Probability Of Winning Craps On First Roll
In a game of craps, a player loses on the roll if a 2, 3, or 12 is tossed on the first roll. What is the probability of losing on the first roll?
1 Answer
Explanation:
In the game of craps you roll two dice. If on the first roll the sum is 7 or 11 you win. If it is 2, 3, or 12 you lose. With anything else that becomes your “point”. If you roll your point again before your roll a 7 you win. The first step in calculating the probability you will is to find the probability that you will “make your point.
- On the first roll, the player can win by rolling 7 or 11, or lose by rolling 2, 3, or 12. Find the probabilities of these five outcomes and put the answers in the below table. If the player rolls initially a 4, 5, 6, 8, 9 or ten, he keeps rolling.
- Date: at 12:12:12 From: Doctor Anthony Subject: Re: Independent repeated trials Hi Ekal - We start by finding the probability of winning at the first roll, that is obtaining 7 or 11: Pr(7) 1 6 6/36 = 1/6 probability of getting 7 2 5 3 4 (use fractions rather than decimals) 4 3 5 2 6 1 Pr(11) 5 6 2/36 = 1/18 probability of getting 11 6 5 So the probability of winning on first roll.
Let's look at the possible rolls we can get with 2 dice rolled, with the 2, 3, and 12 - the losing numbers - coloured in red:
Probability Of Winning Craps On First Roll Play
And so out of 36 possibilities, there are 4 that will cause a loss on the first roll.
So that's
Related questions
Here we explore the specifics of craps strategy, dice setting, the types of craps bets and betting systems, and the theoretical probability of winning. We review the probability of certain dice outcomes, and suggest the best (and worst) bets depending on your goals for the game.
Craps Strategy Guide
The best craps strategy is to learn how to play the game, understand the basic bets which offer the best odds, and stick with those bets. Betting systems which involve raising and lowering your bets based on some arbitrary criteria are a bad idea, because they don’t affect your chances of winning. Hedging your bets is also a bad idea.
Craps is a straightforward casino game, but the bewildering number of bets, many of which are the worst bets in the casino, make it a profitable enterprise in every casino. The strategy that we recommend involves being realistic and having fun while you play. This page looks at some strategies to avoid, and it also explains which legitimate strategic choices actually make sense mathematically.
Learn How to Play
The best way to learn how to play crap is to take one of the free classes that the casino offers, but keep in mind that they won’t go into a lot of detail about which bets are best and which bets are worst. They will show you how the action works, though, so if you’re new to the game, taking one of these classes is a great idea. There is no faster way to learn the game, in fact.
Know the Best Bets
The number of good bets at the craps table is relatively small. If you stick with these bets, you’ll minimize the house edge, which is the percentage of every bet that you’re mathematically expected to lose over the long run. Don’t be fooled by short-term standard deviation. In the short run, which is longer than you think, anything can happen. The smart play is still to stick with the bets offering the best odds.
The only good bets at the craps table are the following:
- Pass
- Don’t Pass
- Come
- Don’t Come
- Free Odds
- Place Bets on 6 and/or 8
The house edge for pass and don’t pass bets (and for come and don’t come bets) is only 1.41% and 1.36% respectively. That means that the casino expects to win $1.41 (or $1.36) for every $100 you wager at the craps table. Compared to the house edge of 5.26% at the roulette table, this is a lot of entertainment for very little money. And if you take the free odds when they’re available, you can reduce the house edge even more.
On the other hand, most of the proposition bets on the craps table have a house edge of 6% or more. Some of them even have a house edge in the double digits. Don’t waste your money placing bad bets. You’re better off spending that money on a show or something.
Don’t Forget to Claim Your Winnings
One common error that neophyte craps players make is forgetting to pick up their winnings from the table. If you leave it on the table, it’s considered part of the action on the next roll, so be sure to pay attention and claim your winnings when you want them.
Craps Systems Don’t Work
Any number of bogus systems involving changing the size of your bets based on a variety of criteria are available. For example, one system might have you increase your bets when the shooter wins. The stated goal of such behavior is to increase the amount of money you have in action during a shooter’s hot streak.
The reason this doesn’t work is called the gambler’s fallacy. This is the name mathematicians give the idea that previous events have an effect on subsequent events, when in reality, these events are independent of each other.
The thinking goes like this. If a shooter has won four times in a row, he’s hot, and you should bet more in order to take advantage of his winning streak. The fallacy is that the fifth roll’s math has no relation to the previous bets.
Sizing Your Bets
You should bet an amount you’re comfortable with, and you should also put as much money into the free odds bet as you can. For example, if you’re playing in a casino that allows double odds, then bet 1/3 of what you’re comfortable with on the pass line, and bet the other 2/3 on the free odds. By doing so, you’re reducing the house significantly, because the odds bet has no house edge—it pays out at true odds. This turns a good bet at the craps table into a great bet.
