chart. Except for 7. It’s unique in its ‘probability of occurring’ and is also, as it turns out, the most favorable of all the possible sums that a person can get in a throw of the dice as indicated by its ‘% of chance’ (16.7%).
So if you sit at a table and throw a pair of dice, the most likely sum to come up will be 7 (about 16.7% of the time), next most likely are 6 or 8 (column 1), each of which occur 5 times out of 36 (or about 13.9% of the time). And so on.
You might ask: Why is 7 the most likely sum to show up?
To put this probability-of-a-sum-of-7 thing mathematically you’d describe and write it this way:
Since it doesn’t matter what number you roll with the first die, any possible number will do, any number will be fine. So the probability of getting the number you need on the first die is 6/6. This means it does not matter what number comes up. Any one of the 6 choices out of 6 will work.
So we roll the first die and a number comes up and (I'll say it again) it doesn’t matter which one.
Now we roll the second die.
So we’d write this down in an equation like this:
6/6 X 1/6 = ‘the-odds-of-a-seven-coming-up-on-a-roll-of-a-pair-of-dice’
Another way of looking at all this is with the law of averages or commonly called the ‘Law of Large Numbers’. In probability, when you do something for a long time, with a great amount of repetition, the results always start to drift to an average.
If you toss a coin only 10 times you might get 3 ‘heads’ and 7 ‘tails’ even though the mathematical odds are 50% for ‘heads’ and 50% for ‘tails’. This is because the sample is just too small to arrive at an accurate display of the probabilities. But if you toss the coin a million times, you’re likely to see results closer to the average of 50% of the time ‘heads’ and 50% of the time ‘tails’ within the calculated margin of error. It always goes this way when dealing with a large number of trials. It’s a law of nature actually. Things always drift towards an average. You can count on and expect this always.
The average number on a single die is 3.5.
That’s why 7 is the most probable of the combinations --- because it’s the average of that particular probability system!
Now we’ve looked at the 7 combo let’s look at a different sum. What if you wanted to roll an 8 during your Craps game? If the first die rolled to a 1 then it would be impossible to get a total sum of 8 from any number that would turn up on the second die. Unlike the 7 sum, to get an 8 overall, the number the first die falls on IS important. What if you needed to roll a sum of 6 during your Craps game and the first die rolled to a 6? It would now be impossible for anything to turn up on the second die to give you a sum of 6 overall. This is why 7 is the most probable roll and all other numbers, greater and lesser, start decreasing in probability of occurrence.
The odds for these other combinations are easily calculated with math equations just as the 7 combination was and will conform exactly with the probabilities listed in my chart above. Let me show you.
To mathematically calculate the odds to get a roll of 9, for example, would go this way:
Now we simply multiply the probability of one die against the other:
4/6 X 1/6 = 4/36 = 11.1% = the-odds-of-rolling-a-sum-of-9!
What’s cool of course is that these probabilities are a function of a pair of dice regardless of what game they are being used in. For example, if you were playing a game of Monopoly and were in a position of buying a hotel for some properties that your opponent could possibly land on in his next move, you’d now have some valuable information about the odds of a pair of dice, to help you place your hotel wisely.
If your opponent was 2 spaces away from one particular square of your property, 3 hops away from another and 7 from another square of yours, it would be a smart move to put your hotel on the property 7 hops away from him. It’s more likely this will come up with his next roll of the dice than the 2 or 3 roll.
Are you starting to see that understanding the underlying principles of things gives you a slight advantage in life?
So let’s move on to the game of Craps now…
Craps used to be played a lot in the back alleys, streets and on riverboats in the U.S. before the days of casinos and gambling halls. Roman soldiers are even known to have played it among themselves during their free time. In modern times it is a common popular game at any casino that you visit and is curious for its seemingly strange and complicated rules. At first the rules seem arbitrary and peculiar. As we soon will see they developed for a very good and exact reason.
The rules are actually pretty straightforward:
So, on a first roll of a round, a 2, 3 or 12 is bad while a 7 or 11 is good. Any other sum sets off a race of rolls to match that sum before rolling a 7. That's about it.
Of course betting is involved.
People will wonder where these strange rules came from. Well it’s our old friend --probability-- of course. The long-term odds of any gambling game should be weighed in the casino’s favor to guarantee they continue to make a profit from their operation. Their business relies on this principle. It isn’t too important for them to win big from their customers but rather to win consistently over a long time in exact proportion to the odds of the particular game. And make no mistake, the odds always favor the House. But it is also very important that the odds are never too heavily weighed against the customer otherwise it would discourage players and after a while no one would want to play.
After working on the odds of Craps at its various stages I realized some interesting things.
Craps is a cleverly designed game in that its odds vary for and against the player depending on the stage of the game they are in. In the beginning of a round the odds work out to be largely in the player’s favour and this no doubt is the allure of the game. It draws you in and pumps your confidence up so that losing isn’t noticed because it’s a gradual bleeding. It almost creeps up and grabs you when it’s too late to do anything about it but it can be so slight you may hardly notice it’s happening.
