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.
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, heads and tails. 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 a tail. 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 quite wrong. It is rooted in a common misunderstanding of probability and statistics, the way the brain works, and 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 number 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.
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 50/50.
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. 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 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.
Remember that in games of luck where the probability of winning or losing on each 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.) History does not affect the outcome.
To avoid possible confusion, please note that the other articles that may remain in this node were both written before I wrote this, so their references do not point to what I have written here. When eraser_ writes "... like you said ..." he must be referring to some other WU that has since disappeared. When s_alanet writes that " ... stupid gamblers believe this fallacy ..." he is not referring to what you read here, but to a different and possibly incorrect description of the gambler's fallacy that is no longer in this node.
Some will be tempted to point out that Blackjack, and most other 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). A player can affect the odds of winning by playing better and by remembering what cards have been played. The gambler's fallacy does not apply to such games, because the probabilities of the outcomes are not fixed. 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.
The comments by s_alanet to not really speak to the gambler's fallacy. They are also largely incorrect in what they do say. True professional gamblers do not play casino games, which are designed to give an 'edge' (higher probability of winning on each play) to the house. Pros play games like poker, where the only edge comes from the skill of individual players. Also, no slot machine has a predictable 'payoff schedule'. That would be stupid for the owner. The payoff schedule is simply the set of probabilities assigned to individual combinations, and thus to the machine as a whole.