Getting Along with the Other Players
We’re big believers that your #1 goal at a gambling table should be to have as much fun as possible. If you win money on top of that, then that’s just gravy. So here’s a craps strategy recommendation that focuses almost exclusively on having fun, proper etiquette, and it’s aimed at new players:
Avoid the don’t pass and don’t come bets.
The odds are slightly better on those two bets, but it’s probably not worth it, because you’re betting against the shooter and most of the rest of the table. Some people might find that to be too confrontational for their tastes.
Besides, isn’t it more fun to root for someone to win, especially if everyone else is? Rooting for someone to lose is just a downer. And that’s bad strategy.
Craps Dice Setting
Setting the dice in craps is (theoretically) an advantage play strategy. Advantage players try to use skill to get an edge over the house in casino games. Dice control fans claim that someone can develop enough skill at throwing the dice that they can affect the odds just enough to give the player a slight edge over the casino.
Casinos have some experience dealing with advantage players because card counting has been a thorn in their sides since the 1960s. They’ve established a large number of countermeasures to thwart card counters, in fact. They ban players suspected of counting cards. They change the rules of the game to make it harder for counters to get an edge. They increase the number of decks and/or use automatic shuffling machines to eliminate a counter’s edge.
So the question a skeptic might ask is this: If setting the dice in craps works, why haven’t the casinos taken any measures to prevent dice setters from taking them for all they’re worth?
The rest of this page ignores the question of whether or not dice setting actually works. Instead, it provides an overview of how to learn to control the dice. Perhaps with enough practice, you can demonstrate mathematically that dice control is a viable advantage play technique.
Dice Setting 101
According to the dice control experts, anyone who can throw the dice can learn how to control the dice with enough practice. The three things to focus on when learning how to control the dice are alignment, delivery, and grip.
Alignment refers to how the dice line up with the craps table. They should line up as if they were sitting flat on the table.
Delivery means the actual throw. You’re required to throw the dice hard enough to hit the opposite wall. Your goal as a dice setter is to toss the dice gently enough to just barely hit the wall. Most players try to give the dice some backspin. You want the dice to stay together as they fly through the air.
Grip might be the most important aspect. You should use a gentle grip while making sure that the dice are kissing each other. They should have absolutely no space between them.
Remember that your goal isn’t to roll a specific total. Your goal is simply to tilt the odds in your favor by avoiding rolling certain numbers. You don’t want to “crap out”.
Get a replica of a casino craps table built in the basement or garage of your home, practice, and track your results until you’re confident that your dice throwing skills provide you with an actual edge over the casino. This will require a large number of trials to be statistically valid, and you should probably enlist the help of a confederate in order to keep track of the numbers.
Betting Strategies When Setting the Dice in Craps
Before you can apply your newfound skills in a casino, you’ll need a bankroll. This is the amount of money that you’ve set aside strictly for the purposes of gambling. You should be comfortable taking this money out of your lifestyle budget. You don’t want to risk losing your rent money or your car payment at the craps table.
You should also keep in mind that even if you succeed in getting an edge over the casino, there is still an element of chance involved. Skilled dice setters don’t claim to be able to shoot with 100% accuracy, and if you get on a losing streak, you can go broke. (Card counters deal with the same issue.)
Once you have the size of your bankroll established, you can decide on the size of your bets. Dice control experts claim to be able to get an edge of about 1% over the house, which is similar to the edge card counters are able to achieve. Your goal is to maximize your profits while minimizing your chances of going broke.
According to the Kelly Criterion, the best way to do this is to size your bets as a percentage of your bankroll equal to the edge you have over the casino. So if you have a 1% edge over the casino, you should bet 1% of your bankroll. So if your bankroll is $10,000, you should place bets of $100.
The best bets to make are the pass line and come bets with as much odds as the casino will let you take. Betting on the 6 or the 8 are also good choices. You don’t have to do anything complicated when placing the bets—just go with the wagers that offer the best odds for any craps player, regardless of whether or not they’re using an advantage technique, and you’ll be fine.
I’ve seen some websites which claim that big winners at your table will draw unwanted attention from the casino. The assumption is that even though you’ll be making reasonably small bets in order to stay under the radar, other bettors might bet big and win based on how well you’re shooting.
I’ve never seen or heard of any casino barring a dice shooter for being too good at the game, though. I’ve never even seen anyone claim such a thing. It’s not impossible, but my guess is that it’s unlikely enough that you needn’t worry about it.
