The following information on gambling systems comes from a number of sources including research literature, books on how to gamble, websites selling gambling systems, discussions with gambling counsellors, interviews with problem gamblers (Turner et al., 2002), and personal observations in casinos and other gambling venues. Note that some books on how to gamble promote systems that are based on erroneous beliefs about random events (e.g., Jones, 1994), yet other books on how to gamble (e.g., Ortiz, 1986) give accurate warnings to their readers about the flaws in many systems.

By their very nature, games of chance involve variations in outcome from play to play; streaks occur, and it is possible in the short term for a player to come out ahead. Some gamblers translate this experience of winning into a belief that they can “beat the odds” by the way in which they play or by the system they use. Many of these systems appear to be quite reasonable and are logically derived from intuitively sensible ideas. If we dig deep enough, though, we find that they involve misconceptions about the nature of random events and probability. The following systems are not restricted to games of chance, but are often used in games of skill as well.

### Systems Based on Misconceptions About Independent Events and the Law of Large Numbers

The following systems are based on misconceptions about the independence of events and the law of large numbers:

- betting on numbers that have not come up very often;
- looking for “pregnant” machines that are due to pay out;
- betting repeatedly on the same numbers in a lottery or in roulette; and
- staying at the same machine.

These systems share the premise that the different numbers in a game come up regularly, and thus, if a number has not come up recently, it is somehow “due.” The flaw in all of these systems can be summed up by the phrase “the dice have no memory.” Numbers or machines can neither be due nor overdue to win. People expect random numbers to be consistent, but erratic (see the discussion of the representativeness heuristic in Kahneman and Tversky, 1982; the discussion in the introduction to the book is particularly relevant). If a person is asked to write down the likely pattern of heads and tails in a series of coin tosses, he/she will write a sequence that alternates erratically between heads and tails but it will not usually include long streaks of heads or tails. For example:

HTHHTTHTHHTTTHTHTHTHTHHTHTHHHTHTHHTHTHTHTTTHTHHH

However, in reality random chance is fundamentally uncertain or erratically erratic. The sequence could quite possibly consist entirely of heads.

Suppose a person has a bag containing 4 black marbles and 1 red marble. If the 4 black marbles are drawn out, it would be reasonable to suppose that the next marble drawn would have to be red. If you did not put the marbles back after drawing them, the next marble would definitely be red. But if you replaced the marbles after each draw, would a red marble still be due to occur? On the face of it, drawing a red marble seems reasonable because, in the long term, roughly 1 in 5 draws will be red. However, if each marble were placed back into the bag after every draw and the bag shaken, each draw would be independent. There would be no relationship between one draw and the next. Theoretically the same black ball could be drawn every time. The red ball might never come up.

According to the law of large numbers, when independently random events occur, as a sample increases, the relative occurrence of events will gradually come to reflect the true probabilities of those events. If we recorded the occurrence of black and red marbles after drawing marbles from the bag (and replacing them) several hundred times, the occurrence of red marbles would be close to 1 in 5. Many people mistakenly derive from this fact the notion that the numbers somehow correct themselves. That is, after 4 black marbles, the next should be red. As stated earlier, this would be true if you did not put the black marbles back in the bag. However, the play of slots, lotteries, roulette and dice games all mimic systems in which the numbers are drawn from an infinite population of “marbles,” the equivalent of putting the marbles back into the bag after each draw.

If the marbles were not put back, then the population of marbles would change with each draw and the draws would not be independent (random without replacement). Most gambling involves independent draws of random numbers. The exceptions are some card games in which the cards are drawn without replacement from a limited population (52 cards for a single deck; 312 cards for 6 decks). As a result, the probabilities of a particular card appearing shift each time a card is drawn. If after 10 cards have been drawn 2 kings have appeared but no ace, the relative chance of an ace being drawn increases from 4/52 (7.7%) to 4/42 (9.5%), whereas the probability of a king being drawn decreases from 4/52 (7.7%) to 2/42 (4.8%). However, even after such cards have been drawn, an ace is still not “due” to come out of the deck. Random draws without replacement affect the relative probability of an ace compared to a king, but one still cannot tell if a king or an ace will be drawn as the next card.

Beliefs that staying at the same machine, betting on the same number or betting on numbers that are due will increase a player’s odds appear to assume that the population of possible outcomes is changing with each bet. This notion is not entirely irrational, but is a misapplication of one notion of random chance (random without replacement) to a situation in which it does not apply (random with replacement). Perhaps this misapplication is a result of the early experience people have with random chance through card games. Systems such as card counting, which work with games like blackjack, offer nothing to people playing games of pure chance. Unfortunately this fact does not stop people from writing books or even computer software to track lottery numbers to predict which numbers are due to win (see Turner, Fritz, & Mackenzie, 2003).

