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[[File:Extrapolating.png|link=Xkcd|frame| [[Alt Text|By the third trimester, there will be hundreds of babies inside you.]]]]
{{quote|''"There are three kinds of lies: [[Lies, Damned Lies, and Statistics|lies, damned lies, and statistics.]]"''
It has been generally demonstrated that, because human brains are wired toward pattern detection, we are lousy at intuitively interpreting statistics; this is the main reason why casinos are viable businesses. Trying to do anything to curb this problem often results in the worship of the [[Random Number God]], or beliefs like:
* '''The hit/miss belief''': "A hit ratio below 25% is hopeless and a hit ratio above 75% is guaranteed. Everything else is a crapshoot."
:There is a 1 in 4 chance of hitting any one of them, without aiming (or missing them all, which is possible in roulette wheels with a 0).
▲Not so much. There are four groups of 25% in 100%. Go ahead and count them. We'll wait.
* '''[[Gambler's Fallacy|The Gambler's fallacy]]''': All probabilities should somehow "even out" while you're playing. For example, if the computer has a hit chance of 50%, and hits, that's okay. However, if it then scores another hit right away, [[The Computer Is a Cheating Bastard]]. In truth, it just happened to be the way the "dice" fell. As is often stated, "dice have no memory."
* '''Naive Combination of probabilities''': Given the probabilities of two events, people will often simply either add them or multiply them. Generally speaking, calculating the combined probability is much more complicated. For example, if someone accused a group of 100 people of taking drugs, each person would be 1%. Accusing 4% of adults, and 4% of children, if the group is half of each, would be 4 people, not 8.
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{{examples|Examples of how this plays out in storytelling:}}
* ''[[
* [[Dave Barry]] once joked that he always flew on the airline with the most recent crash, on the assumption that it wouldn't be "due" for another one.▼
* [[Mark Twain]]'s ''Life on the Mississippi'' contained the following proof of what you can do with statistics:▼
{{quote|In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.}}▼
* In [[Lois Duncan|Lois Duncan's]] ''[[A Gift of Magic]]'', the psychic protagonist, Nancy, is given a standard test to detect telepathic abilities. She is asked to pick, without looking, all the white cards out of a deck of cards filled with an equal amount of black and white cards. Because she wishes to hide her ability, she picks all the black cards so that she would get all the "wrong" answers and fail the test. The examiner sees right through Nancy's ploy because there is an equal probability of picking only white or only black cards and explains that if she really wanted to screw up the test, she should have picked a roughly equal amount of black and white cards at random.▼
** That might not have helped. People are ''horrible'' at generating random numbers, so even if she picked equal (or near-equal) numbers of black and white cards, a more sophisticated analysis of her picks would reveal what she was doing, most likely by identifying a lack of runs of a single color (see fallacy #2 above). It might ''delay'' the recognition of her ability, though...and unless it were ''blatantly'' obvious what she was doing, it might leave enough doubt to prevent others from being certain.▼
==
* A high school science teacher on ''[[The Daily Show]]'' [http://www.thedailyshow.com/video/index.jhtml?videoId=225921&title=Large-Hadron-Collider thought there was a 50/50 chance] of the LHC creating a black hole and causing [[The End of the World as We Know It]]. His rationale? It could happen, or it couldn't happen, therefore there was a 1 in 2 chance of the apocalypse. [[You Fail Nuclear Physics Forever]] is also involved.▼
** Correspondent [[John Oliver]], who was conducting the interview, then suggested that he and the teacher try to breed after the end. The teacher replied that this was impossible, as both were male, but Oliver insisted it would either happen or not happen, a one-in-two chance!▼
* In the ''[[Corner Gas]]'' episode "Security Cam", Karen figures that there's a 50% chance of a riot breaking out in downtown Dog River, using exactly the same reasoning.▼
* On ''[[The O'Reilly Factor]]'', Bill O'Reilly argued that life expectancy was lower in the US than in Canada because the US has ten times as many people, and therefore has ten times the [https://web.archive.org/web/20120513170725/http://mediamatters.org/mmtv/200907270052 number of accidents.]▼
* On the second episode of ''[[Burn Notice]]'', Michael guessed that a conman's former cellmate didn't drink, which made some sense in the context if he was genre savvy to those sorts of questions, but his explanation didn't: that he just guessed because the man either drank or he didn't, a fifty percent chance. So either Michael's estimate of teetotalers among the male prisoner population is extremely optimistic, or he needs to take a stats class.▼