Summary
Dice control is a technique that tries to turn the game of craps from a game of pure chance into a game of skill (like darts). It’s questionable whether or not it’s a realistic advantage play technique, but it’s theoretically possible. Learning how to set the dice involves experiment with your grip, your delivery, and the alignment. Having a reasonable sized bankroll and sticking with the best bets on the table will help someone maximize their chances of turning this into a viable casino strategy.
Craps Bets Explained
It’s hard to believe now, but just a few decades ago, craps was the biggest moneymaker in any casino. Over the last fifty or so years, it’s gradually declined in popularity, while slots and blackjack have becoming increasingly popular. One of the reasons for this decline is the seeming complexity of the game, which features a multitude of what seem like bewildering bets.
The truth is, though, that the best bets on a craps table are the simplest bets, so you don’t even have to learn all the betting types to play. If you stick with the basics, you can have lots of fun, and your money will last a long time.
This page describes those bets along with some of the more exotic (and not necessarily recommended) other bets.
The Pass Line Bet
This is one of the most basic bets in the game. You place a pass line bet before the come-out roll, which is the first roll of the dice in a round of play. If the come-out roll is 7 or 11, the pass line bet wins. If it’s a 2, 3, or a 12, the pass line bet loses.
Players love this bet because it’s rooting for the shooter to win. It’s also one of the best bets at the table, with a house edge of only 1.41%.
The pass line bets pays out even money.
Don’t Pass Line
This is the opposite of the pass line bet. If the come-out roll is 7 or 11, the don’t pass line bet loses. If it’s a 2, 3, or a 12, the don’t pass line bet wins.
The house edge on the don’t pass line bet is marginally better than the house edge on the pass line bet. It’s 1.36%, which is 0.05% better. You’d have to be betting $2000 or more per hour to even notice the difference, and even then, in the long run, you’re only gaining $1 per hour over the pass line bet.
Players who stick with the don’t pass line bet are called “wrong bettors”.
If any number other than 2, 3, 7, 11, or 12 comes up, a point is set. We’ll discuss that more soon.
The Come Bet
If the pass line and the don’t pass line bets were the only two options available, anyone could learn craps in moments, but it wouldn’t be such an interesting game. That’s why the come bet is available, too.
The come bet basically treats the roll of the dice after a point has been set as if it were another come-out roll. It’s treated the same as a pass line bet.
So if a shooter is trying to make a point, and you’ve made a come bet, then that bet pays out just like it would if this were a come-out roll instead an attempt to make a point.
The Don’t Come Bet
It doesn’t take a rocket scientist to figure this one out. If the come bet is the equivalent of a pass line bet, then the don’t come bet is the equivalent of the don’t pass line bet. Just like the come bet, it’s a bet against the shooter trying to make the point, but it treats that roll just as if it were a come-out roll.
Taking Odds
This the best bet in the game. In fact, it’s almost always the best bet in the casino. The trick is that you can only take odds after you’ve made an initial pass line or come bet.
Taking odds is a bet that can be made any time a point is established. The amount you bet in this case wins if the point is rolled before a 7 is rolled.
And unlike every other bet in the casino, the taking odds bet pays out at true odds. There is no house edge.
Casinos limit the amount you can wager on taking odds as a multiple of the amount you wagered on the initial pass line or come bet. The higher the multiple, the better deal this bet becomes.
If the point is 4 or 10, taking odds pays out 2 to 1.
If the point is 5 or 9, taking odds pays out 3 to 2.
If the point is 6 or 8, taking odds pays out 6 to 5.
Laying Odds
This is the opposite of taking odds, and it pays out when the shooter gets a 7 before rolling a point. You can only lay odds if you placed a don’t pass line or don’t come bet first, and like taking odds, you can only wager a particular multiple of that bet. The casino has house rules about the max you can bet.
The payouts for this bet also pay out at true odds, but since you’re placing the opposite bet, the payouts are different, as follows:
If the point is 4 or 10, laying odds pays out 1 to 2.
If the point is 5 or 9, laying odds pays out 2 to 3.
If the point is 6 or 8, laying odds pays out 5 to 6.
Place Bets
Besides the bets already mentioned, the only other good bets at the craps table are place bets. You can make a wager on a place bet on any of the point numbers at any time. The point numbers are 4, 5, 6, 8, 9, and 10. If the number you chose is rolled before a 7, then you win. If a 7 rolls before your number, then you lose.
These bets do NOT pay out at true odds. The house edge varies based on which number you make a place bet on.
If you make a place bet on 6 or 8, then the casino pays out 7 to 6, giving the house an edge of 1.52%.
If you make a place bet on 5 or 9, then the casino pays out 7 to 5, giving the house an edge of 4%.
