This is the first of a series of challenge posts. I’ll pose a problem in the hopes of convincing the wise Internauts to come forth with solutions. I intend the problems to be do-able rather than mind boggling: simply intriguing problems that I’d love to know the answer to but haven’t found the time yet to work through. Think of it as Web 2.0 enlightenment mixed with good old fashioned laziness. Or think of it as Yahoo! Answers, blog edition.

Don’t expect to go unrewarded for your efforts! I’ll pay ten yootles, plus an optional and unspecified tip, to the respondent with the best solution. What can you do with these yootles? Well, to make a long story short, you can spend them with me, people who trust me, people who trust people who trust me, etc. (In lieu of a formal microformat specification for yootles offers, for now I’ll simply use the keyword/tag “yootleoffer” to identify opportunities to earn yootles, in the spirit of “freedbacking”.)


dice So, on with the challenge! I just returned from a pit stop in Las Vegas, so this one is weighing on my mind. I’d like to see an analysis of strategies for playing craps that take into account the variance of the bettor’s wealth, not just the expectation.

Every idiot knows the best strategy to minimize the casino’s edge in craps: bet the pass line and load up on the maximum odds possible. The odds bet in craps is one of the only fair bets in the casino, so the more you load up on odds, the closer the casino’s edge is to zero. But despite the fact that craps is one of the fairest games on the casino floor, it’s also one of the highest variance games, meaning that your money can easily swing wildly up or down in a manner of minutes. So on a fixed budget, craps can be exceedingly dangerous. What I’m looking for is one or more strategies that have lower variance, and are thus less risky.

So that this challenge is not vague and open ended, let me boil this overall goal down into something fairly specific:

The Challenge: Suppose that I walk into a casino with $200. I arrive at a craps table that has a $5 minimum bet and allows 2X odds. I’m looking for a strategy that:

  1. Has at least some chance of making a profit (otherwise, why bother?), and
  2. Maximizes the expected amount of time (number of dice rolls) that my $200 will last.

I prefer if you ignore the center bets in your analysis. Bonus points if you examine what happens with different budgets, table limits, and/or allowed odds. Another way to motivate this is as follows: I have a small fixed budget but want to hang around a high-limit table for as long as possible, because I get a better atmosphere, more drinks, and a glimpse of life as a high roller.

As an example, here is a strategy that appears to have very low variance: On the come out roll, bet on both the pass line and the don’t pass line. If the shooter rolls 2, 3, 7, or 11 you break even. If the shooter rolls 4, 5, 6, 8, 9, or 10, you’re also guaranteed to eventually break even. The only time you lose money is when the shooter rolls a 12 on a come out roll, in which case you lose your pass line bet and keep your don’t pass bet (i.e., you lose half your total stake). There’s only one problem with this strategy: it’s moronic. You have absolutely no possibility of winning: you can only either break even or lose. One thing you might add to this strategy to satisfy condition (1) is to take or give odds whenever the shooter establishes a point. Will this strategy make my $200 last longer on average than playing the pass line only?

For bonus points, I’d love to see a graph plotting a number of different strategies along the efficient frontier, trading off casino edge and variance. Another bonus point question: In terms of variance, is it better to place a single pass line bet with large odds, or is it better to place a number of come bets all with smaller odds?

To submit your answer to this challenge, post a comment with a link to your solution. If you can dig up the answer somewhere on the web, more power to you. If you can prove something analytically, I bow to you. Otherwise, I expect this to require some simple Monte Carlo simulation. Followed of course by some Monte Carlo verification. :-) Have fun!

Addendum: The winner is … Fools Gold!

40 Responses to “Challenge: Low variance craps strategy”

  1. [...] – An analysis of strategies for playing craps that take into account the variance of the bettor’s wealth, not just the expectation. [...]

  2. tk says:

    if you want to make your time last, surely making a single bet and rolling until it is settled is the way to go. i would lay or place a single number. expectation may be lower than line bets with full odds, but you won’t have to wager as much.

    if you want to bet on every roll, maybe you could alternate between pass and don’t pass with full odds. you might want to alternate in such a way so that you are always trying to have the same number of points for each, but it could also depend on which points.

    the biggest danger here is dealer error. though that might work in your favor, too.

    i’ll leave the analysis to someone else…

  3. Mohammad Mahdian says:

    Hi Dave,

    I don’t know much about craps, but there is a 1965 paper by Tom Ferguson that solves a very similar problem for general MDPs:

    http://www.math.ucla.edu/~tom/papers/ruin.pdf

  4. Fools Gold says:

    A Statistical Characterization and Comparison of Selected Craps Money Management and Bet Selection Systems. Ken Elliot Kbeiico. Columbia, MD 21046

    As to the ‘max odds’ question: the mathematical answer seems to be that you should never make a line bet on which you are unable or unwilling to take full odds. Now ofcourse this can make the experience short and sweet if you lose that first bet or the first few bets. 100x odds are often publicity promotions with most players sticking to five or ten times odds, yet Casino Royale (LV-strip) offers one dollar line bets and 100 times odds. Best strategy: $606 PutBet. After three hits, take it down and leave the flat bets for the dealers.

