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Gambler's Fallacy
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The Gambler's Fallacy is believing that a random outcome is affected by previous outcomes, or believing that sequences of random events have memory (such as rolling dice, flipping coins, drawing cards, and pulling slot machines). Basically, it's the idea that events which are independent and random will occur in "streaks". For instance, this fallacy occurs if you believe that a coin which has just landed on heads ten times in a row is now more likely to get tails next time to even things out. Or alternatively, heads is on a roll and will appear next time too. Both of those beliefs are fallacious, because every coin flip has a 50/50 chance of being heads or tails, no matter what came before it or how many times. Psychologically, this fallacy tends to come from the fact that the odds to replicate a pattern do go up cumulatively. The probability of rolling 20 on a d20 twice is 1/400, the same as any expected sequence of two numbers. The probability of rolling the first is 1/20, and the probability of rolling the second is also 1/20. The fallacy occurs when someone assumes that once they've rolled two 20s in a row, it's less likely than usual (< 1/20) that they'll get another 20. In reality, once they've rolled two 20s in a row, it's just as likely as ever (1/20) that they'll roll a 20 again. This also, most notably, works the other way around - if they've lost many bets in a row, they aren't any more likely to win the next bet. Psychologically, what you're doing is inventing desirable patterns that fit with the events you observe despite the patterns not really being there at all. A similar misinterpretation is that if an event has a probability of 1-in-n, then you are guaranteed a success if you make n attempts. As an exaggerated example, the probability of a "heads" on an unbiased coin is 1/2, therefore, flipping a coin twice is guaranteed to get at least one "heads." This is not true.note This is sometimes called the fallacy of equipartition, as if there are n equally probable outcomes, this line of thinking suggests that if you run the procedure n times, you'll get every outcome exactly once each. Another factor is that many people confuse "a number of independent events" (where any of a number of permutations will do) with "a series of independent events" (where only one permutation will do). If we flip a coin twice, we have a 50% chance of getting heads and tails in some order (heads-tails and tails-heads; the other two possibilities are heads-heads and tails-tails). But if we specify that we want the series to be "heads-tails", the probability that that particular series will come up is only 25% — the outcome tails-heads no longer fits the criteria. (Of course, any series has the same chance of coming up. You have as much chance of flipping heads-tails as you do tails-heads, heads-heads, or tails-tails; namely, 25%.) Also, stuff really does even out over time. Just not in the way some people might think. Say that you have flipped a coin and you have had 4 heads and 1 tails come up. Heads has come up 80% of the time. Now, you get the "normal" (more common) sequence, where 5 heads and 5 tails come up, bringing a total of 9 heads and 6 tails. You then have only 60% heads, so while this is a smaller number, it didn't exactly "even out." To explain the above in another way, flip a coin 10 times, and the chances that heads was flipped 4 times or more is 82.81%. Flip it 1000 times, and the chances heads was flipped 400 times or more is 99.99999999%. But even if it was less than 400, the next flip will still be 50/50: long-term odds predict the general trend of many results, not what will happen in a specific instance.note This is known in probability theory as the Law of Large Numbers, which says, for example, that the difference between the observed frequency of a perfectly fair coin landing heads up and the theoretical frequency of 1/2 becomes negligible in the long run, even if it isn't always zero. This is also the reason why playing a high number of low-stakes games in Casinos increases the chances of the house making money; the house advantage only affects who wins a small percentage of the time, but this advantage "evens out" over the long haul. Unless you're a good card counter, taking advantage of free stuff, or just enjoy playing, you're more likely to be successful with a small number of high-stakes events. The simplest explanation for why this fallacy can occur is that the human mind is trained by evolution to recognize patterns as a survival instinct.For example...If you get sick every time you eat a certain food, you learn the pattern of getting sick and eating the food; thus, you avoid eating the food so you stop getting sick. Alternatively, if you see five people cross a street when the street light is green, and all five people get hit by cars, you can recognize the pattern of the light's color, the traffic's movement, and the injuries on other people. Thus, you know not to cross the street when the light is green. The trouble is, the human mind can often find patterns in things where there simply isn't a pattern that can be replicated, such as in a game of chance. A coin might come up