If you make a place bet on 4 or 10, then the casino pays out 9 to 5, giving the house an edge of 6.67%.
Buy and Lay Bets
The buy bet is the same as a place bet, only it pays out at true odds instead of at the payouts for a place bet. There’s a catch, though. The house charges a 5% commission (or vig) on this bet. This makes the house edge on all of those bets the same: 4.76%.
It doesn’t take a rocket scientist to figure out that you should never place a buy bet on 5, 6, 8, or 9, but you should never make a place bet on 4 or 10, either. You’d have to be a fool to give the casino that much of an edge over you without getting something else in exchange.
Lay bets are the opposite of buy bets, and, like the buy bets, the casino charges a 5% commission in exchange for paying out at the bets’ true odds. You’re betting that a particular number will NOT be rolled before a 7 comes up.
Put Bets
Put bets aren’t allowed in all casinos. In fact, casinos in Atlantic City and Connecticut don’t even offer this option, but it’s not a big deal, because this isn’t such a great idea for the player anyway.
A put bet allows a player to bet on the pass line after the come-out roll. The player gets to choose her point. Of course, you can achieve the same effect with a buy or place bet.
Proposition Bets
Many of the bets on a craps table are proposition bets. Just as the bets listed above offer some of the best odds in the casino, the bets listed below offer some of the worst odds in the casino. Most of the proposition bets are one-roll bets–the outcome of the next roll determines a win or loss.
The Field Bet
The field bet is popular with beginners to craps, but that doesn’t make it a good bit. It only SEEMS like a good bet. A bet on the field is a bet that the next roll will be one of the following: 2, 3, 4, 9, 10, 11, or 12.
This seems like a good bet because it pays out on 7 out of 12 numbers.
The house usually pays out even money for 3, 4, 9, 10, or 11, and it usually pays out 2 to 1 for a 2 or 12. Some generous casinos even pay out 3 to 1 for a 2 or 12.
The house edge for the usual payout structure is 5.56%, but at casinos which offer the 3 to 1 payout on the 2 or 12, it’s only 2.78%.
Big 6 and Big 8
This is the same thing as a place bet on 6 or a place bet on 8, with one exception. The big 6 and the big 8 bets only pay even money. (A place bet on these numbers pays out at 7 to 6.) This is what gamblers call a sucker bet. This one change makes the house edge on these two bets 9.09%. Compared with a 1.52% house edge, big 6 or big 8 are such lousy bets that you’d almost think they’d be illegal.
Hardways
A hardway bet is placed on an exact combination of dice, and these combinations are pictured on the betting surface of the craps table. There are four possible hardway bets:
- Hard Four (2-2)
- Hard Six (3-3)
- Hard Eight (4-4)
- Hard Ten (5-5)
Hardways are won if the combination is rolled before a 7.
The Hard Four and Hard Ten bets pay out 7 to 1, and they have a house edge of 11.11%.
The Hard Six and Hard Eight bets pay out 9 to 1, and they have a house edge of 9.09%.
Any Craps
This is a one-roll bet that the next total will be 2, 3, or 12. The bet has a payout of 7 to 1 and a house edge of 11.11%.
Any 7
This is a one-roll bet that the next total will be 7. This bet pays out at 4 to 1 and has a house edge of 16.67%.
Eleven or Three
These are two separate bets, but they’re more or less the same. Each bet is a one-roll wager that the next total will be 11 (or 3). In either case, it pays out at 15 to 1 and has a house edge of 11.11%.
Twelve or Two
These are also two separate bets that are more or less the same. Each bet is a one-roll wager that the next total will be 12 (or 2). Either way, it pays out 30 to 1, and the house edge is 13.89%.
Horn Bet
This is a one-roll bet on the following four numbers: 2, 3, 11, and 12. You have to place this bet in a multiple of 4, with 25% of the total on each of those numbers. If any one of these numbers is rolled, you win. The house edge on this one is 12.5%.
Horn High Bet
This is like a horn bet, but it’s made in multiples of 5 instead of 4. 20% of the bet goes on three of hour numbers, while the other 40% of the bet is placed on the high number that the players chooses. For example, if you bet $5 on this one, you might put $1 each on the 2, 3, and 12, and then put your additional $2 on the $11. The house edge is huge and depends on which of the high numbers you choose.
C&E
Probability Of Winning Craps On First Roll Dice Game
This is an abbreviation for “craps and eleven”. It’s a one-roll wager that the next number rolled will be 2, 3, 11, or 12.
Hop Bet
The hop bet is a one-roll bet that the next roll will be a particular combination. For example, you might bet on a 2-2 coming up next, which is called a “hopping hardway”. You could also bet on another combination, like 6-4. Most of these bets pay 15 to 1, but the 3-3, 4-4, and 5-5 pay 30 to 1. The house edge on the former is 11.11%, and the house edge on the latter is 13.89%.
