Analysis Of A Roulette Strategy
    By Tommaso Dorigo | October 17th 2009 10:37 AM | 15 comments | Print | E-mail | Track Comments
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    As Hank explains in a recent article, when he visits a Casino he plays the Roulette. His simple strategy consists in betting on a single colour, doubling the bet every time he loses; when he wins, he starts back with the minimum bet.

    Such a strategy is not going to make you rich, but no strategy does at the Roulette, especially the American one which has both a "0" and a "00" -two neutral numbers thrown in to enhance house odds. The idea of doubling every time is that eventually the colour you bet on is bound to appear, and you will win back all your fiches, plus one. Example: lose 1 - lose 2 - lose 4 - lose 8 - lose 16 - win 32: the sum is 32-16-8-4-2-1, which is exactly one fiche.

    There is, of course, one problem with this strategy: roulette tables have a maximum bet, which prevents you from continuing the doubling ad infinitum, even if you had that much money to throw on the table. Moreover, a very rare fluctuation with a uninterrupted, arbitrarily long streak of odd-colour occurrences, is bound to make you broke if you play for long.

    While one cannot change the table rules, and thus has to live with a maximum bet, the rare fluctuations can be kept rare by playing shorter games! If one defines before playing a fixed exit strategy -say, leave the table upon winning a certain number of fiches-, this is going to make it easier to prevent the killing streaks to occur during your game.

    I decided to put together a simple program to extract winning probabilities with the above strategy. The program simulates the outcome of reds, blacks, and neutral numbers in 10,000,000 sessions of up to 200 wheel turns -the typical duration of a night-long game. The user may define the number of fiches after which he or she leaves the table (less than 200, of course, since we can win at most one fiche per turn of the wheel with our strategy), and the maximum bet allowed. Below I detail some results.

    If the table has a maximum bet of 100 fiches, we get:
    max 200 rolls, quit at  +10: won 93.3435%, avg win= -9.2293
    max 200 rolls, quit at  +20: won 87.1386%, avg win=-17.4491
    max 200 rolls, quit at  +50: won 71.2252%, avg win=-36.6622
    max 200 rolls, quit at +100: won 56.9104%, avg win=-51.7284
    With a maximum bet of 1000 fiches, we get instead:
    max 200 rolls, quit at  +10: won 99.1284%, avg win= -9.2005
    max 200 rolls, quit at  +20: won 98.2628%, avg win=-18.2627
    max 200 rolls, quit at  +50: won 95.7134%, avg win=-44.6285
    max 200 rolls, quit at +100: won 91.6532%, avg win=-78.0571

    And with a maximum bet of 10000 fiches (if you can afford it!), we get:
    max 200 rolls, quit at  +10: won 99.876%, avg win=-11.8301
    max 200 rolls, quit at  +20: won 99.751%, avg win=-23.5798
    max 200 rolls, quit at  +50: won 99.379%, avg win=-59.0183
    max 200 rolls, quit at +100: won 97.935%, avg win=-107.22

    You may notice several things from the numbers above. First of all, the obvious thing that, as the maximum bet is increased, the chance of going home positive increases. Second, the average win is always negative. This should also not surprise you much: on average, the Casino wins!
    The third thing to note is that the probability to end up winning is a decreasing function of your greed. The more you want to win, the more likely it is that you lose. If you decide for a very conservative exit strategy (say, go home if you win 10 fiches), then you have very good odds of making it, especially with a high table maximum.

    This strategy is not for the faint hearted: during a bad streak you might have to put 1000 units on the table, in order to try and win back the original unit you bet at the start of the streak. Mathematically this makes sense, but from a psychological point of view, it is rather absurd.

    For those who want to know more about how the above numbers arise, I have prepared three graphs. The first one shows the distribution of fiches won (the total gain) at the end of each session -assuming, as we have, that we either stop when we win 50 fiches, or that we go on for 200 turns of the wheel. We have also assumed that when we reach the maximum bet and we lose, we continue betting that maximum amount until we win. Here is the distribution:

    As you see, there is a large peak at wins of +50, but there is a long tail of very bad losses. The average ends up at -44 tokens, but there are non-zero chances of losing really large sums.