During a conversation between the first author and a regular gambler, the gambler emphatically stated that, if in the long term the odds of a coin toss coming up heads equalled 50/50, after a streak of 20 heads in a row there must be a bias to get the final total to equal 50/50, even if it is only a small bias. The truth is that no bias is needed. After a million tosses, there might still be 1,000 more heads than tails, but the ratio of heads to tails would still approximate 50/50. Short-term deviations from the true odds are not corrected; they are washed out as more events occur. The average occurrence of an event will rarely exactly equal the true odds, but will be correct to several decimal places after a few thousand events have occurred.

Suppose a series of coin tosses starts with 10 heads in a row (100% heads), and then another 90 flips are added with 46 heads and 44 tails. After 100 flips the total number of heads would be 56 (56%). The average has regressed towards the true mean of 50% (from 100% to 56%), even though the subsequent 90 flips showed no bias towards tails. In fact, this “correction” occurred even though in the subsequent 90 flips there were slightly more heads than tails.

If something is random with replacement, the only thing that a 1-in-5 chance tells the player about the next event is that it is a 1-in-5 chance. No matter what has happened before (e.g., 10 wins in a row or 1,000 losses in a row). If the chance of winning is 1 in 5, then the probability of a win on the next draw is still 1 in 5.

### Systems Based on the Misconception that Random Patterns Are Reliable Predictors

Systems based on the belief that patterns that appear in random events can help predict future patterns include the following:

• the search for “hot” or lucky tables or lucky seats;

• selecting “hot” or lucky numbers;

• the belief in lucky players;

• betting on numbers that come up a lot; and

• the search for biases, patterns or sequences.

These systems are the opposite of systems based on the belief that infrequent numbers are due to come up, but the same players often endorse both types of systems. The premise of the systems under discussion here is that there is a reason that a particular number has been coming up frequently. For example, the number might be lucky or the lottery balls might be biased toward a particular number.

As stated above, many people do not understand the independence of random events, instead they expect random events to always occur according to the true odds, that is, to be consistently erratic and self-correcting. When an outcome occurs more often than expected, people infer that there is a bias towards it or that the particular outcome is a lucky one. As one player with a gambling problem interviewed by the authors said, lottery numbers are “random, but not truly random.” He based his belief on patterns he saw in the winning numbers drawn. In essence, we invent beliefs about luck, biases or hot numbers because random numbers sometimes appear to be predictable. Many people are unwilling to accept that random events can fail to be consistently erratic.

A belief in biases may be the result of observing that random chance does *not* correct itself. Unfortunately this accurate observation can lead people, not to reject the belief that random results somehow self-correct, but instead to adopt a second, parallel theory: the bias or luck theory of chance. In essence, gamblers who have applied an incorrect model to predict random events and found that it does not always fit their experience, turn to luck or bias as an explanation for deviations from their model. The idea of luck is needed only because they started out with a faulty model of random chance.

#### Looking for Patterns and Sequences

Gamblers tend to search for complex patterns in lottery, slot machine and VLT game results. For example, a person might search for numbers that “predict” the next winning number. One individual struggling with a gambling problem told us how, after tracking the *Pick 3* lottery numbers for a three-year period, found that whenever a 5 was drawn, on the next draw, either a 0 or a 9 would be drawn (this is not exactly his system, but describes his method for determining what bet to place). With this information, he believed he was able to narrow down the number of possible tickets to help him select the winner. This strategy is logical and evidenced based, but will nonetheless simply not work.

#### Looking for a Bias

The search for a bias involves doing “historical research” into previous outcomes of a specific game on the assumption that patterns indicate that there is a defect in the device (e.g., a tilt of the roulette wheel, a repeated pattern in a computerized random number generator). Casinos encourage players to look for patterns by posting past numbers and providing players with note pads and pencils at the roulette table to keep track of the numbers. In contrast, casinos do not allow the use of note pads at the blackjack table where keeping track of past numbers might actually be an effective strategy. The popularity of bias systems may be due to the success of blackjack card-counting systems, where diminishing stacks of cards actually produce opportunities for bettors to improve their chances.