As you can see, the best bets at the craps table are the simplest bets. Stick with those, and you’re playing one of the most exciting and least expensive games in the casino. The more complicated bets are for suckers. Avoid them like you’d avoid angering a large bull.
Odds for Dice Outcomes in Craps
When someone discusses craps odds, they’re discussing one of two things—the odds of rolling a certain number, or the payout for a particular bet. Odds are one way of describing a probability, but they’re also a way of describing how much a bet pays.
This page explains both types of craps odds. We refer to payout odds as the number that a bet pays off, and true odds as the probability that a given outcome will appear. The difference between the true odds and the payout odds is the house edge, which is the number that explains how the casinos stay so probable.
The Basics of Probability
Probability is that branch of mathematics that deals with the likelihood of something happening (or not happening). An event’s probability is always a number between 0 and 1, but that number can be expressed in multiple ways.
A simple example of a probability is a coin flip. The probability of the result being heads is 0.5, because half the time, that’s what will happen. 0.5 can be expressed also as 50%, ½, or 1 to 1.
That last expression of the probability is the one we’re most concerned with on this page, because that’s an expression of the odds.
The equation for calculating a probability is to divide the number of ways something can happen by the number of total ways it could happen. When rolling dice, you can calculate the odds of rolling a 1 by dividing 1 by 6. There are 6 possible outcomes, but the one you want to know is the chance of rolling a 1.
1/6 can also be expressed as 0.167 or 16.7% or 5 to 1. When expressing a probability as odds, you compare the number of ways something won’t happen with the number of ways something can happen. There’s only 1 way to roll a 1 on a single die, but there are 5 ways to roll something else, so the odds are 5 to 1.
When you’re calculating multiple probabilities, you add the probabilities together when you want to know the odds of event A OR event B happening. You multiple them by each other when you want to know the odds of event A AND event B happening.
For example, if you want to calculate the probability of rolling a total of 2 on 2 dice, you would multiply the probabilities of rolling a 1 on the first die by the probability of rolling a 1 on the second die. 1/6 X 1/6 = 1/36, which can be represented as odds of 35 to 1. (You’re calculating the odds of rolling a 1 on die A AND the odds of rolling a 1 on die B.)
On the other hand, if you wanted to know the probability of rolling a 1 on either of the two dice, you’d ADD the two probabilities together, and you’d get a result of 1/6 + 1/6, or 2/6, which can be reduce to 1/3. That would be expressed in odds as 2 to 1.
True Odds
When you discuss the odds of something happening, you’re discussing the true odds, or the probability, that something will happen. The difference between the true odds and the payout odds is what creates an edge for the casino. Casinos wouldn’t make a profit if they paid bets off at their true odds; they’d only break even. And like any other business, casinos exist to make a profit.
Payout Odds
So every bet in a casino pays out at less than true odds, except for one, which we’ll discuss later. For example, if you make a bet on something that has a 3 to 1 chance of happening, and the casino pays out at 2 to 1 on that bet, the casino will make a profit in the long run.
Suppose in a mathematically perfect simulation that you place four bets of $1 each on something that has a probability of occurring of 3 to 1. You would win once and lose three times. If you lose $1 on your three losses, and you win $2 on your single win, how much money did you net? You lost $1.
For every dollar that you wagered in that scenario, you lost an average of 25 cents.
That’s the house edge in a nutshell.
The House Edge
The house edge is usually expressed as a percentage of each bet that you can expect to lose over the long run. In the example above, the house edge was 25%, which is huge.
The house edge on most casino games is between 1% and 10%, but in craps, you’ll find some of the best bets and some of the worst bets in the casino.
The Best Odds in Craps
The best bets at the craps table are the ones with the lowest house edge, and luckily, those are also the simplest bets you can make. These bets include the pass bet, the don’t pass bet, the come bet, the don’t come bet, taking odds, and laying odds.
The house edge on the pass bet and the come bet is 1.41%, which means that for every $100 you wager, you should expect to lose, in the long run, an average of $1.41.
The house edge on the don’t pass and the don’t come bet is 1.36%, which means that for every $100 you wager, you should expect to lose, in the long run, an average of $1.36.
When you take odds or lay odds, your bet pays out at true odds. This means the house edge is 0, making this the best bet in the casino. The only catch is that in order to take or lay odds, you have to make a pass or don’t pass bet first.
The Worst Odds in Craps
The worst bets at the craps table are the complicated bets. They have the highest house edge, and when we say the house edge is high, we mean that it’s staggering.