** A bit more complicated: from the way the conversation was going, it was clear that the conman was testing Michael on whether he really knew the former cellmate. So the odds weren't whether the cellmate was a drinker, but whether the comment about the drinking was a "test question" or not.▼
* A common mention on the show ''[[Hell's
** Also commonly used by wrestling commentators when discussing multi-person matches. They frequently claim that the champion in a 4-way match only has a 25% chance of retaining his title, with no regard to comparative skill levels or possible alliances between the participants. Of course, since the outcome is predetermined, it tends to be much more common for the champion to retain his belt. These sort of statistical predictions are even more stupid in matches like the elimination chamber where the final competitor to be released would clearly have a huge advantage if other other factors were equal.▼
* In the ''[[Law and Order]]'' episode "Coma", McCoy tries to ease Kincaid's conscience about subjecting a comatose victim to a high-risk surgery in order to remove a possibly trial-winning bullet. Subverted in that he's perfectly aware that it's bad statistics.▼
{{quote|''"Well, I see one of three things happening: she gets better, she gets worse, she stays the same, and we get strong evidence. Two out of three ain't bad."''}}▼
* Invoked in ''[[Survivor]]'' - As the players in the game dwindle, Probst tells them that they have a "one in ''x'' shot at winning the million dollars." The way he mentions this, it sounds like the winner of challenges (and at the game ''period'') is chosen at random, when it actually isn't. (You can argue that if you're in the final six with [[The Load]] and someone who the jury ''hates'', you would have a one in ''four'' shot since the jurors would not vote for them.) Justified in that he [[Fridge Brilliance|does this to motivate the players]], and it's part of his "character".▼
* The [[Fox News Channel]]'s fondness for flashy graphics to engage the viewer's attention occasionally lends itself to a few mistakes. Such as [http://pics.blameitonthevoices.com/112009/fox_news_math_fail.jpg a pie chart where the total breakdowns add up to 193%], or [https://web.archive.org/web/20110615070455/http://www.mathfail.com/scientists-poll.jpg this poll with a breakdown that adds up to 120%]. Either with the pressure of the rush to get on-screen information ready by showtime, those responsible have little time to double-check their work; or, [[They Just Didn't Care|they care more about making a quick impression on the viewer than ensuring accurate information]].▼
* ''[[Cox and Box]]'': In the (sometimes cut) gambling number, the titular characters roll nothing but sixes on their dice, leading them to suspect the other is cheating. Although they both ''are'', no dice-weighting is quite ''that'' good.▼
* Deconstructed in ''[[Rosencrantz and Guildenstern Are Dead]]'' by [[Tom Stoppard]]. A coin flipped nearly a hundred times comes up head each time, and they try to figure out how it's happening. Two explanations Guildenstern develops are divine intervention and random chance.▼
** Hamlet himself (though in his [[Hamlet|own play]]) provides the in-universe explanation: "The time is out of joint". Presumably this affects the law of probability somehow.▼
** Of course it is [[Meta Fiction|that kind]] of play so the canonical explanation might well be that Stoppard is doing it.▼
== [[Tabletop Games]] ==▼
=== Card and Dice Games ===
* Any game of chance - but most especially any game which uses dice - will find players who think the ''right'' decision is the decision that agrees with the dice as they rolled after they have rolled. For example, in [[Monopoly]], you may decide to build houses when you see your opponent will land on your monopoly a throw of 6, 7, or 9 on two six-sided dice. (This is not an error: no monopoly on a standard Monopoly board is spaced so you would land on it on a 6, 7, or 8, though if there were one, it would have higher odds than the above combination.) Anyone with half a clue as to how the game works and basic probability theory realizes that's about as lethal a situation as your opponent could be in (for a single monopoly), and would build. Yet if your opponent throws a 12, and bypasses your entire trap, your decision was just as reasonable as before. It just didn't pan out. This sort of fallacious thinking holds for:
** Naive poker players, who fold a bad hand only to see it turn around later (in a game with community cards)
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*** In Video Poker, the advantage is extremely small, if present at all. On average, it takes three solid years of perfect play to break even.