In some games—roulette, for example—it is theoretically possible to find a game with an actual bias (see Barnhart, 1992; Bass, 1985). A wheel might have a slight warp or not be correctly levelled. According to Barnhart (1992), many teams of wheel trackers have successfully profited from biased wheels. The most successful attempts, however, were criminal scams that involved tampering with the wheel. One European group paid a factory worker to insert the wrong size screws in the frets that divide the numbers, making some frets slightly loose (Barnhart, 1992). In the early 1980's, a group of engineering and physics students at University of California at Santa Cruz attempted to beat the wheel by using a concealed computer device to predict the outcome of the wheel (Bass, 1985). Their scheme ultimately failed due to the practical limitations of using a concealed computer in a casino. Another variation on this theme is to track how the croupier throws the ball to determine his/her “signature” (Bass, 1985; see also Barnhart, 1992). To counter this possibility, some casinos require that their employees not look at the wheel when they throw the ball.

Most bias tracking systems fail to take into account the complex and chaotic effects of initial uncertainty, and the extent to which random numbers often mimic patterns by pure chance. Such systems rarely consider the possibility that an apparent bias is in reality simply the result of random chance. Detecting bias may be possible in theory; in practice it is extremely difficult, particularly if the player has to do it undetected and without the aid of a computer (see Ortiz, 1986). In addition, in today’s casinos electronic sensors record every roll of the wheel, so that the casino would most likely detect an apparent bias (real or random), from either the wheel or an employee, before a player could have a chance to profit from it. Thus bias systems are really a thing of the past (Ortiz, 1986). Bias systems are also used in lotteries, horse races, blackjack and other types of gambling (see Turner, Fritz, & Mackenzie, 2003, for examples).

### Jamming the Machine

Some players believe that they can win by jamming the buttons of a VLT or slot machine so that it plays continuously. In the past this was done by using a toothpick to hold down the spin button. Some manufacturers have redesigned their machines to prevent jamming. In Britain, however, some machines now have an auto-play button for the gambler’s convenience (Griffiths, personal communication). In some cases, jamming the machine may be based on a misunderstanding of the meaning of the payback percentage or on the belief that the machine is due to pay out. In other cases, it may be an attempt to “crack the code” of the random number generator by observing the results and deciphering patterns. Technically speaking, jamming the machine reduces the randomness of the play by eliminating the variation in the delay between spins; however, the sequence of random numbers generated may be longer than 4 billion numbers, so it would cost a fortune to crack the code (see Turner & Horbay, 2004).

### Incremental Betting Strategies

Some common incremental betting strategies are the following:

• increasing bets after a loss;

• decreasing bets after a loss; and

• increasing bets after a win.

Understanding incremental betting strategies is particularly important because, unlike many other types of systems, they do increase the chance of winning—in the short term. The flaw in these strategies is the belief that it is possible to know when to quit. When incremental systems fail—which, given enough time, they invariably will—the gambler is left with a very large loss. Although they have no effect a gambler’s long-term success, these systems manipulate random numbers in a manner that leads to a strong illusion of skill (Turner & Horbay, 2003).

#### Martingale, or “Doubling Up”

Perhaps the most common, and also potentially most disastrous, of the incremental betting strategies, the Martingale system involves doubling the size of the bet after a loss. This is most often used with even money bets (bet $1, win $1) such as blackjack, reds vs. blacks on a roulette wheel, or pass-line and come bets in craps. Turner (1998) has shown that the outcome of this system is often positive; that is, most of the time a player wins soon enough to make back their losses. When this does not occur, however, the results can be devastating. One gambler described how he amassed $90,000 by doubling after each loss. Then he hit a losing streak and lost it all, plus another $250,000. Turner (1998) also found that, if the belief that random numbers correct themselves were true, this system would be very successful.

*Table 3. Sample betting sequence (Martingale)*

Bet |
10 |
20 |
10 |
10 |
20 |
40 |
80 |
160 |
320 |
640 |
1,280 |
10 |
10 |

Outcome |
lose |
win |
win |
lose |
lose |
lose |
lose |
lose |
lose |
lose |
win |
win |
win |

Profit |
–10 |
10 |
20 |
10 |
–10 |
–50 |
–130 |
–290 |
–610 |
–1,250 |
30 |
40 |
50 |

In the series shown in Table 3, the player wins 5 times and loses 9 times. In spite of losing more often than he wins, at the end of the session he is up $50. Theoretically, if a person had an unlimited bankroll and there was no limit to the size of bet, he/she could make money with this system. In practice this seldom happens because (1) most casinos have upper and lower limits on bet sizes, and (2) the gambler is likely to run out of money and thus will not be able to continue betting.