  5. Fools Gold says:

    “Better” depends upon goals and preferences. Few people actually trek to Vegas to put their entire bankroll on that first bet despite what mathematicians say is optimal strategy. More bets means more entertainment time, more free drinks, etc. The answer would also depend upon bet progression strategies that you intended to employ. If all you are ‘comfortable’ with is a table minimum bet with five times odds, then go for that. A line bet followed by two come bets would be adequate to stay in the game with your bankroll and catch any ‘hot streaks’. Hedge systems don’t make sense. Grinders don’t really do well anyway.

  6. Fools Gold says:

    what you want to maximize is the overall mean length: that is the number of rolls before exhausting your bankroll or making your pre-determined win-goal.
    At 5 dollar bets with two times odds, Hoyle’s Press and Oscar would be the two best systems, though Ponzer, in third place, would be preferable overall.
    Best thing would be to play at a slower table where there are lots of grinders who will slow things down even more. That maximizes your time at the table more cheaply than switching to a Patrick Right system.

  7. Fools Gold says:

    The reference in comment4 carried out simulations to 800 rolls. I would ofcourse question whether this is really enough.

    When a statistics professor had a pair of shaved dice manufactured to precision standards he found that the ‘edge’ could not be detected in fewer than one million rolls, so I wonder if any Monte Carlo simulation would really prove anything.

  8. Fools Gold says:

    URL for comment Four:

    http://www.conjelco.com/downloads/elliott-paper.pdf

    A statistical Characterization and Comparison of Selected Craps Money Management and Bet Selection Systems. Ken Elliott.

  9. Fools Gold says:

    Elliott paper is also available in:
    Finding the Edge: Mathematical and Quantitative Analysis of Gambling (Institute of Gambling & Commercial Gaming) (Hardcover)
    by Olaf Vancura (Editor), William Eadington (Editor), Judy Cornelius (Editor)

  10. Don Favero says:

    I wrote a book on this subject. After interviewing 50 dealers across America, I applied their low limit stategies to several multi-month craps tours. The answer you are seeking is called the Fremont Street Grind. Simply it is a sure way to live a high roller life on a very low limit gaming budget. Amazon.com or Barnes and Noble…. Craps Across America. Check it out.

  11. Fools Gold says:

    Re: post ten:
    Fremont street grind is more commonly known as Anything But Seven and has no relationship to the problem posed in the posting at all and ofcourse since there is no line bet there is no way to be the shooter.
    $10 field
    $15 Place 5
    $18 Place 6 & 8

    2,12 pay $20
    3,4,9,10,11 pay $10
    5,6,8 pay $11
    7 loses $61

  12. Don Favero says:

    There is a way to play the Fremont Street Grind and be the shooter as well. If I am the shooter and I roll a 5, 6, or 8, I immediately consider playing a level of the FSG. With odds behind the pass line for the point number, I place the other two numbers for the lowest table minimum, and then place the lowest minimum on the field. Then if any number is rolled, I win something, be it a small amount. In my case I may use the winnings to build up the odds behind the line wager, or I may continue in the grind if I am rolling numbers. Of course I may press the place bets as well, depending on whether I am up on the casino, even or behind. So for a minimum of money, I am getting constant action and table time credit. This is way to cover all numbers but the seven for less than the typical $27 across. At a $5 dollar table the FSG is an approach that many players use. The key is not to stay in the grind but to take all place bets down when a profit is assured even it is as little as $5 or $10. You can play hours this way without serious risk. Most players stay in the grind too long, thinking that a “7″ will never show. Of course it always does.
    I am not suggesting that the FSG is the complete answer, but it is one way to play the table on a low limit gaming budget.
    If you hang on Fremont Street in downtown Las Vegas or stop in at the sole table at Ellis Island on Koval Lane, behind Bally’s, then you will see the Freemont Street Grind being played by the majority of players at the table. These players may leave the casino just $25 up, having downed 10 free beers and comped enough for a free buffet later in the eveing. That’s living in Vegas, baby!
    The other approch is to play both the pass line and don’t pass….simply hanging for comps or just playing odds behind the line or laying odds on the don’t pass line. This is the slowest grind in craps, but at a choppy table to kill time until a streak occurs, it is a way to maintain your position. This pass line – don’t pass line bet is also a way to teach a new player how Craps works, since money is only at risk during the comeout roll — on a 12 roll. You could cover the 12 for $1 just to hedge the pass line bet, but truly it is not worth it.