as heads ten times in a row if you flip it ten times, but that was entirely random chance; the coin has no more chance to come up heads on any flip than it does tails. (That is, of course, assuming you're not using a Two-Headed Coin.) Note that Gambler's Fallacy applies only to systems that both have no memory, and are explicitly known to be fair. Drawing cards without replacement (read, deck now has "memory") does alter the probabilities of the next cards drawn, and if you do not explicitly know that the event being tested is fair, you can use things like n-heads-in-a-row to draw conclusions of bias in the system (see Non-examples and Theatre sections below). See also Random Number God and Artistic License – Statistics. Compare Sunk Cost Fallacy. The Gambling Addict may tell themselves this lie. Games which implement a Bad Luck Mitigation Mechanic avert this trope. |
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Frieren of Frieren: Beyond Journey's End often falls prey to Chest Monster treasure traps despite having a spell that can detect them with 99% accuracy. If anything, using that spell makes her more reckless since she believes that because it's not 100% accurate, a "great mage" like her will "see through that one percent" to pull off "historic discoveries" with enough tries. So far, all this logic has gotten her is being chomped and slobbered over by these mimics. | |
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Discussed at length in Rosencrantz and Guildenstern Are Dead, in which Rosencrantz flips a coin 85 times in a row and gets heads every time. Guildenstern suggests that it shouldn't be surprising since each coin has an equal chance of coming up heads or tails. Neither Rosencrantz nor Guildenstern is satisfied with this explanation. (Technically, Guildenstern is right in that given a fair coin, a series of 85 heads is exactly as probable as any other single series of that length [namely, 1 in 2^85]; however, if a coin should actually land the same 85 times, it's a good reason to believe that such a coin [or flip] is NOT fair.) | |
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The Simpsons: In "The Mansion Family", Mr. Burns needs to go to the hospital for a while and thinks he should get Homer to watch over his mansion while he's gone. When Mr. Smithers points out that this is a bad idea because Homer screwed up everything else Mr. Burns has ever asked him, Burns responds by saying that since Homer failed so many times, he's due for a good performance. Needless to say, Homer screws up again; pirates steal Burns' yacht, and his pet monkey was mauled in a knife fight. In "Homie the Clown", Krusty loses a lot of money betting against the Harlem Globetrotters, because he figured that "the Generals were due". In "Margical History Tour", the retelling of Henry VIII's life shows Henry (Homer) meeting Anne Boleyn (Lindsay Naegle), who touts her track record of bearing sons. So, they marry, but she produces a daughter, And Henry has her beheaded for it. |
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This is the fate of the cat in the Looney Tunes short "Early to Bet." The bite of the Gambling Bug sends the cat to the bulldog taunting him to play gin rummy, only to lose every time and being forced to perform physical humiliation stunts as penalties. The Gambling Bug even lampshdes it: | |
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The Guild of Gamblers on the Discworld operates on the rule Never Give A Sucker An Even Break. The Guild exists largely to regulate how far its members can mark cards, what sort of spring-loaded devices they can keep up their sleeves to insert a fifth Ace, and how deeply billiard balls may be shaved. They're also across the street from the Alchemists' Guild, which in itself illustrates this. The Alchemists' Guild can't blow up again, can it? (...Yes, it can.) |
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This one also shows up among players of World of Warcraft, in particular with the rare dragon whelp pets that drop out in the world, with many players assuming the more you kill the whelps, the higher your chance grows of finally getting a drop, while in reality the chance was initially independent of each past or future kill... but because people refuse to accept that improbable does not mean impossible or certain (or simply because the large variation in time required was annoying), World of Warcraft developers actually modified this detail to conform to players' expectations. Your chances for a drop do gradually increase the more you kill. This was also implemented because of some "kill this mob and loot this item off them" quests, where the drop rate was not 100%. Some of these quests had unusually low drop rates, and you could spend an hour or more trying to finish a quest. With this change, the chance goes up and up with each "loot", and then resets itself to the default rate with every success. | |
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In Song of Susannah, King fictionalizes his own nearly fatal car accident. Before it happens, his Author Avatar is shown musing that since a similar accident happened in the area recently, the odds of something like that happening to him have dropped to almost zero. He doesn't say why, but he certainly doesn't say it's because people will be really careful now. | |