The craps table features countless proposition bets of varying complexity, but here are a few examples of bets with bad odds in craps.
The Big 6 and Big 8 bets offer a house edge of 9.09%. That’s absurd when you consider that you can place the same bet as a “place bet” and only face a house edge of, at most, 6.67%.
Hardway bets also offer lousy odds. The house edge is either 9.09% or 11.11%, depending on which hard total you’re wagering on.
The Any Seven bet is another doozy. The house edge on this one is a whopping 16.67%.
Any time you find a game with bets with a house edge that ranges between 1.36% and 16.67%, you should educate yourself about which bets are offer the best odds and which ones offer the worst odds.
Theoretical Probability of Winning Craps
The game of craps is unique in a couple of ways. For one thing, the game offers some of the best bets in the casino. For another, it also offers some of the worst bets at the same time. Most casino games either have a high house edge or a low house edge; craps has both.
Another intriguing aspect of craps is that it’s the only casino game where you can bet on something NOT happening. In all other casino games, the only event that you can bet on is the one that happens. You can’t bet that red won’t come up in roulette; you can only bet on black or on one of the green zeroes.
Both of these make for interesting uses of probability in analyzing the game.
Possible Totals and Their Probabilities
In craps, there are only 12 possible totals, but the probabilities of the various totals vary significantly. That’s because, as a probability problem, each total (except for 2 and 12) can be achieved in multiple ways.
Here’s a list of each of the totals along with the ways of achieving them and the probability of getting each of them:
2 is only possible if you get a 1 on both dice. The probability of that happening is 1/36. This is calculated by multiplying the odds of getting a 1 on the first die with the probability of getting a 1 on the second die. As a math problem, it looks like this: 1/6 X 1/6 = 36.
3 is possible in 2 different ways. The first is to get a 1 on the first die and a 2 on the second die, and the second is to get a 2 on the first die and a 1 on the second die. This makes a total of 3 twice as likely as a total of 2, which is a probability of 2/36.
4 is possible in 3 different ways. You can get a 1 on the first die and 3 on the second. You could also get a 3 on the first die and a 1on on the second. Or you could get a 2 on both dice. That’s 3 times as likely as getting a 2, which means the probability is 3/36.
5 is possible in 4 different ways. You can get a 1 and a 4, a 4 and a 1, a 2 and a 3, or a 3 and a 2. That’s 4 times as likely as getting a 2, which means the probability is 4/36.
6 is possible in 5 different ways. You can get a 1 and a 5, a 5 and a 1, a 2 and a 4, a 4 and a 2, or a 3 and 3. That’s 5 times as likely as getting a 2, which means the probability is 5/36.
7 is the most likely outcome when rolling two dice. There are 6 ways to get a total of 7: a 1 and a 6, a 6 and a 1, a 2 and 5, a 5 and a 2, a 4 and a 3, and a 3 and a 4. This is 6 times as likely as rolling a 2, which means the probability is 6/36.
The rest of the totals correspond accordingly. A total of 8 has the same probability as a total of 6. A total of 9 has the same probability as a total of 5. A total of 10 has the same probability as a total of 4. A total of 11 has the same probability as a total of 3, and a total of 12 is exactly as likely as a total of 2.
Expressing these Probabilities
Expressing a probability as a fraction is just one way to express it. Another way is to express a probability as a decimal. You could also convert that to a percentage, which is very common and quite intuitive. Finally, you can express probabilities in odds format.
Understanding that something will happen once every 36 times is intuitive enough when looked at as a fraction. But 2.78% is also an intuitive way of looking at it. Poker players probably prefer to look at that number as odds, which are 35 to 1. (There are 35 ways to NOT roll a 2, and only 1 way to roll the 2.)
Understanding the House Edge
Probability matters because of the house edge. Casino games are rigged, as it turns out, but not in the way you think. The reason casinos are profitable is because they never pay bets out at their true odds of happening.
For example, if you place a bet that a 2 is going to come up on the next roll of the dice, the odds of winning are 35 to 1. But this bet pays off at 30 to 1.
So suppose you bet $1 on that total 36 times in a row, and you saw mathematically perfect results. (This won’t happen in the short run, but the more rolls of the dice that are seen, the closer the results will mirror the mathematical expectation.)
You’ve bet a total of $36. You won one of those bets, and you got $30 out of the deal. The difference of $6 is your net loss. That difference represents the house edge, and it’s usually expressed as a percentage. The house edge on this particular bet is 13.89%, which means that you can expect to lose $13.89 for every $100 you wager.
This number might be meaningless, but you can use it to calculate how much entertainment you see for your dollar. The reality is that if you play any game with a house edge long enough, you’ll eventually lose all your money to the house. The trick is to maximize the amount of entertainment you get out of the deal.