  13. Fools Gold says:

    Well, we are straying quite a bit from the blog’s original question which is limited to determining an optimal strategy involving linebets/oddsbets and proving it by monte carlo methods. The Elliott paper that I have cited provides data exactly on this topic, the Fremont Street Grind is unrelated to it but is, for some players, an interesting grind play.

    An odds bet may be fair in its payout but is of no benefit to the player unless he was intending to wager that money anyway. If any portion of the odds bet is ‘scared money’ for that player, the bet is of no use to him.

    There is a difference between maximizing financial gain and maximizing the time playing that game given a limited bankroll. Grinders who play for comps on room and board and stay at the table all day long are quite different than players who want to win big and leave.

  14. And the winner is … Fools Gold! The Elliott paper is nearly exactly what I was looking for, exceeding my requirements in many respects, if not quite answering 100% of the optional bonus questions. I award some bonus points for directly addressing the pass/don’t pass strategy I mentioned. In sum, I award Fools Gold Y$20 yootles, Y$10 for the best answer and Y$10 bonus tip. Thanks, Fools Gold!

    Thanks also Don Favero for your answers and link to your book: looks very related and interesting, just not quite as dead on an answer as the Eliot paper. Also, thanks Mohammad Mahdian for another interesting/relevant paper, though again not quite as directly applicable to the specific scenario at hand.

  15. Fools Gold says:

    “…Maximizes the expected amount of time (number of dice rolls) that my $200 will last.”

    Thank you for the Yootles. I don’t know what I will ever be able to do with them, but its a nice honor and I am sure that all participants in this thread enjoyed their activities.

    I would ofcourse now like to point out that although amount of time and number of dice rolls are surely linked there are different influences on the two separate measures. The casino likes a fast game because their money is “per decision” on a bet rather than “per bet made” and only the line bets are “contract bets” that can not be taken down. The question in this thread was posed for a 5 dollar minimum table with 2x odds. Casinos often have the lowest open table raised to a ten dollar minimum bet and many, many casinos require even greater minimum bets during busy times. A gambling day-boat or cruise ship has a captive market and will usually offer only 2x odds. Real world casinos on dry land where the players are free to go elsewhere usually are forced to offer much better odds than 2x. Some casinos offer a ‘chairs only’ table but limit the odds to 10x although at the ‘standees table’ the odds would be 20x to 100x. Even the famous grind-joint, Casino Royale, will have a one or two dollar minimum line bet at 20x odds but will require a five dollar minimum line bet for the player to take 100x odds. Some Henderson and Reno casinos also have similar quirky policies on odds bets.

    The player who selects a ‘grinder joint’ with ‘break-in dealers’ can get higher “time at table” values than a player making the exact same bets in an upscale casino with more experienced dealers because “time at table” is quite different than “number of rolls”. Some players seek out relatively empty tables while some enjoy a more crowded one, such things do affect the number of rolls that will take place.

    A betting strategy that encompasses more than just 2x odds would be a far more “real world” exercise. Ofcourse a ‘real world’ strategy would also have to consider: tips, beverages, dealer and player mistakes and comp strategies.
    This would be similar to the world of Video Poker machines with payout tables that exceed 100percent: a statistician milking such a machine must play perfect strategy and a perect strategy will often require a player to intentionally lose a particular hand so as to gain the machine’s maximum payout table.

