One of the biggest myths about gambling is that games of luck can be overcome with methods such as varying bet sizes to manipulate the odds. Depending on how the specific system works, in the long run, you will either sacrifice a few large losses for many smaller wins, or many small losses for a few larger wins. In the long run you can only lose at any game of pure luck. The more you play, regardless of what form of betting system you use, the ratio of money lost to money bet will always approach the same constant. This is known in mathematics as the Law of Large Numbers. Betting systems are just mathematical voodoo that will never survive the test of time. If you don’t believe me here is what the Encyclopedia Britannica declares under the subject of gambling:
“A common gamblers’ fallacy called ‘the doctrine of the maturity of the chances’ (or ‘Monte Carlo fallacy’) falsely assumes that each play in a game of chance is not independent of the others and that a series of outcomes of one sort should be balanced in the short run by other possibilities. A number of ‘systems’ have been invented by gamblers based largely on this fallacy; casino operators are happy to encourage the use of such systems and to exploit any gambler’s neglect of the strict rules of probability and independent plays.”
Probability and Measure (second edition) by Patrick Billingsley provides specific proof that betting systems do not win in the long run. On page 94 he states that, “…There are schemes that go beyond selection systems and tell the gambler not only whether to bet but how much. Gamblers frequently contrive or adopt such schemes in the confident expectation that they can, by pure force of arithmetic, counter the most adverse workings of chance…” After proving a detailed explanation regarding the proof, he concludes (page 95), “…Thus no betting system can convert a subfair game into a profitable enterprise…”
The Martingale (double after every loss) system
Every week I receive two or three e-mails asking me about the betting system by which a player doubles his/her bet after a loss. This system is generally played with an even money game such as the red/black bet in roulette or the pass/don’t pass in craps and is known as the Martingale. The reasoning behind those who believe it is that by doubling your bet after a loss, you will always win enough to cover all past losses plus one unit. For example if a player starts at $1 and loses four bets in a row, winning on the fifth, he will have lost $1+$2+$4+$8 = $15 on the four losing bets and won $16 on the fifth bet. The losses were covered and he had a profit of $1. The fallacy is the promise of guaranteed winnings by using the Martingale. Occasionally the player will lose several bets in a row and will reach a point where he doesn’t have enough money to double.
In order to prove this point, I created a program that simulated two systems, the Martingale and flat betting, and applied each by betting on the pass line in craps (which has a 49.29% of winning). The Martingale bettor would always start with a $1 bet and start the session with $255 which is enough to cover 8 losses in a row. The flat bettor would bet $1 every time. The Martingale player would play for 100 bets, or until he couldn’t cover the amount of a bet.In that case he would stop playing and leave with the money he had left. In the event his 100th bet was a loss, he would keep betting until he either won a bet or couldn’t cover the next bet. The person flat betting would play 100 bets every time. I repeated this experiment for 1,000,000 sessions for both systems and tabulated the results. The graph below shows the results:
As you can see the flat bettor has a bell curve with a peak at a loss of $1, and never strays very far from that peak. Usually the Martingale bettor would show a profit represented by the bell curve on the far right, peaking at $51, however on the far left we see those times when he couldn’t cover a bet and walked away with a substantial loss. That happened for 19.65% of the sessions. Many believers in the Martingale mistakenly believe that the many wins will more than cover the few loses.
In this experiment the average session loss for the flat bettor was $1.12, but was $4.20 for the Martingale bettor. In both cases the ratio of money lost to money won was very close to 7/495, which is the house edge on the pass line bet in craps. This is not coincidental. No matter what system is used in the long run this ratio will always approach the house edge. To prove this point consider the Martingale player on the pass line in craps who only desires to win $1, starts with a bet of $1, and has a bankroll of $2047 to cover as many as 10 consecutive losses. The table below shows all possible outcomes with each probability, expected bet, and return.
The expected bet is the product of the total Pragmatic bet and the probability. Likewise, the expected return is the product of the total return and the probability. The last row shows this Martingale bettor to have had an average total bet of 11.81172639 and an average loss of 0.16703451. Dividing the average loss by the average bet yields .01414141. We now divide 7 by 495 ( the house edge on the pass line) and we again get 0.01414141! This shows that the Martingale is neither better nor worse than flat betting when measured by the ratio of expected loss to expected bet. All betting systems are equal to flat betting when compared this way, as they should be. In other words, all betting systems are equally worthless.