Calculating Expected Losses per Hour
Suppose you’re playing craps, and you’re making the bet on the 2 over and over. The house edge is 13.89%. Assume that you’re able to make 40 bets per hour at $2.50 per bet. That’s $100 per hour you’re putting into action.
At that rate, you’ll lose $13.89/hour over the long run.
Compare that with another game, roulette. Let’s say that you’re playing on a standard American roulette wheel, where the house edge is 5.26%. You can assume 40 bets per hour here, too, and if you assume the same amount per bet, you’re looking at losing $5.26 per hour over the long run.
Roulette seems like the better game, right? But that’s just because we picked one of the worst bets at the craps table.
The best bet at the craps table is the don’t pass bet, which has a house edge of only 1.36%. Now you’re looking at an expected loss of $1.36 per hour. Heck, you can drink a couple of free cocktails an hour and wind up having a lot of fun for very little money at that rate.
Understanding probability and the house edge is the first step to becoming an intelligent gambler.
Craps Betting Systems
Betting systems of any kind are exercises in futility. No manipulation of your betting amounts can change the fact that all casino games have an insurmountable edge over the house. This includes craps betting systems.
The big error that most betting systems make is assuming that the odds of something happening change because of what happened previously. This is true (to an extent) in a game like blackjack, but that’s because the deck has a memory—once a specific card has been played, it can’t be played again until the deck has been re-shuffled.
In other games, like roulette and craps, each outcome is an independent event. Just because someone has been rolling hot all night doesn’t mean she’s statistically likely to continue rolling hot. It also doesn’t mean that she’s due for a loss. The odds of each outcome of each roll remain the same regardless of what happened on the previous one, two, three, or even thirty rolls.
The rest of this page explores and examines some craps betting systems.
Hedge Bets
Some of the worst systems you’ll come across involve what’s called “hedge betting”. These bets are supposed to reduce your risk. For example, you might try to hedge your bet on a pass line bet by also placing a bet on any craps. This seems like a can’t lose system, but once you examine the math, the system falls apart.
Suppose you make a pass line bet of $20. You have 8 ways to win, 4 ways to lose, and 24 results which are essentially “no result”. After 36 rolls, on average, you’ll have won $20 eight times, lost $20 four times, and seen a “no effect” the rest of the time, for an average win of $80.
Now suppose you make a pass line bet of $20 and a $5 bet on “any craps”. You now have 12 ways to win and 24 ways to lose. You’ll only win an average of $60 over 36 rolls with this system.
You’re better off just sticking with the pass line bet, obviously, right?
The Rothstein System
Another craps betting system that might be familiar to some players is called “the Rothstein system”. It’s just a variation on a popular betting system called the Martingale System, and it’s equally worthless.
The way this system works is that you begin by placing a single betting unit on the pass line. If you lose, you bet again, but this time you bet three units. If you lose again, you bet seven units. In other words, any time you lose, you double your last bet and add one unit.
If you win after your first bet, you’re up by one unit. If you win after your second bet, you’re up by two units. And so on.
Each time you win, you go back and start over again with one unit.
The Watcher System
We’ve also seen this one called “the Patience System”. Your goal is to win $10 a day with this system. You need a bankroll of $500 minimum for this system.
This time, instead of betting the pass line, you’re betting don’t pass. To use this system, you watch the game until there have been four successful pass line bets in a row. Then you bet $10 on don’t pass, because surely by now the dice are bound to lose, right?
This system is flawed because it doesn’t take into account the fact that each roll of the dice is an independent event. The probability on the upcoming roll doesn’t change just because of the four rolls prior.
People who like this system like to point out that the odds of the dice winning five successive passes are 31 to 1. So the idea is that you’ll win $10 31 out of 32 times. The advice this system promotes is also to quit playing for the day once you’ve won.
What if you do lose, though? The system has a plan for that, too. You double your bet on don’t pass. The odds of six successive passes are 63 to 1, and if you win this time, you win your money back and then some. And if you lose again, you double your bet again, because the odds of seven successive passes are 127 to 1.
Every progressive betting system faces the same problem. Eventually you’ll wind up with a long enough losing streak that you won’t be able to make your next bet in the progression because it’s over the house maximum bet.
Also, you’re not betting that five (or six, or seven) successive passes will be made. You’re only betting on the next pass. The odds of winning are slightly less than 50%, regardless of what happened on the previous rolls.
Hot and Cold Betting Systems
The idea behind these kinds of systems is to bet the pass line when the shooter is hot, and to bet the pass line when the shooter is cold. One common way to approach this system is to bet the pass line.