    There are shortcuts to estimating the chances of ‘Gambler’s Ruin’ versus making a predetermined profit goal:
    With (Y;Z) the result of a one-unit pass line bet with Beta-times odds can be represented by
    X := Y + BetaZ
    which has moments
    m1 = −7/495
    m2 = 1+784/495B + Bsquared
    m3 = −7/495+103/165Bsquared +17/30Bcubed
    m4 = 1+1568/495 B + 6Bsquared +538/99 Bcubed +599/300Btothe4th

    The results which first appeared informally in the Las Vegas Advisor (October 1996), tell us something remarkable about the effect of odds bets in craps. …”

    I’ve found another reference to the same work:
    “…For example, a pass line bet with double odds is such a wager. We are concerned with the probability that, in an independent sequence of identical wagers of this type, the gambler loses $L$ or more betting units (i.e., the gambler is ruined) before he wins $W$ or more betting units. Using an idea of Markov, Feller established upper and lower bounds on the probability of ruin, bounds that are often very close to each other. However, his formulation depends on finding a positive nontrivial root of the equation $\varphi (\rho)=1$, where $\varphi$ is the probability generating function for the wager in question. Here we give simpler bounds, which rely on the first few moments of the specified wager, thereby making such gambler’s ruin probabilities more easily computable. … ”

    Ofcourse some players do not want to maximize time at table, they want to maximize their chance for financial gain and have only a constraint that their time at table must last long enough for a r/f/b comp. (Some of those retirees in Henderson have not paid for a room, meal or drink in over a dozen years. Their goal is to reach liver-ruin before they reach gambler’s ruin).

    This thread has been an interesting exercise but at 2x odds it is not quite a “real world” scenario. The very popular 3x4x5x odds now offered help speed up the game (which casinos like) and really do not damage the player all that much either. More commonly encountered odds of 10x and 20x would be real world scenarios whereas the much rarer 100x odds are really more often thought of as “advertising come-ons” rather than real opportunities.

  16. Fools Gold says:

    “…the biggest danger here is dealer error. though that might work in your favor, too.”

    This upthread comment about dealer error introduces some real world factors: dealer error, player error, tips to dealers and to waitresses. Biased coins and biased dice can be tested by graduate students studying statistics but the recording errors made by the observing graduate students is often greater than the error introduced by the biased coin or biased dice. So I would conclude that any ‘dealer error’ or ‘player error’ would have a negligible effect.

  17. FoolsGold says:

    “The Challenge: Suppose that I walk into a casino with $200. I arrive at a craps table that has a $5 minimum bet and allows 2X odds…”
    The Real World: Five dollar tables are rarely available during ‘prime time’. 2x odds would be unusual nowadays. Most casinos have to offer higher odds simply as a matter of competition. The very popular 3x4x5x odds would probably be encountered today.
    200 is okay but atleast a thousand dollars is much better. 10x and 20x odds are usually available without too much searching and the gimmicky 100x odds are available in limited locations. The odds affects variance, not outcome but the goal is to maximize profit over a reasonable amount of time, not merely to maximize time at table.

  18. FoolsGold says:

    I wonder.

    We all know that craps is a negative expectation game since (excpet for the free odds bet) each and every bet has a house edge.

    However, most of the money that the house keeps is not due to the famed “house edge” but is instead due to poor money management by the players who voluntarily choose to make bets that have a high house edge or are more likely to deplete their bank roll.

    If in order to determine the effect of cocked dice on a game a mathmatician had to carry things out to a million rolls then any ‘edge’ is not the main factor in casino profits.

    We walk in and hope that our ‘short run’ number distribution is profitable even though we know that the law of large numbers will be aiding the house over the long run. Is there anything that will really help a player overcome the fact that the casino is the one with the huge pile of chips?

  19. slipperytoad says:

    I am a poker player, but will play craps at times. Mostly, I play under the exact conditions you outlined, with the same goals.

    Place your 40 chips on the left side of your tray. Flat bet all 40 – 1 at a time, placing the “winnings and original bets” on the right side.

    If you have more than 40 chips after 40 decisions pocket your winnings and begin again.

    If you have less than 40, bet 2 chips as many times as you can. Again placing the winnings and bets in the opposite side of the tray.

    You can do this a third and fouth time if necessary, 3 and 4 chips respectively.

    Amazingly it works quite often.

    NOTE: I reduce my wagers back to 1 after I have a profit and begin the process again

    ALL WAGERING SYSTEMS FAIL WHERE FLAT BETTING FAILS

  20. slipperytoad says:

    I alwas play don’t pass without odds. This may not be optimal but it keeps the wagers managable

  21. FoolsGold says:

    >I always play don’tpass w/o odds.
    Odds .. .. House Edge
    0X 1.36 percent
    1X 0.682%
    2X 0.455%
    3X 0.341%
    3-4-5X 0.273%
    5X 0.227%
    10X 0.124%
    20X 0.065%
    100X 0.014%

  22. FoolsGold says:

    >I am a poker player,
    bluffing won’t help at craps.
    >ALL WAGERING SYSTEMS FAIL WHERE FLAT BETTING FAILS
    Well, I think all wagering systems fail period!
    >slipperytoad says:
    >I alwas play don’t pass without odds.
    Okay.
    Slipperytoad chooses to play at 1.36 percent house edge when he could by laying odds at the almost universally available 3x4x5x tables be playing instead at 0.273 percent house edge.

