Does the doubling system in roulette really work?
The doubling system is sometimes referred to as the Martingale doubling system or the roulette doubling system. The Martingale system has some history and is a concept or idea in mathematics which can be applied as a method of gambling in roulette but the subject here is specifically "DOUBLING" in any 50/50 gambling situation.
For those of you who do not know of the doubling system it is pretty simple to grasp. The idea is that you are betting on a 50/50 bet like the toss of a coin. So each time you bet you stand to either win or lose with equal probability. And the bet is 1 to 1. I.e. you pay £1 and if you lose you get nothing and if you win you get £2. It's like you and the other player both pay £1 each. Whoever wins gets the £2.
So even odds and even money.
The simple rule is that if you won last time you bet £1 and if you lost last time you double the stakes.
The idea of the doubling system is that you have a basic stake of £1 and you play a round. If you win that's fine. If you lose you double your bet on the next round. If you lose again you double your bet again. You carry on doubling your stake until you win at which point you start with a £1 stake again.
If you pay £1 and lose it then by betting twice as much the next time you stand to win your stake back and your previous lost £1 plus £1. If you lost on the second bet your net losses are £1 + £2 = £3 so you bet £4. If you win you recoup your losing streak plus £1. If you lost three times in a row before a win you have lost £1 + £2 + £4 = £7 so you bet £8 and if you win you get your £8 stake back and your winnings of £8 which pays for the previous three bets totalling £7 plus £1 profit.
Given that the odds of winning are 50/50 then over a long time you would expect approximately equal wins and losses. On average you will win half your stake every round. Over a long period this is exactly what happens.
This logic is very sound. It works!
Well it works in theory. But there are a number of reasons why it is impractical. Some people claim that it doesn't even work in theory but they are wrong. It does. But in practise you have to have a fairly big bank account to cover the longer sequences of losses that are bound to occur.
It is tempting to think that to see a coin turn up heads four times is pretty rare and that would need £16 to cover it. Hmm. That sounds okay. But what if it turns up heads 8 times or ten times in a row. Well 8 times would need £256 and the even more unlikely event of ten losses would require £1024 to cover it. This is getting worrying. But surely the chances are so unlikely that the result would be the same ten times in a row that it is still worth giving it a go. Well there seems to be some misconception by humans as to what "unlikely" means. Most humans put it a little too close to the concept of impossible. If something is "impossible" then it will NOT occur. If it is "unlikely" then it "WILL" occur! If you consider that the chances of getting 10 heads in a row are 1 in 1024 then if you toss a coin 10240 times you would expect to see that sequence about ten times. It is about as "LIKELY" to occur once every 1024 times as it is "UNLIKELY" to occur when you try it once.
Here is a detailed examination of how the doubling system works.
For the purpose of looking at doubling in detail we have had a piece of software written to simulate series of games and to generate graphs of the results. You can download the software from the DOUBLING SIMULATOR page and use it along side this explanation as a good illustration of what happens in betting sequences. The images are taken from the software but are reduced in size to save space on this page and the server and because it is the "impression" rather than the "detail" that is important. You will get a far better impression by running the software yourself and seeing the figures involved and the regularity of certain patterns.
Here are three graphs illustrating a sequence of 100 bets of £1 on 50/50 odds.
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| Graph 1 | Graph 2 | Graph 3 |
Not very surprising. In the first illustration the gambler never went below zero and won £23. In the second case the account dipped to -9 and ended on -3 and in the last one the account dipped by 20 but ended on -12. What we expect with continuous single unit betting on a 50/50 bet is an average of 0. What happens if we run the simulator for 10,000 games? ...
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| Graph 4 | Graph 5 | Graph 6 |
This is perhaps a little surprising. The fluctuations are still quite dynamic over 10,000 games. The first wins about £125, the second loses £140 and the third wins about £25. So over 10,000 bets the winnings and losses are quite moderate and stay in line with an overall expectation of averaging around zero. What is perhaps a little more surprising is what happens if we run an average of 10,000 accounts over 10,000 games. Effectively running 100,000,000 (one hundred million) games...
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| Graph 7 | Graph 8 | Graph 9 |
What is surprising is that the quality of the graphs remain the same. The same proportional fluctuations. The range has increased probably by a multiple of about 1000 but the nature of the graph remains consistent. They are fractal!
Now things get a bit more interesting.