Here is another experiment I conducted earlier which proves the same thing as the experiment above. This one is played against roulette teasting three different systems. Player 1 flat bet a $1 each time. He was not using a betting system. Player 2 started a series of trials with a bet of $1 and increased his wager by $1 after every winning bet. A lost bet would constitute the end of a series and the next bet would be $1. Player 3 also started a series of bets with a bet of $1 but used a doubling strategy in that after a losing bet of $x he would bet $2x (the Martingale). A winning bet would constitute the end of a series and the next bet would be $1. To make it realistic I put a maximum bet on player 3 of $200.Below are the results of that experiment:
Total amount wagered = $1,000,000,000
Average wager = $1.00
Total loss = $52,667,912
Expected loss = $52,631,579
Ratio of loss to money wagered = .052668
Total amount wagered = $1,899,943,349
Average wager = $1.90
Total loss = $100,056,549
Expected loss = $99,997,018
Ratio of loss to money wagered = .052663
Total amount wagered = $5,744,751,450
Average wager = $5.74
Total loss = $302,679,372
Expected loss = $302,355,340
Ratio of loss to money wagered = .052688
As you can see the ratio of money lost to money wagered is always close to the normal house advantage of 1/19 =~ .052632 . In conclusion varying of bet size depending on recent past wins or losses makes no difference in outcome in the long run than always betting the same.
A third experiment
An Old Timer’s Guide to Beating the Craps Table is a betting system that makes big promises about turning the craps tables into your own personal cash register. I offered to test his system for free. The results are posted here.
The cancellation betting system
Despite all my warnings about betting systems, readers continually ask me to suggest one. To satisfy those who enjoy playing systems I have done a full explanation and analysis of the cancellation betting system.
Don’t waste your money
The Internet is full of people with betting systems for sale that usually make promises of beating the casino at games of luck. Those who sell these systems are the present day equivalent of the 19th century snake oil salesman. Under no circumstances should you waste one penny on any gambling system. Every time one has been put to a computer simulation it failed and showed the same ratio of losses to money bet as flat betting. If you ask a system salesman about this you likely will get a reply such as, in real life nobody plays millions of trials in the casino. You’re likely to also hear that his/her system works in real life, but not when used against a computer simulation. It is interesting that professionals use computers to model real life problems in just about every field of study, yet when it comes to betting systems computer analysis becomes “…worthless and unreliable…” as the salesman of one system put it.
A more basic argument against systems is that if they worked so well why are the salesmen selling them in the first place? If I had a system with just a 1% edge, I could easily turn $1,000 into a $1,000,000 within a year, and keep going. I certainly wouldn’t want any competition to create undo suspicion. If you ask them about this, the answer will usually be that they are trying to transfer money from the evil casinos to give to nice simple folks as yourselves. Gee, how nice of them!
Gambling systems have been around for as long as gambling has. No system has ever been proven to work. From an inside source I know that system salesmen go from selling one kind of system to another. It is a dirty business by which they steal ideas from each other, and are always attempting to rehash old systems as something new.
Here are some examples of system salesmen who try to take advantage of the mathematically challenged. Please don’t write me with requests for additions to the list. There are hundreds of sites like these on the Internet, and this list is just a sampling. Frequently these sites vanish in the middle of the night, or suddenly direct traffic to a porn site. Please do let me know if any of these links don’t work or take you to other than the intended place.
The Wizard of Odds Challenge
Despite all the evidence I have supplied I continue to get challenges via e-mail from people who believe their individual systems are the exception to the rule and really work. These challenges usually come with a request that I waste my time writing a free simulation program to prove or disprove the author’s system. Contrary to what some may think, I do not sit around all day with nothing to do but test betting systems. So in an effort to give the true believer a fair chance to prove me wrong, I will bet your $2,000 against my $20,000 that your system won’t win a negative expectation game over a billion trials. Here are the specific rules: The system must be clearly explained. It should not have vague run on sentences, but explain exactly how much to bet at each and every step, and under exactly what conditions. There can be no guesswork or human judgment involved. I must be able to code it into a computer.
The system must have a betting range of no more than 1 to 1028 units. So a bet of at least 1 unit must be placed on every trial and the maximum bet must be 1028 units.
The system must be played on a game based entirely on luck like roulette or craps. Card games are not out of the question, but my test will assume a freshly shuffled deck for every trial. The system must also be based on a game with a negative expectation, again, such as craps or roulette.
This must be an honest challenge. Please do not bother me if you want to exploit unusual rules, like a cruise ship crossing from international to US waters, or something meant to exploit bias in random number generators.
The $2000 fee must be paid up front.
I will then program the system in C++. A random simulation will then be performed of one billion bets resolved. If the player shows a profit at the end you win, if the player shows a loss I win. Should you win, I’ll state publicly, on my main page, that your system defies the laws of probability and that I was wrong.
The results of my simulation are binding. If you feel I did not run an honest simulation, I will provide the source code and you can have your own experts challenge it.