    This extends his TimeAtTable but denies him various Comps and winings. It is clearly not optimal but does it make a great enough difference to the bottom line? Remember, as a Don’tPlayer he will have to Lay the odds, not Take the odds, so he will need a bigger bankrole that otherwise.

  23. Do you believe in dice control? Do you believe in winning craps systems? Do you think you can gain even a slight advantage over the casino using dice control or winning systems? If you answered “Yes” to any of the questions, then you’re playing in Fantasyland, not in reality and you’re setting yourself up to be a big-time loser. If you want to truly learn all aspects of the game and have a legitimate chance of actually beating the casino, then you must learn to play casino craps in reality. What’s the secret? Visit my site to read a free sample chapter from my new Ebook about the absurdity of dice control. Also, read the free informative articles at my site. Do you want a legitimate chance of winning, or are you going to let your burning desire to beat the house take control of your common sense? Don’t be fooled by bogus claims that dice control works or winning systems exist. Be smart. Play smart. Learn how to play casino craps the right way.

  24. FoolsGold says:

    >Do you believe in dice control?
    No. Not one whit. And if it DID exist, it would still add zilch.
    >Do you believe in winning systems?
    Sure. Get Lucky!! Short term variance and pure luck can result in a win!
    This thread was not about some sort of winning system, just a mathematical exploration of how to more optimally utilize various bankroll factors so as to maximize time at table.

  25. Dice control? No. Dice influence? Absolutely. It’s a well researched and proven fact. FoolsGold is correct when he says short term variance and pure luck can result in a win. But by influencing the outcome of the roll you can reduce both the volatility and the variance of the game. Skilled DI’s have more winning sessions, larger wins, and smaller losses when the remaining volatility and variability kick in.

  26. Joe Goldman says:

    Learn to play craps the right way without getting duped into buying bogus systems or wacky dice-setting schemes. Check out Learn to Play Casino Craps for a great eBook that doesn’t give false hope and doesn’t promise you can beat the house out of millions of dollars. It doesn’t offer wacky dice-setting schemes from bogus dice control specialists to overcome the house advantage. It does, however, teach the math so you can easily learn and understand that the game is designed for you to lose. This eBook also challenges the silly notion that a player can affect the outcome of a craps decision by mastering dice control through dice setting. The fact is that there’s no true long-term winning strategy for craps. What’s the secret to craps? If the game is designed for players to lose, why do knowledgeable players bother with it? How do knowledgeable players learn the optimal mix of essential variables that allows them to walk away with a smile and money left in their pockets? The secret is in this eBook.

  27. I’m glad Joe Goldman mentioned the math of the game. Hard to believe he’s peddling a “Learn to play craps” e-book for $9.95 when all of that basic information is available FREE on just about any craps-related website you’ll find. “The secret is in this e-Book.” Give me a break. There aren’t any secrets in this game. You want to talk about e-books – I’ve got an e-book you can download as well. Only mine is free. You can find it at this link:

    http://www.crapsfest.com/modules.php?name=Downloads&d_op=viewdownload&cid=7

    Now, there are lots of folks who understand the math of the game who will tell you that dice influencing is a reality. Rather than focus on the guys who are teaching DI (dice influencing) including myself – consider Stanford Wong. Wong is a statistician who is best known for his blackjack play and the many books he has written on card counting, etc. But he’s also a well known dice influencer and has written extensively on the subject. Wong will tell you that DI works.

    Rather than take my word (or Wong’s) that dice influencing is a reality I’ll offer this challenge. Get a pair of casino dice and put together some sort of practice rig that will allow you to toss the dice and get the same “random” results you’d get in the casino. That means using new certified perfect dice (about $5 a pair) and real pyramid rail rubber on the back wall of the rig.