In roulette the roulette wheel always has that little pesky green zero sitting quietly out of the way on the wheel. In American roulette wheels they have two extra green "zero" slots - the zero and the double zero. As most people will realise this tips the odds a little in the casinos favour. But how much? Well strictly mathematically the single zero changes the odds from 18/36 (50/50) to 18/37 that you will win , say, a red bet. So 19 times out of 36 you will lose. It is still close to 50/50 and here are three illustrations of the effect the one zero has on single unit gambling over 100 games...
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| Graph 10 | Graph 11 | Graph 12 |
Not very different from the 50/50 graphs above. That is because they are close to 50/50 bets. But over a long sequence it can be seen very clearly what is happening. Below are graphs of 100 games with 100 accounts, 1000 games with 1000 accounts and 10,000 games with 10,000 accounts in that order...
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| Graph 13 | Graph 14 | Graph 15 |
It soon becomes clear that the net effect is very securely in the casinos favour.
Now to examine what happens when we use the doubling technique. Here are the three examples of 100, 1000, & 10,000 game sequences using the doubling system with 50/50 odds...
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| Graph 16 | Graph 17 | Graph 18 |
So long as you can always place a bet you are guaranteed to win an average of half your stake per game. It is very consistent and very satisfying. What is perhaps more interesting is what happens when the casino puts the zero in the wheel to upset the odds. Well the graphs below show the results of 10,000 accounts running 10,000 games for no zero (50/50), one zero (18/37) and two zeros (18/38) games respectively...
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| Graph 19 | Graph 20 | Graph 21 |
The pattern is very nearly the same. The subtle difference being that as the odds go down from 50/50 to 18/37 and then 18/38 the net winnings are reducing by the same proportions and the number of big losing streaks are increasing slightly.
For the first graph the winnings are 49,921,243 which is about 50,000,000. That is 50 million pounds won in 100 million games or 50% of the stakes played. For the second graph 48,249,686 is very close to 18/37ths of one hundred million and the third graph is 46,655,564 which is very close to 18/38ths of one hundred million. So this doubling method really does work.
There are, however several other issues to consider in the real world. The first thing to consider is your bank balance. You can only make a bet for a certain sum of money if you can secure that amount of money. And money is hard to come by. With this software I have frequently seen sequences of 30 losses costing the gambler 1,073,741,824. That's over a billion pounds and well outside the realistic range for most of us. (Bankers respectfully excluded.)
Not only does the real world place a limit on your ceiling so to speak but just to ensure miracles don't happen the casinos impose limits on the stakes allowed. In most casinos for normal people these limits are relatively low. Here is the effect on the gambling with 50, 500 and 1,000,000 limits imposed respectively...
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| Graph 22 | Graph 23 | Graph 24 |
The limit has re-introduced the 50/50 random element. There is a reason for this in the mathematics and I am not about to prove it here but in essence the frequency of losing streaks that are hit where the stake limit is exceeded will be proportional to the amount of wins. I don't know if the long term probability is still that you win. I expect it will match exactly 50/50. But as soon as you introduce the zero to offset the odds look what happens to the same three game set ups as above...
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| Graph 25 | Graph 26 | Graph 27 |
As most roulette players realise the little green zero is there to set the odds in the houses favour. But what I find interesting is how solidly it does it. Playing with the simulation software and seeing the reliability of the suction of money out of the gamblers pockets is surprising... and not surprising at the same time.
Overall the conclusion is that the doubling method does not work in practice because of the limited resources available to you either because of your bank balance or because of the stake limit imposed by the casino. What is interesting is that it does overcome the negative effect of the zeros added in to the game to lend advantage to the house when there is no limit to the stake.
If you have any questions or comments please contact me.
Additional Info:
Sam Baker has written an interesting page on this subject called Simulating Roulette with a Doubling System where he has developed a some JavaScript to illustrate gambling sequences on the page.
The Wizard of Odds is a reputable qualified actuary who revels in the analysis of gambling. You can find much more in depth analysis of the Martingale system on his Betting Systems page and specifically on his Ask the Wizard - The Martingale Betting System page. This site is seriously worth bookmarking.
VagasClick.com is an interesting site all about gambling and they have a good article on the Martingale Betting System which is suggestive that it can be a winning strategy.
There is a page on Wikipedia about the Martingale System with some rather complicated looking maths too.
Ian Sharpe has written an article called Martingale madness: a gambling system tested and supplied some JavaScript simulation of Martingale sequences in a roulette.
James Woodger has developed a neat roulette simulator which demonstrates both the Martingale and the Reverse Labouchère methods.
Also: For nutty ways to select your lottery numbers see Zyra's Novel and Amusing Ways of Selecting Your LOTTERY NUMBERS
