A friend of mine once employed the gambler’s fallacy for significant profit. The means by which he did so involve a certain game that used to be played on cruise ships. The game: called Horses (or, alternately, Horse Race) involved a track divided into eight lanes and about thirty squares in each lane. Each person would place a set amount of money on a particular horse and the horses would be advanced by means of one roll of the dice per set time period (often a day). The pool would then be divided between all those who had bet on the horse that reached the end of the board, representing the finish line first. Due to the gambler’s fallacy, very few people chose to bet on the horse that had won the preceding race. By always betting on that horse, my friend derived a statistical advantage great enough to ensure that, over a long time frame, he consistently made rather than lost money. This money was usefully applied to the purchase of alcoholic beverages.

The same statistical trick can be employed by any wily gambler. Just remember the fair bet equation: if the probability of winning (expressed as a fraction of 1) multiplied by the payout to you in case of victory equals a number greater than one, the bet is ‘fair’ meaning that you will make money doing it over and over or, at least, you will not lose money in aggregate. Clearly, with Horses, the fair bet equation only works if the size of the pool divided by the number of people betting on your horse (lower than the others due to the gambler’s fallacy) multiplied by (1/8) is greater than one. By simply making this calculation before each round my friend’s continued supply of spirituous beverages was ensured. The same tactic can be applied to any game with a set probability of winning: rolling dice, flipping coins, etc.

The gambler's fallacy is the incorrect belief that the outcome of any particular event in a series of independent events whose outcome has a fixed probability is influenced by the outcomes of previous events in that series. What happens the next time doesn't depend in any way at all by what happened the last time.

Take coin tosses for example. If we toss a coin spinning into the air and let it fall to rest on a hard surface, that is an event. This event has two possible outcomes: the coins comes to rest head up or tail up (it could conceivably come to rest on its edge, but that is so unlikely that we can ignore the possibility). Because there is nothing to favor one over the other, the two outcomes have the same probability of happening, and that probability is 50% (100% divided by two). In a series of coin tosses, each event is identical, so the probabilities of the outcomes are the same for each event in the series (fixed probability). That is true regardless of the results of the previous tosses. In other words, nothing that has already happened before a particular coin toss has any effect on whether that toss will come out heads or tails. History doesn't affect the future. The gambler's fallacy is our tendency to believe that a string of heads, for example, somehow makes it more likely that the next toss will produce tails. The longer the string of heads, the more we will tend to think that "The next one's gotta come up tails."

The tendency to think this way is general and fairly strong, despite being entirely wrong. It is rooted in a common misunderstanding of probability and statistics, in the way the brain works, and in the belief in luck (which might be considered a kind of folk theory of probability).

Probability theory asks us to believe that for a very long series of coin tosses the number of heads will be nearly equal to the number of tails, and that the closer the number of tosses gets to infinity, the more nearly equal will be the numbers of heads and tails. Most people understand that quite well, and it is rather intuitive even for those who have not studied the theory. The problem arises when we apply this long-term 'balancing out' effect to the short term. It just doesn't work that way. The effect demands a large number of trials (sample events).

Our brain, like all brains, functions basically as a pattern recognizer. The basic value of knowing the world around us is to be able to predict events and thus increase our chances survival. So recognizing patterns in space and time is a crucial ability. It is also a sub-intellectual process, and usually comes into our consciousness automatically or as a 'gut feeling'. The brain is very good at pattern recognition, but the problem with applying pattern recognition to random events of fixed probability is that there is no pattern to be recognized. The sequence of toss outcomes is therefore unpredictable. Seeing a string of ten heads in a row, one might bet that the next will be a head too, because 'it's on a roll'; alternately, one might bet on a tail, because long strings are an abnormal (infrequent) pattern. Both are wrong thinking (or feeling) because the probability of the next toss is always exactly 50/50, not the slightest bit more or less.

Then there's luck. A person's sense of luck, good or bad, is hard to explain, but it seems likely to be connected to those subintellectual or even subconscious feelings that our brain's attempts at pattern recognition leave us with mixed up with hope or some sense of personal deserving. I don't think much in terms of luck myself, not seriously anyway. I do consider myself to rather unlucky when it comes to guessing random outcomes or at 'games of luck' (slot machines, lotteries, etc.) and I consider my wife to be lucky at them. I couldn't try to tell you why, other than it just seems that she tends to win, whereas I tend to lose. The idea of luck seems to be baked into us all. The gambler's fallacy is seen in people's feeling of luck, too. After a string of 'bad luck', many people will develop a strong anticipation of imminent 'good luck' and continue playing, thinking "I can't lose this time. After losing again, that feeling only intensifies, compelling the player to continue. After a string of successes, however, many people will also continue playing, thinking that they are 'on a roll' and can't lose.

We must remember that in games of luck (chance, or random outcomes) where the probability of winning or losing on each particular bet is the same (roulette, slot machines, etc.), it does not matter how many times you've won or lost in the last n number of bets. The probability of winning the next time is exactly the same. (It's important to note that in casino games, that fixed probability of winning is less than 50% for the players. Casinos don't gamble, and their certain profits depend on the large number of bets made 'against' them.) History does not affect the outcome.

Notes:

Blackjack, and most other multiplayer card games, involve skill as well as the luck of the draw, and the cards are not dealt in a fully random process; rather they are dealt randomly from a finite set of possibilities (the boot). When the boot is small enough, like a single deck, players can calculate probabilies for individual cards turning up (events). A player can affect the odds of winning by remembering what cards have been played and making better decisions on that basis, taking into account probabilities and potential payout. The gambler's fallacy does not apply to such games, because the probabilities of the outcomes are not constant over the duration of the game (the events are not independent). It does apply exactly to both roulette and slots, though, even if they are crooked. In the case of a crooked table or machine, the odds are still fixed, it's just that the odds are lower than you believe them to be. Casinos make more than enough money without cheating.

No slot machine has a predictable 'payoff schedule'. That would be stupid for the owner. The payoff is simply the set of probabilities assigned to individual combinations. The machine's payoff probability will always be less than 0.50 and is the same for each play.

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