    Pick a dice set you can find quickly and easily. I suggest using a hardway set with the five pips on top and the 6-1 on the left and right sides of each die. Grip the dice the same way every time – for starters you can go with the thumb in the back, the pinkie and index fingers on the side faces of the dice and the middle and ring fingers on front. We call that a five-finger or lock grip. It’s not the best grip but it’s easy to execute quickly and consistently. Pick a spot on your practice rig layout about six to eight inches from the back wall. Get a piece of chalk and put an “X” on that spot. Now start lobbing the dice to that spot. The idea is to get the dice to that spot with the softest toss possible. Toss the dice the same way over and over and over. Once you get to the point that the dice are flying through the air together and striking the wall softly at approximately the same time you are ready to start tracking. Go to my Crapsfest website at http://www.crapsfest.com, click on “downloads” then click on “roll trackers.” Download the latest copy of BoneTracker. Note that BoneTracker is open source and free, but requires that you have MS Excel or Open Office to operate. Once the BoneTracker program opens set up your dice pre-set arrangement in the front page. The default is for the V-3 set so you’ll need to input 5-5 as the “up” numbers and 4-4 as the front numbers. The rest of the numbers will automatically be inserted. Start the roll tracking macro and start tossing the dice. You’ll need to differentiate between the right die and left die and sometimes it’s difficult to keep track of the dice as they fly through the air. Most DI’s use one red die and one green die to simplify tracking, but you can simply mark the one pip on one die with an “L” and the other with a “R” and track it that way. BoneTracker is set up to allow you to track 720 rolls per book, so toss 720 and review the results. The program will tell you if you have one die off axis, both dice off axis, or are hitting primary or secondary numbers. As you advance in your skills you can use that info to diagnose grip and toss issues to improve your skills. For now, don’t sweat it. Just toss, get in a groove, and track. In the end you’ll want to toss about 10 books of rolls – 7200 tosses – to get a result of any statistical significance. An SRR of 6.0 is “random” and if your results are random they should come very close to this number. If they are significantly different either way – say from 5.5 to 6.5 – then you are exhibiting influence over the dice. You can capitalize on those numbers using the dice set as is – or you can go into the transposition function on the program and look at your results with other dice pre-set arrangements. With an SRR of 5.5 using the hardway set you may find a “mutant” set that will give you an SRR of 9.5 or more. It all depends on the characteristics of your toss.

    Okay, the latest version of BoneTracker is 2.22. You can download it at this link:

    http://www.crapsfest.com/modules.php?name=Downloads&d_op=viewdownload&cid=1

    While you’re over there at the CrapsFest website take a minute to browse the articles. Yeah, you’ll see that I do seminars, sell books, dice, videos, etc. But the articles and downloads mentioned are available for free. You can also visit my other website – http://www.axispowercraps.com and look at some of the primary dice pre-set arrangements and some other details on the game – also available free.

    Virtually anything you need to know to learn dice control on your own is available somewhere on-line for free. I encourage you to learn everything you can about it without spending a dime. If, at some time, you decide you want to learn more about DI then buy a book. I recommend picking up something from half.com or used from Amazon. There are several good ones out there. The best IMHO is Mad Professor’s Crapshooting Bible. Check it out.

    If you decide you want some individual or small group training on dice control we do offer that. Watch the websites for details on upcoming craps clinics. No, I’m not one of the “golden girls” of craps. You won’t find me making unsubstantiated claims on TV, radio, or in magazines. But I will step up to the table with any student who attends one of the seminars we do and play a live session. That’s where the rubber meets the road.

  28. FoolsGold says:

    Flogging a dead horse?
    I’d like to re-invigorate this thread and thus propose that Mr. Pennock consider revising the challenge to be more in keeping with the conditions encountered in the real world.

    The original challenge involved a 2x odds table, such as might be found on a cruise ship wherein the gambler has no way of seeking a different casino. A more real world approach would focus on 3x4x5x odds which is encountered in about half the casinos in Vegas or a 5x table or perhaps a 10x table.
    Also, despite the faltering economy, I think a real world challenge should involve a table at which the minimum bet is ten dollars. Sure one can trek to Vicksburg, MS and play at a One Dollar table with 20x odds or Biloxi, MS and play at a 3.00 table but most players will encounter 10.00 tables, not 5.00 tables.

    >I prefer if you ignore the center bets
    Agreed. That 16.67% house edge on BigRed would be tough on any gambler and most gamblers would ignore the center bets.
    >I have a small fixed budget but
    >want to hang around a high-limit table.
    Limits are for the casino’s benefit and are designed to prevent a gambler from gambling with the casinos money by following a lucky streak.
    >I get a better atmosphere, more drinks,
    >and a glimpse of life as a high roller.
    Agreed. I might even add that better dealers at the higher limit tables might make fewer mistakes. Ofcourse more of those free drinks might induce player errors as well.