If the shooter wins, you continue to bet the pass line. As soon as the shooter loses, you switch to betting on the don’t pass line. You switch back as soon as the shooter wins again. The goal is to eventually catch someone on a winning streak (or a losing streak) and win lots of money during that streak.
This would work great if the house didn’t have a total on the don’t come bets that don’t win or lose. The percentage of those will eventually eat up any edge you might have gained using this system.
The Best System Is No System at All
Going with the best odds is the best craps system you can use. That’s easy enough to do. Just bet the pass line (or the don’t pass line), and whenever you’re able to take or lay odds, take or lay the maximum you can. This reduces the house edge dramatically. It doesn’t eliminate it, but your entertainment dollar will stretch a lot farther.
Martingale System – Progressive & Reverse
The Martingale System is arguably the most famous betting system in history. It’s an example of a “progressive betting system”. In this case, progressive means that bets get progressively larger until the player has a win—you shouldn’t confused the word’s use with its use when discussing slot machine jackpots, for example.
The Martingale System is easy enough to learn to use. Find a gambling game which offers an even money bet. The pass line bet in craps would be one example, as would betting on black or red in roulette. (This system is especially popular with roulette players.)
Then place a one unit bet. If you win, pocket your winnings. If you lose, double the size of your next bet; i.e., bet two units. If you win this time, you’ve won back the bet you lost on your first bet plus one unit. If you lose, you double up again on your next bet, this time wagering four units.
The system is designed to result in a profit of one unit after each inevitable win.
An Example of the Martingale in Action
Here’s a fictional example. You bet $10 on the pass line at the craps table and lose. On your next bet, you bet $20, and you lose again. On your third bet, you bet $40, and you win.
You lose $10 on the first bet and $20 on the second bet, for a total loss of $30. When you win on the third bet, you’re up $10.
Does the Martingale System Work?
At first glance, this seems like an unbeatable betting system, and it actually would be unbeatable if it weren’t for two things:
- The fact that you don’t have an unlimited bankroll, and
- The fact that casinos have betting limits.
It might seem unlikely that your next bet would be so high that your bankroll couldn’t cover it, but let’s look at the actual numbers. Assume you have a gambling bankroll of $5000. How many times in a row would you have to lose before you ran out of money?
The progression looks like this: $10, $20, $40, $80, $160, $320, $640, $1280, $2560, $5120…
So you only have to lose ten times in a row in order to need to bet more than your entire bankroll in order to continue using your system. But what’s worse is that all those bets were eating into your bankroll before that point, so you’re actually out of money after nine bets, not ten.
Some might think that with even money bets it’s extremely unlikely that you’d ever lose nine times in a row on an even money bet, but it happens more often than you think. And when it does happen, it’s a devastating loss.
But that’s not the only problem with this system. The other problem is betting limits. Casinos don’t let you wager any amount you want to. They have a maximum bet (and a minimum bet) at any table.
It’s not unusual to find a table where you have a minimum $10 bet and a maximum $500 bet. You only have to lose six times in a row in order to be unable to continue your progression.
So on a practical level, the Martingale doesn’t work. You’ll never have an unlimited bankroll, even if you find a casino with extremely high betting limits.
The other problem is that eventually you’ll have to place such a large wager that you’re putting a tremendous amount of money into action of a tiny net win. Do you really want to place a wager for $320 in order to see a net win of just $10? Common sense would tell you that’s a bad deal.
The Reverse Martingale
A related betting system is the “Reverse Martingale”, which is also sometimes called the “Anti-Martingale”. In this variation of the system, you increase your bets when you’re winning, and you reduce your bets when you’re losing. The theory is that you’ll benefit from your hot streaks while minimizing the damage from your inevitable losing streaks.
The Reverse Martingale proceeds from a mistake in logic called the gambler’s fallacy. This is the belief that if something has happened several times in a row, it becomes less likely to happen again, and vice versa. The reality in almost all gambling games is that each bet is an independent event, and the odds don’t change.
In other words, if you spin a roulette wheel five times in a row, and it lands on red each times, the odds of the ball landing on black the sixth time are still the same. They don’t increase or decrease based on the previous events.
All casino games have a mathematical edge over the player. No betting strategy changes that mathematical fact. Betting systems like the Martingale can be fun in the short run, and they might even occasionally show a small profit. But if you play a negative expectation game long enough, you’ll eventually see a net loss. That’s how casino gambling works.
This isn’t necessarily a bad thing. You just need to understand it so that you can have the right attitude about gambling. It’s entertainment, and you should treat it as such. Don’t pin your ability to make a living or pay your bills on the false hopes offered by progressive betting systems.
On the other hand, if playing using a progressive betting system makes the game more fun for you, then by all means indulge yourself. Gambling is supposed to be fun.