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In Normal or lower difficulties XCOM: Enemy Unknown, a shot fired by a soldier after another soldier missing a 50%+ shot is more likely to hit... Because the game cheats in your favor and secretly adds an additional 10% chance to the next shot, and does so until a shot hits. It also reduces alien shots accuracy by 10% after a shot hits, until a shot misses. The game stops cheating in Classic or Impossible difficulty, throwing off veteran Normal players. | |
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The Twilight Zone (1959): In the gambling-themed episode "The Fever", the arrogant Franklin falls prey to the titular fever, a gambling addiction so horrendous that he doesn't get a wink of sleep for two straight days. One of his professed reasons to keep at it is the gambler's fallacy — in an effort to break even, he doesn't stop playing the slots until the slot machine breaks from overuse with his dollar coin inside it, causing him to suffer a nervous breakdown. | |
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In the second Revival Round of Liar Game, Nao falls into this, thinking that after her opponent had won a one-in-four chance gamble three times in a row, that it would be incredibly improbable for her to win a 4th time, meaning that she should bet all-in. It's actually pretense for a trap. Nao had been fooling her opponent into thinking that she had a tell so that she can trick her into a massive loss. Nao was probably well aware of the actual odds but needed justification for her seemingly stupid bet. | |
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On My Name Is Earl, Earl mentions a Noodle Incident wherein he lost a series of Rock-Paper-Scissors games to a monkey. The monkey threw "Rock" several times, and just when Earl decided to throw "Paper," the monkey threw "Scissors." | |
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Ultra Fast Pony has a doubly-fallacious example. First, Twilight is treating a not-remotely-random system (namely, hiding from a dangerous killer) as if it were random—and then, within that system, Twilight says a plan that's failed once is therefore due to succeed soon. | |
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In The Langoliers, the characters are faced with the mystery of how most people on their airplane have vanished while they slept. They are only saved because one of the passengers still present can fly the plane, which is, of course, an unlikely coincidence. At the point where they still assume that the same has happened to other planes in the air, one of them reasons that the odds anyone else has survived it like them are minuscule because it happening a second time have now become as unlikely as it happening twice since it already happened once. (As opposed to: it's unlikely to happen twice, but if the unlikely already happened once, that doesn't affect future odds.) | |
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NUMB3RS: In the pilot, Don, watching a baseball game with his father, comments that a batter is "due" a big play after going four games without a hit, a concept which Charlie is quick to refute, only for the player to end up making a big hit. Subverted as it turns out that Don knew what was going to happen because the game is a replay and he had read the recap in the paper, and was just taking the opportunity to yank Charlie's chain a bit. In another episode, Charlie asks a group of people to spread out in a random pattern. The people move to positions that are all equidistant from each other, but within the space that is available. Charlie points out that what they made was pseudo-random spacing, since a truly random pattern would have some points that are closer to each other, not evenly distributed. |
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A Pokémon has a 1 in 4096 (before Gen VI, 1 in 8192) chance of appearing as shiny. Given any number of encounters, it's still very possible to never see a shiny, as the shiny odds per random encounter never increase or decrease barring very specific exploits the game barely informs you of. | |
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Darths & Droids. The Munchkin gamer Pete, facing a situation where a dice roll of 1 would be disastrous, encourages Annie to use one of his dice: "I've pre-rolled the ones out of it." The Rant explains that, beforehand, Pete had carefully prepared a number of 20-sided dice that had rolled two 1s in a row, and placed them in a special, roll-proof container. Since the chances of rolling three 1s in a row is only 1 in 8000, surely rolling another 1 from these pre-rolled dice is almost impossible, right? A bit later in the comic, one of those pre-rolled dice actually does come up as 1. Pete's reaction? | |
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In a Peanuts comic, Lucy uses this fallacy to convince Charlie Brown to try to kick the football again. | |
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Bay of Angels: Caron, a gambling addict who plays roulette, keeps a notebook and writes down numbers. He believes that he can predict what numbers are going to come up on the roulette wheel by tracking what has come up before. | |