    >If you can dig up the answer somewhere on the web, more power to you.
    I may have made a mistake by digging up an answer that was already available. That answer was based on computer simulations of an insufficient number of rolls. I just thought it would be easier to analyze an existing article than engage in Monte Carlo simulations.
    Ofcourse perhaps the easiest thing to do is to accept that no professional craps players have ever emerged from the real world quagmire of Negative Expectation and that therefore Monte Carlo simulations are unnecessary because the Real World of craps shooters have been conducting them for years.
    The shareware program WinCraps would probably suffice and costs only twenty dollars, so perhaps it could be used.
    Its clear that an altered challenge might spark additional simulations but might also achieve results that differ little from the article that has already been cited.

  29. lol Craps says:

    I think my favorite strategy is betting on the don’t pass line and then making the free odds bet. I believe this is the best strategy to use in the house. I also have a craps blog if you are interesting in more craps info.

  30. Ron Lowe says:

    I’m an expert at craps-I have a real solution that would force casinos to close their craps-it is just so hard to lose using my method–I will only play on a 5 to 1 free odds table or better–and I live close to myrtle beach sc and the boats do offer 5 to 1 free odds on your line bet–first I advise never to play on a table that does not offer at least 5 to 1 free odds–my method can only lose if no one makes a pass out of at least 7 attempts, maybe 8—I’ve been going to Atlantic City since when just Resorts was the only casino there–I have seen cold tables but it is so,so,so rare not to see a point number made out of 7 attempts-out of approx. 200 times I’ve been to AC I’ve seen it happen maybe twice-my method even allows a small amount of your total bet to go on the hardways 2,4,6,8 if it’s the point number only but other than that you just play the free odds–there is no such thing as a 100% winning everytime in craps-however my method is the closest anyone will ever come to it–I have studied craps-practiced craps,researched craps,played casino craps for 40 years, and there is no other way to win on a consistent level–heres the best free advice I can give to anyone playing any casino game–never raise your bets when winning–only raise your bets to win while losing–casinos love to see you giving them a chance to get it back like that but they hate giving you a chance to win like that–good luck—chess52@ymail.com

  31. Play by increasing the betting increments indefinitely, so your lost $20 bet will become a $40 bet. After only seven rounds of losing in which you start with a $5 minimum and double every time, you will end up betting $320 on a single bet.

  32. Larry Thompson says:

    Here is my solution to grinding out profits at craps…..and, IT WORKS!

    THE DAILY OR SESSION FOLLOW MODE WITH SWITCH-PROTECTOR

    This is a neat little grinder that can be used at the craps table for making steady, daily profits.

    1. Use an adaquate bankroll and use the MM of “up & up & back to flat bets”.

    2. Pick your daily or session side by going to the table and get your first decision and that will be your “primary side” for the day….or each session.

    3. All you do then is “follow” your daily//session side and get +3 units for that session……and if a losing streak of three in a row hits you, all you do is switch to that side until it gets you your +3 units…or, it switches you back to your ‘primary’ side with a loss. Remember….hit & run is the real key to beating these games!

    Thats all there is to it….and it works! Doesn’t need any more superlatives……IT SIMPLY WORKS!

    If you have questions, feel free to email me and I’ll do my best to help. NOTE: I do not sell systems……I just hate seeing people throw their money away, trying to “gamble”. So all I’m doing is sharing what I use almost every day!

    Enjoy……

    Larry:-)

  33. William Mouncey says:

    Sorry the contest is closed, but this would have won it. Your requirements are: $200 stake, maximize playing time, and a possibility of a win, on a table with no more than 2x odds. Okay here it is: The RADARBRUCE system.
    Part one of the system is the come-out. Play the pass and don’t pass together, but here’s the twist: put twice as much on the pass as the don’t pass. So, a one-unit bet on a $5 table would be $10 pass/$5 don’t pass. Now you are playing the ONLY favorable bet in the casino, the craps come-out bet. Remembering the non-payoff for the DP on the bar roll (let’s say it’s the twelve), then there are 8 ways to win 1 unit (six 7′s and two 11′s) and 3 ways to lose 1 unit (one 2 and two 3′s), with 1 way to lose 2 units (one 12). The net odds is 8:5, but the payoff isn’t 5:8, it’s 8:8, or even money60% better than the actual odds.
    Part 2 of the system is the point. When a point is rolled, either put double odds on the don’t pass, giving you a net one-unit bet on don’t pass, or don’t do anything, giving you a one-unit bet on pass. That’s the beauty of this system. It’s flexible. Now the question is when to hedge, and when not to hedge. Well, for starters, look at the shooter, and use your noodle. If nobody is making anything, and there’s very little money on the table, hedge every roll. If he’s a dice setter, don’t hedge, he’ll probably make a few points before he craps out. If he’s not a dice setter, and there’s some money on the table, then use the actual payout ratio for points of 5:6 for the 6/8, plus 4:6 for the 5/9, plus 3:6 for the 4/10 points, equaling 18:30, or 2:3. so vary the hedging on and off in a 2:3 ratio, and the payoff should average out over a large number of runs to 50:50, or no net win. Except for the fact that the come-outs are paying 1:1 on a 8:5 bet, and you have your chance of making at least some money. To make money, but with a higher variance, hedge every bet, and you will meet the house odds in the long run, about 2.88% for the pass-Don’t pass system, because the bar roll does not pay off, at a 1:36 rate, or just under 3%.
    So there’s the RADARBRUCE system. Pass/Don’t pass at a 2:1 ratio, and hedge the DP with double odds 60% of the time, and you should at least break even. But watch for dice setters, and ride them into the sunset. Simple, low variance, and anybody with 200 bucks can do it.

