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.
| Possible outcomes of Martingale up to ten
losing bets |
Number
of losses |
Final
outcome |
Highest
bet |
Total
bet |
Net
outcome |
Probability |
Expected
bet |
Expected
return |
| 0 |
Win |
1 |
1 |
1 |
0.49292929 |
0.49292929 |
0.49292929 |
| 1 |
Win |
2 |
3 |
1 |
0.24995001 |
0.74985002 |
0.24995001 |
| 2 |
Win |
4 |
7 |
1 |
0.12674233 |
0.88719628 |
0.12674233 |
| 3 |
Win |
8 |
15 |
1 |
0.06426732 |
0.96400981 |
0.06426732 |
| 4 |
Win |
16 |
31 |
1 |
0.03258808 |
1.01023035 |
0.03258808 |
| 5 |
Win |
32 |
63 |
1 |
0.01652446 |
1.04104089 |
0.01652446 |
| 6 |
Win |
64 |
127 |
1 |
0.00837907 |
1.06414175 |
0.00837907 |
| 7 |
Win |
128 |
255 |
1 |
0.00424878 |
1.08343900 |
0.00424878 |
| 8 |
Win |
256 |
511 |
1 |
0.00215443 |
1.10091479 |
0.00215443 |
| 9 |
Win |
512 |
1023 |
1 |
0.00109245 |
1.11757574 |
0.00109245 |
| 10 |
Win |
1024 |
2047 |
1 |
0.00055395 |
1.13393379 |
0.00055395 |
| 10 |
Loss |
1024 |
2047 |
-2047 |
0.00056984 |
1.16646467 |
-1.16646467 |
| Total |
1.00000000 |
11.81172639 |
-0.16703451 |
The expected bet is the product of the total 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.
Another experiment
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:
Player 1
- 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
Player 2
- 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
Player 3
- 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. 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.
Also be warned that there are many others out there selling get rich quick
gambling schemes that claim they are not betting systems. These sites usually
throw out lots of fancy physics words like "chaos" and "fractals" but display
no evidence they know what these words mean. I have in the past listed some
such sites above but got angry letters claiming I shouldn't criticize what
I don't understand. Personally I feel that every method claiming an easy way
to beat the casinos is a scam and I don't need to understand whatever the secret
is. However to be totally fair I'll only list betting systems above since those
have been mathematically debunked by computer simulations. If anyone did find
a true easy way to beat the casinos why aren't they getting rich doing it?
Want some more examples? Visit Shameful
Sellers of Crap.
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 craps, roulette, or baccarat under ordinary
United States rules.
- The $2000 fee must be paid up front.
- After the fee is paid I will 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.
Please, Don't Write
The most frequent topics about which people try to debate with me are about betting
systems. I have said everything I have to say about them above. Those wasting
my time trying to argue that the dice or roulette ball has a memory will get
a brush off at best, or, more likely, no reply at all. If you really want to
discuss this topic one on one with me, you will have to pay $200 per hour for
my time, with a retainer fee up front.
| |
|
|
Better known as "The Wizard of Odds", Michael Shackleford
uses math and computer analysis to determine optimal playing strategy for
all casino games. His work can be found on his website, WizardofOdds.com
with player strategies and probabilities on most of the casino games. Michael
lives in Las Vegas and in his spare time likes to college license plates
and gamble.
You can read more about Michael's work at his website, WizardofOdds.com
|
|