As a rule, a player will use this system until they hit one of those “impossible” long losing streaks and then will then get burnt badly. The danger of this system is that it often does work in the short term, so players quickly become convinced that it will always work. In therapy sessions, it is a good idea to find out if a client is using this system because the system is rewarded so often that a client might never truly understand its fundamental flaw, even after losing $100,000. One gambler that the authors interviewed was so utterly shocked when his system stopped working that he was grasping at all kinds of alternative explanations for what had happened to him—not realizing that the explanation was in the system itself.

#### D’Alembert

The d’Alembert system is a milder or more conservative version of the Martingale system. Instead of doubling after a loss, the player only increases his bet by a single unit. Table 4 illustrates the betting sequences that might be followed by a player following a d’Alembert system.

*Table 4. Sample betting sequence (d’Alembert)*

Bet |
10 |
20 |
10 |
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
70 |
60 |
70 |

Outcome |
lose |
win |
win |
lose |
lose |
lose |
lose |
lose |
lose |
lose |
win |
win |
lose |
win |

Profit |
–10 |
10 |
20 |
10 |
–10 |
–40 |
–80 |
–130 |
–190 |
–260 |
–180 |
–110 |
–170 |
–100 |

With this system the gambler is less likely to reach the house’s upper limit and is also less likely to go bankrupt. However, the gambler is also less likely to be able to recoup his losses after a moderately long losing streak. The flaw with this system is that when the player encounters a particularly long losing streak (as is likely if play continues long enough), the player will lose more than he/she will win (as shown in Table 4). The d’Alembert system does not have the extreme wins or losses of the Martingale system, but in the long run the player is still most likely to lose money.

#### Pushing a Winning Streak

Gamblers who follow a strategy of “pushing” their wins increase their bets after a win rather than after a loss. On the surface, this is less harmful than the Martingale and d’Alembert systems, but ultimately leads to giving back all that is won. Attempts to prevent this by “locking” in a win by reducing the bet after a series of 2 or 3 wins (Patrick, 1986) are in the end also doomed to fail. “Locking in wins” leads to more conservative play, but it also produces a strong illusion of success, so it is unclear if such a system is in the end helpful or harmful.

#### Other Systems

There are a large number of variations on these basic systems. One, the cancellation system, involves starting with a series of numbers, using these as the basis of bet sizes, calculating new increased bets based on those that do not win, and then crossing out those bets sizes that do win. These systems can be remarkably complex and in the end a complete waste of time. Other variations involve hedging one’s bets by placing an additional bet on some other option. For example, one gambler described his plan to run the d’Alembert system, placing “banker” bets in a game of baccarat but also hedging his bet by betting a smaller amount on the “player” hand (see description of baccarat). The idea is that if he loses on one side, he makes up for the loss on the other side. This strategy does increase the frequency of winning in the same way as buying multiple lottery tickets does, but in the long run it leads to greater losses. The flaw with all these systems is that, in the long term, it is simply not possible to beat the house edge by increasing bets.

### Beliefs and Attitudes

In addition to following systems, gamblers engage in many other activities they think will help them win. These include concentrating on winning, keeping a positive attitude, being aware of their gut feeling, and looking for lucky numbers, places or things. Some gamblers regard these behaviours as part of the skill of gambling. Some gamblers even consider having or finding luck to be a skill. In one sense, these mental activities are skills. If you were working on your career, maintaining a positive attitude and putting in a lot of mental effort would be very helpful. Mental attitude may indeed assist in games such as poker where skill does play a role.

Gut feelings may, in fact, be based on past experience. That is, something feels right because it somehow reminds us of previous circumstances (see Reber, 1993). When the past experience is informative, gut feelings can be accurate guides to behaviour (e.g., a police detective might have a gut feeling that a particular suspect is lying), but in gambling, gut feelings are irrelevant because the past is irrelevant to a specific bet or play. The ability to pick out what works and stick to it is a useful skill. When events are random, however, past success has no bearing on future success.

It is theorized that superstitions are often the result of chance occurrences that reinforce the superstitious belief (see Skinner, 1953, for additional comments). On one occasion, the first author was at a slot machine and just about to press the spin button when a friend distracted him for a couple of seconds. When he looked back at the machine he had won $5. After that he started hoping to be distracted again. Call it luck, skill, a system or a gut feeling, it will not help the player win. The simple truth is that in the long term nothing can help someone beat a game of pure chance.