  34. William Mouncey says:

    I meant to write the actual odds for points was 24:36, not 18:30. Forgot to proofread. The theory still stands, though.

  35. @William: Thanks for the clear and detailed strategy and explanation. Indeed, this one seems like a promising contender. I even like the name.

  36. joe says:

    with all the strategys out there, theres no certainty, this blog is great, peels back the surface to reveal a fantastic world… within the game. i have a blog which touches on the topic of craps but does not go to the extent this great blog has http://www.casino-table.com. Will save this site for future and reccomend to would be players or established players alike.

  37. Zack says:

    I’m only 19, but I go to the casino more than I should… I think. My first game of choice was poker. On my 19th birthday I entered the casino with $45, the minimum buy-in was $50. I went to the roulette table, put $5 on a colour, hit, and headed for the poker room. The cards were just coming my way. I left with $400+. Impressive right? Went the next night with all that money and left with $50 again. This was the start of my interest in casino games. I landed a job at a national merchandising company as a shipper, and I was told that as a shipper, you get fired for 1 mistake (that was a scare tactic, you got a 1 mistake grace… then they fired you). There was a man named Patrick who worked there for 10 years, and my boss told me he made 0 mistakes in that 10 years. So, my boss, myself, Patrick and another co-worker set up a poker night, a $10 buy-in tournament style. So we were playing for about 2 months like this until one day Patrick logged into his account (his casino rewards account) and he had comps upon comps upon comps. He has free rooms (Mon-Thurs) and $400 toward restaurants, Not to mention free meals almost every night. So I asked him what he did, and he said he shot craps. At this point I had no idea what the hell that was, but it caught my attention. I went with him one night, and we were there for 15 minutes. He won 20 and walked. He did this EVERY day, for the past 10 years. And I didn’t fully understand what his bets were (for I didn’t know what a crap was) and starting replicating him. This has nothing to do with making $200 last, but everything with taking money and comps from the casino. First of all, I buy in for $500-$800 (in the formula which they award comps, the bigger the buy in, the bigger the comps, for the most part) and I bet the minimum on 6 and 8. You will never be able to beat the 7, ever, no way, in the long run. And statistically, 6 and 8 are suppose to appear before a together. Obviously you risk a little more, but I read a comment about hit and run, and that IS the way to go if you want to consistently win. I may be shooting myself in the foot by mentioning this, but I failed 2/3 math classes in high school, but I know proper bets when I see them. You hit a $6 bet on 6 or 8, you take down $7. Then take down all your bets (unless you’re shooting and you have in on the pass line). Then wait for the next round (whether they make a point or 7 out). Wash, rinse and repeat. I’ve walked making just a $4 profit, because I knew that if I stayed there I wouldn’t walk with a profit at all… but I did win a free buffet, so it was like a $27 profit. Slow and steady gets her ready. And thank you for all who posted on this thread, it really allowed me to think about stats in general.

  38. gamblingblog says:

    craps is such a risky game, i can understand the fascination for it, but still lol, you have to be brave!

  39. Your blog inspired me! I took the RADARBRUCE system I mentioned above, and incorporated the best parts of into my book, “Own Big Red.” It’s available on Amazon.com at the link shown. The RADARBRUCE strategy is not the whole system, just one facet of it. I would appreciate some constructive reviews by your readers. Thanks for the inspiration.

  40. Mac McCrery says:

    Check out Industrial dice .com: I think we have found another way to play craps.Commisiner of gambling[state of nevada] says he would be inclined to grant a liecince if I do a couple of things. Mail me after you see the .com

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