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Many Warhammer gamers or tabletop roleplayers will tell you that this is absolutely true. Others will perform astounding feats of Mathhammer in mid-game and tell you exactly how many units of each side should die in an assault, what is the expected variance, and whether or not the assault makes sense in terms of points of enemy units destroyed versus own losses. | |
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In The Thin Man, Dorothy Wynant breaks up with her fiancé out of fear that their children will grow up to be murderous psychopaths, as she believes her father to be. Her brother Gilbert assures her that he's been reading extensively on Mendelian genetics and that her children only have a one in four chance of being murderous psychopaths, so as long as she stops at three, she'll be fine... then he realises the first one might be the murderous psychopath. Setting aside the many things he gets wrong about Mendelian genetics in his speech, there's nothing that says exactly one child out of four would be a murderer. | |
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Zero Punctuation brings this up by name in Yahtzee's review of Grand Theft Auto V. Yahtzee theorizes that the game constantly switches between three different protagonists in the belief that the player would have to find at least one of them to be likable. Yahtzee, however, ends up equally disliking all three main characters. | |
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Kingdom of Loathing's "adventure queue" remembers the last 5 combat and non-combat encounters you've had in each area, and if the normal selection method picks one of them, it'll reject it 75% of the time and pick a new one. | |
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Stories: Path of Destinies: One of the endings of the games rely on this. Coming under possession of the Skyripper, a weapon so powerful that could potentially destroy the universe, Reynardo wonders what are the chances the worst case scenario might happen, and that they are 1 in 128. Reynardo becomes convinced that as long as he uses the weapon only once, everything is fine, despite being explained by both Calaveras and Zenobia that this is not how statistics work, Reynardo chooses to fire the weapon still, sure that the weapon's first shot is safe. This ends up destroying the universe in that timeline. | |
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In the Friends episode in which the gang go to Las Vegas, Phoebe notices an old woman who always seems to win a jackpot at a slots machine immediately after Phoebe has left the machine in question. Ross explains that this is a lurker: someone who waits for a slots player to have a losing streak and leave the machine, only to "cash in" on their jackpot. Of course, slots machines are entirely randomized. | |
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In Only Fools and Horses, after beating Boycie at poker, Del Boy offers him double or nothing on the spin of a coin. Boycie's response is "I've beaten you on a spin twice, Del. By the law of averages, you've got to win this time." (The coin isn't fair, as it happens, but Boycie doesn't know that.) | |
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Played straight in the first Major League movie, and played with in the second. In the Big Game at the end of the first movie, manager Lou Brown elects to have Rick Vaughn pitch to Heywood, the star power hitter on the Yankees, despite the fact that Heywood has consistently hammered Vaughn's pitching all season long. When catcher Jake Taylor voices surprise at this, Brown just says that he has a hunch that Vaughn is due for a good outing against Heywood. Fortunately for the team this works, although Taylor does his best to assist Vaughn by psyching out and distracting Heywood with some Trash Talk. In the second movie Taylor, who has taken over as interim manager, elects to have Roger Dorn, who hasn't played all season, pinch hit against a pitcher that Dorn is notoriously terrible against. The decision flies against logic so badly that the team's announcer can't come up with any rationale for the decision to the audience, admitting he has no clue why Taylor would do this. However, it turns out not to be Taylor believing that Dorn is due for a good performance or gambling, but instead exploiting the fact that the pitcher would always famously pitch "inside" to Dorn (throwing the ball very close to a batter to make it difficult to hit the ball with any power), and Taylor wants Dorn to let himself be hit by an inside pitch, since being hit by a pitch counts the same as getting a hit in baseball. (At least as long as the umpire doesn't believe you allowed yourself to get hit intentionally.) |
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Gambler's Fallacy | |
| Gambler's Fallacy / int_fa5e90fd | comment |
City of Heroes utilises a system called the Streak Breaker. This mechanic in the attack calculations sometimes forces attacks that would normally be misses to instead hit, based on the current number of misses in a row versus your chance to hit. In short, it breaks streaks of misses. | |
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