We know, from my little chart way back, that the odds of getting the preferred 7 or 11 on your first throw of the dice is 6/36 and 2/36 respectively or 8/36 collectively. That works out to about a 22.2% chance of winning immediately. Rolling a 2, 3 or 12, on the other hand, and losing immediately, carries with it a 4/36 chance collectively or 11.1%. Isn’t it slick how the chance of winning is designed to be almost exactly twice the chance of losing? You have odds of 2 to 1. You are twice as likely to win than lose on your first throw. This is a sweet deal for the player and it pulls them into the game.
But the House ain’t worried. It’s all about the big picture to them.
So far so good.
The 22.2% chance of winning and the 11.1% chance of losing on the first throw account for exactly 33.3% of the chances of what will come up on the first toss of the dice or 1/3. That leaves all the remaining numbers that could come up on a first throw such as 4, 5, 6, 8, 9 and 10. You can do the math yourself and see, from my chart, that their combined individual probabilities equal 66.7% or the other 2/3 of the equation. So you have a 66.7% chance of not losing with these numbers, but not winning either. If you get one of them, it becomes your ‘point’ and you are now allowed to scoop up the dice for another throw to see if you can throw a match to that ‘point’ before you throw a 7.
Notice that now throwing a 7 has become a bad thing. If you throw one, you immediately lose and worst of all, 7 is of course the most likely number to come up. The tables have suddenly turned in favor of the House at this point. Hopefully your ‘point’ is a 6 or 8 since these are the next most favored numbers to come up in a throw, after 7. If your ‘point’ is a 10 or a 4, you’re up against hard odds indeed as you hope to roll them before you roll a 7. (Refer to my chart to see the odds).
Now this is where things get a bit complicated. If you roll a 4, 5, 6, 8, 9 or 10 on your first roll of a round, the probability of winning on the next throw depends on what your ‘point’ equals of course. Calculating the probability of winning now requires computing a complicated sum of different probabilities to obtain the various ‘point’ values, multiplied by the probabilities of winning once you know your ‘point’.
I undertook this task (I’ll spare you the page and a half of calculations) and when all the fractions have been multiplied and sums have been completed, it turns out that the overall probability of winning at Craps is equal to 244 chances in 495. This equals 49.2929%. Put another way, the chances of winning are just a smidge below a fair 50-50 bet.
In reality this means that if you play Craps over a long period of time on average you will win $10 exactly 49.2929% of the time and you will lose $10 exactly 50.7071% of the time (assuming you're making $10 bets each round). So your average winnings per bet, overall, could be figured out to be:
$10 X 49.2929% minus $10 X 50.7071%
So as you play round after round at the Craps table you slowly lose about 14.1 cents for every 10 bucks you bet.
Wow! you say…
When the odds are so slim between winning and losing like that, the guy with the deeper pockets can endure all the ups and downs and variables and ultimately outlast the guy with less money in his jeans. And that’s exactly what happens, every day. You may get what seems to be a 'lucky' streak but if you continue playing it will always eventually turn the other way at some point.
To kind of wrap this up I just wanted to mention another, final quirk of the Craps game. It’s a little tougher to grasp but is a very important aspect of the game in regards to probability so I need to bring it up.
In the game of Craps, people NOT throwing the dice (i.e. they aren’t the current shooter or roller) have the opportunity to also place bets on the round, just like the shooter. This is why we often hear a big cheer rise up from the casino Craps table throughout the evening. The spectators aren't expressing great affection for the shooter, they're cheering because their own side bet has just paid off on a particular throw he's made. They can either place a bet in favor of the shooter winning the round or, more interestingly, place one against him, hoping he’ll lose! They can plunk $10 down on a special area of the table designated for anyone betting against the shooter and if he loses the round, these betters immediately win an amount equal to what they bet, from the House!
But hang on… something’s not right here!
Yes, this would be true but as it turns out there's no such luck. They didn’t overlook anything.
You might think, great!
To do that, it looks like this:
Next we need to work out the difference between your previous superior average winnings (which used to equal that of the House before we considered the rule) and the level this average was reduced to by eliminating your opportunity to win by 1 time in every 36 when betting against a shooter. (the odds of a 12 getting rolled)
That difference is calculated this way:
What this is telling us is that your previous rate of return of +14.1 cents on every $10 bet (before the rule was sprung on you) just got reduced by 27.8 cents with the rule.
So finally, your real actual rate of return, for every $10 bet that you make, works out to be:
This means that as you play the game, round after round, placing side bets against the shooter, you slowly lose on average about 13.7 cents for every 10 bucks you bet. It’s a little better odds than the shooter (remember, he's losing 14.1 cents on each of his 10 bucks bet) but it still means your money is slowly trickling away from you and ends up being a total loss over time.
It's a little depressing isn't it?
The trick to gambling is knowing when to hold up.
Always keep in mind two things: The odds always favor the House and It's smart to quit while you're ahead.