    The secon graph shows the distribution of the longest streak of losses. This is what determines the long negative tail in the previous graph:

    The third graph shows the correlation between longest streak (on the vertical axis) and total gain (on the horizontal one). When bad streaks of losses do not occur, we end up winning (the blob of black crosses at the bottom right); when the streaks become longer, we hit the maximum bet, and we start losing money. The longer the streak, the worse it gets.

    I hope this is useful for those who apply Hank's "double it" strategy at Roulette. My personal advice, however, is to avoid Roulette tables altogether. As Hank says, play Poker or Blackjack instead!


    You make two points which  I probably did not highlight enough; don't get too greedy, in that you need to pick a point where you will leave and achieve it and stop.  The 5.26% house edge simply cannot be overcome with any strategy.  The second, which is amplified by your simulation, is that you have to make sure your house minimum and maximum are quite far apart.  If the table maximum is 30X the minimum, you may be in for trouble, since you double your bet every time, but with a minimum of $1 and a maximum of $3000, there is no problem.    A $10 minimum and $3000 maximum gives you 9 consecutive losses to get your money back, just make sure you have $6000 in our pocket if you take that approach.

    You also do not have a dancing head video of yourself gambling - the only thing missing from an otherwise outstanding article.
    Yeah, I really envy your video, damnit!
    A better strategy than the basic Martingale used here, might be to bet on single numbers. While the odds of winning on a single spin are reduced, some quick numbers I ran imply a pretty good chance of hitting +10, (similar to that of the Martingale), with one big advantage: You aren't boosting your bets anywhere near as much, capping out on the end of a 200 spin losing streak with a theoretical bet of 156.

    Allowing for lower limits, you could do a pair of 100-spin runs, which cap at a bet of 9.[1]

    You're still not going to win in the long run, but at least your potential losses aren't anywhere near as bad. (No limit caps at -5582, the 'twin 100' caps at -634.)

    [1] That's all maintaining bets for any net gain on a win, if you bet to get +10 on a single win you risk more but get a much easier to analyze case: Over a max of 100 bets you can reach a bet of 12, have a 93.05% chance of winning at least 10, risk losing 405 and expect to lose 4.19.

    Thank you for your stats to put thing into perspective. I agree with you that the system is not a get rich scheme. However I guess it depends all on the persons intention. If your intention is to make $100 extra cash a day, it's not a bad system. Once you have managed to accumulate a decent pot you could probably invest that money into something more useful.
    Another advantage is that considering that bank saving rates are rather poor this maybe a good way to add some extra tax free cash. Though the strategy only pans out if one is not so greedy, and has modest profit aims.


    You still haven't quite gotten the right perspective. There is one, and only one, system where you can expect to make money from a roulette table:

    Be the house.

    Any betting system a player can come up with trades off the following three things: What is your chance of coming out ahead, how much you will lose on average and how badly are you risking getting clobbered. Increasing your chance of winning means also increasing your average loss and/or how badly you can lose. Martingale systems are particularly bad because you are betting a lot to win a little thus you have a good chance of winning, but when you lose you lose big.

    Remember, it is really easy to spot the suckers at a roulette table: They're the ones playing roulette, (the house PC is over 5%).

    I understand, be the house. I played around with the free versions from the popular casino websites, and for the most part I did win. My experience has been that gambling is not a suitable income path. But for some extra side cash it can work, if you are not greedy and know your limits. I do not how the real life casino wheels work but sometimes I get the feeling that the online free/demos ones are slightly tweaked in the players favour to sweeten the deal.

    Also as mentioned the table limits the max bet, so for the strategy to work there is a celling limit of how much you can make overall. If you know what you are after you can get it. Using a $5000 demo deposit I was able to make $500 in 20min. Not bad, but I have heard that people can get banned from even online casinos if they play it to safe and constantly win a certain amount. I also found that betting above $25 per bid can wipe you out quickly regardless of the deposit or system. This is due to the set table limits, which would stop the system from working after several spins.

