This fallacy occurs when someone believes that the statistical likelihood of an independent event is directly related to what has happened in the past. For example, in flipping a coin, the odds of an individual flip coming up heads is 50/50, but when we have a 10 flips in a row, many will believe that the odds of all of them being the same is lower than any other series of results when it isn't. In this fallacy, the person thinks that the universe will somehow "balance out" and thus often choose the opposite of they've seen.
Gambler’s fantasy is the opposite of the “hot-hand fallacy,” a fallacy that causes people to predict the same outcome of the last event (as opposed to the opposite outcome ). Both fallacies, however, tend to occur in the same people as they both fall victim to the representativeness heuristic (making judgments about the probability of events under uncertainty).
This fallacy is arguably the biggest factor in the success of casinos and lotteries. Research has suggested that those with an internal locus of control, and thus think that their skill outweighs chance, are more susceptible to this fallacy. Additionally, neurological studies have shown that this fallacy is related to sections of the brain that control goal-directed processes rather than those that control decision-making. A key in avoiding this fallacy is to treat each event as if it is the beginning of a new series and not part of a previous one.
There are caveats to this fallacy, however. If the likelihood of an event is not independent, then studying the pattern of results might give a better prediction of what will happen (think unbalanced roulette wheels or loaded dice). Even here, the past results don’t affect future results; they just provide information about the type of results that might be produced. Additionally, if you know that something is going to happen within a particular frame of reference, then the closer it gets to the end of the frame, the more likely it is to occur. This is why card counting systems work; if an ace is drawn, the likelihood that the next card will be an ace has decreased (assuming we are not replacing the cards as they are drawn).
Examples:
Red had come up six times in a row on the roulette wheel, so Greg knew that it was close to certain that black would be next up. Suffering an economic form of natural selection with this thinking, he soon lost all of his savings.
A joke told among mathematicians demonstrates the nature of the fallacy. When flying on an aircraft, a man decides to always bring a bomb with him. "The chances of an aircraft having a bomb on it are very small," he reasons, "and certainly the chances of having two are almost none!"
A similar example is in the book "The World According to Garp" when the hero Garp decides to buy a house a moment after a small plane crashes into it, reasoning that the chances of another plane hitting the house have just dropped to zero.
The most famous example of the gambler’s fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence, although no more nor less common than any of the other 67,108,863 sequences of 26 red or black. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an "imbalance" in the randomness of the wheel, and that it had to be followed by a long streak of red.
For example, people believe that an imaginary sequence of die rolls is more than three times as long when a set of three 6's is observed as opposed to when there are only two 6's. This effect can be observed in isolated instances, or even sequentially
When a teenager becomes pregnant after having unprotected sex, people assume that she has been engaging in unprotected sex for longer than someone who has been engaging in unprotected sex and is not pregnant.
A study by Huber, Kirchler, and Stockl (2010) examined how the hot hand and the gambler's fallacy are exhibited in the financial market. The researchers gave their participants a choice: they could either bet on the outcome of a series of coin tosses, use an "expert" opinion to sway their decision, or choose a risk-free alternative instead for a smaller financial reward. Participants turned to the "expert" opinion to make their decision 24% of the time based on their past experience of success, which exemplifies the hot-hand. If the expert was correct, 78% of the participants chose the expert's opinion again, as opposed to 57% doing so when the expert was wrong. The participants also exhibited the gambler's fallacy, with their selection of either heads or tails decreasing after noticing a streak of that outcome. This experiment helped bolster Ayton and Fischer's theory that people put more faith in human performance than they do in seemingly random processes
This coin has landed heads-up nine times in a row. Therefore, it will probably land tails-up next time it is tossed.
Bill is playing against Doug in a WWII tank battle game. Doug has had a great "streak of luck" and has been killing Bill's tanks left and right with good die rolls. Bill, who has a few tanks left, decides to risk all in a desperate attack on Doug. He is a bit worried that Doug might wipe him out, but he thinks that since Doug's luck at rolling has been great Doug must be due for some bad dice rolls. Bill launches his attack and Doug butchers his forces.
Jane: "I'll be able to buy that car I always wanted soon."
Bill: "Why, did you get a raise?"
Jane: "No. But you know how I've been playing the lottery all these years?"
Bill: "Yes, you buy a ticket for every drawing, without fail."
Jane: "And I've lost every time."
Bill: "So why do you think you will win this time?"
Jane: "Well, after all those losses I'm due for a win."
Joe and Sam are at the race track betting on horses.
Joe: "You see that horse over there? He lost his last four races. I'm going to bet on him."
Sam: "Why? I think he will probably lose."
Joe: "No way, Sam. I looked up the horse's stats and he has won half his races in the past two years. Since he has lost three of his last four races, he'll have to win this race. So I'm betting the farm on him."
Sam: "Are you sure?"
Joe: "Of course I'm sure. That pony is due, man...he's due!"
"Above the roulette tables, screens listed the results of the most recent twenty spins of the wheel. Gamblers would see that it had come up black the past eight spins, marvel at the improbability, and feel in their bones that the tiny silver ball was now more likely to land on red. That was the reason the casino bothered to list the wheel’s most recent spins: to help gamblers to delude themselves. To give people the false confidence they needed to lay their chips on a roulette table. The entire food chain of intermediaries in the subprime mortgage market was duping itself with the same trick, using the foreshortened, statistically meaningless past to predict the future." (Michael Lewis, The Big Short: Inside the Doomsday Machine. W.W. Norton, 2010)
"In baseball, we often hear that a player is 'due' because it has been awhile since he has had a hit, or had a hit in a particular situation.
"Consider the parents who already have three sons and are quite satisfied with the size of their family. However, they both would really like to have a daughter. They commit thegambler's fallacy when they infer that their chances of having a girl are better, because they have already had three boys. They are wrong. The sex of the fourth child is causally unrelated to any preceding chance events or series of such events. Their chances of having a daughter are no better than 1 in 2--that is, 50-50." (T. Edward Damer, Attacking Faulty Reasoning. Wadsworth, 2010
Eric: For my lottery numbers, I chose 6, 14, 22, 35, 38, 40. What did you choose?
Steve: I chose 1, 2, 3, 4, 5, 6.
Maury: Please put all my chips on red 21.
Dealer: Are you sure you want to do that? Red 21 just came up in the last spin.
Maury: I didn’t know that! Thank you! Put it on black 15 instead. I can’t believe I almost made that mistake!
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