    Yes, the free sides of online casinos are often 'juiced' to encourage players to come over to the pay side.

    Also remember that what kills you when using things like Martingale systems isn't that you are likely to lose, (getting a 95%+ win rate isn't hard), it's that when you do lose you lose big, (with a six double cap, one loss wipes out 63 wins, but you have a 1/47 chance of losing). This is what makes many betting systems so tempting, one thinks that one is betting the individual stacks and getting the full-run odds, when you need to either look at individual bets or the full betting runs.

    I did a spreadsheet workout of what the amount of deposit one would need to survive, it appears you need a deposit of ideally upto the 14 losing spin. Because of this, as mentioned before the house limits the bid value, usually depending how much you bet, it works out to be between the 5-11 losing spin. When you get two zeros spins you know it is time to call it a day before you lose it all. I suppose one could play it very safe and bet using only pennies. But that will take forever to win a decent pot.

    So the moral of the story is, unless you have money to burn and accept that you will lose it all, do not expect to the roulette table to be a source of you income.

    Not even then.

    All increasing your cap does is trade off odds for payoff, (and not even an even trade at that). A 14 bet run means that you are taking odds of 1 to 16383 on something that is 1 to 7989. You have exactly the same problem, it only takes one loss to wipe out everything that you have gained and then some. Remember, there is no reason why the losing run won't be your next one.

    Everything on a roulette table is a sucker bet, if you're going to insist on using a Martingale system, at least go over to the craps table where the odds aren't quite as much of a rip-off.

    Hi Rick, funny story a friend of mine thought he could out smart the casinos and use the above mentioned "system" to win. I watched this guys face go from extreme optimism to complete anguish in five minutes. He deposited £1000 pounds and proclaimed how he is going to make £5000 out it in 30 minutes. In the first minute to made around £2500. Now with this amount of winning in such a short period of time I suggested that he quit now. But nope, he was aiming to hit is high target. The first dozen spins were on his side. But then he started getting the unprobable 0's three times, how unlikely was that, very it seems. He wiped out his account in the next several spins.
    What struck him the most was not that he lost all that money, but that the "system" did not work. Luckly my friend earns a high income so losing that much(for the rest of us) was not a big deal. The guy probably spends that much on a good night out. So my friends experience is shocking proof that online real money games are tweaked from the demo practice games. Where once you could detect some patterns in the demo games, the real games have no patterns.

    The funny part, this friend of mine works in banking, go figure.

    Hi, oh I got lost in the thread! it was so long. Roulette is a fun game and I didn't realize that you could actually use some strategy, geez. I feel like there's information overload after reading this article :) Cool people. :D I guess I'll master poker since everyone's in to it but roulette is way much easier to play. :D

    There’s no system that’ll help you win roulette all the time. You’ll have just as much success or better flipping a coin to decide to bet on Red or Black. I mean, that’s why it’s called Gambling. If there was a system indeed, then roulette would be put out of commission.
    Still, there’s smart betting and then there’s just plain old gut feeling and hoping the gambling gods are kind to you this particular time around.

    No one said anything about winning all of the time - the math curve shows that if you plan accordingly based on the house minimum and maximum and don't get greedy (or stupid, and start believing in 'luck) you will make money.   That's my only goal - I want to pay for my vacation so I use roulette to do it.
    Yup, sorry for saying 'all the time'. What I really meant is that on the long run, no progression systems proved to be reliable. If you want to maximize your odds you always have to pay attention to the maximum bet which puts you in the position to place very small bets meaning small winnings and, sometimes, night-long games. I see this strategy safe only if you set yourself a winning target like few bucks a game and you're off to the next.
    When are you planning your vacation? This millenium? :)
    Just joking Hank, I really wish you lots of luck with the strategy. It's sure the plan works if you're careful not to reach the table's maximum bet.

    I also know a few that can do better than this. It's enough to google and study'em.