*** In Black Jack, it requires card counting (and maximizing your bet when the odds are slightly in your favor), and they'll kick you out if you try it. Or they'll reshuffle the deck frequently in the case of Atlantic City casinos, where they can't kick you out. There are also special table rules that messes the available strategies up, like the house hitting on soft 17.
* One popular strategy (called [[wikipedia:Martingale (roulette system)|Martingale]]) in Roulette that is believed to always net you money. The same strategy works the same on any 50% chance double money back bets (or as long as the chance to win is balanced against the payout). The basics of the strategy is to bet 1 on red/black, odd/even, or high/low when you start and if you win, and double the bet if you lose.<ref>For example, losing three in a row then winning means betting 1, 2, 4, 8, which means you've lost 7 and won 8, with a net profit of 1.</ref> The belief is that you will eventually win, and thus win the initial bet.
** To always win, you need an infinite amount of money and time.
** In real casinos there is always a betting limit, which removes the option to double up at some point.
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* In the ''[[Asterix]]'' album ''The Soothsayer'', a centurion is tasked by the roman empire to round up all prophets and soothsayers in order to curb down pagan beliefs that go against roman pantheon beliefs. A conman passing himself as a soothsayer gets caught, and is given a test to see if he can predict a roll of two six-sided dice. He breathes a sigh a relief as he knows his luck is usually awful, and picks (stupidly, statistically speaking) seven, which just so happens to come up on the dice and "prove" him the real deal. He goes on an [[Insane Troll Logic]] demonstration that he picked the right number ''because'' he can't tell the future. The centurion isn't convinced until the soothsayer mentions that the village believe anything he tells them, which makes the centurion offer to let them go if he convinces the villagers to leave.
===
* Among its [[So Bad It's Horrible|''many'' flaws]], ''[[FATAL]]'' says that to determine the probability of an event, you roll two percentile dice,<ref>That's a 100-sided dice, or more commonly two 10-siders with one representing the tens digit, for non-gamers</ref> and if the second one is equal to or greater than the first, you succeed. That means that everything has a flat 50.5% chance of happening. And yes, you're supposed to do this for '''anything'''.▼
▲* ''[[Discworld/The Science of Discworld|The Science of Discworld]]'' books have an arguably accurate but somewhat twisted take on statistics: the chances of ''anything at all'' happening are so remote that it doesn't make sense to be surprised at ''specific'' unlikely things.
▲* [[Dave Barry]] once joked that he always flew on the airline with the most recent crash, on the assumption that it wouldn't be "due" for another one.
▲* [[Mark Twain]]'s ''Life on the Mississippi'' contained the following proof of what you can do with statistics:
▲{{quote|In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.}}
▲* In [[Lois Duncan|Lois Duncan's]] ''[[A Gift of Magic]]'', the psychic protagonist, Nancy, is given a standard test to detect telepathic abilities. She is asked to pick, without looking, all the white cards out of a deck of cards filled with an equal amount of black and white cards. Because she wishes to hide her ability, she picks all the black cards so that she would get all the "wrong" answers and fail the test. The examiner sees right through Nancy's ploy because there is an equal probability of picking only white or only black cards and explains that if she really wanted to screw up the test, she should have picked a roughly equal amount of black and white cards at random.
▲** That might not have helped. People are ''horrible'' at generating random numbers, so even if she picked equal (or near-equal) numbers of black and white cards, a more sophisticated analysis of her picks would reveal what she was doing, most likely by identifying a lack of runs of a single color (see fallacy #2 above). It might ''delay'' the recognition of her ability, though...and unless it were ''blatantly'' obvious what she was doing, it might leave enough doubt to prevent others from being certain.
▲== [[Live Action TV]] ==
▲* A high school science teacher on ''[[The Daily Show]]'' [http://www.thedailyshow.com/video/index.jhtml?videoId=225921&title=Large-Hadron-Collider thought there was a 50/50 chance] of the LHC creating a black hole and causing [[The End of the World as We Know It]]. His rationale? It could happen, or it couldn't happen, therefore there was a 1 in 2 chance of the apocalypse. [[You Fail Nuclear Physics Forever]] is also involved.
▲** Correspondent [[John Oliver]], who was conducting the interview, then suggested that he and the teacher try to breed after the end. The teacher replied that this was impossible, as both were male, but Oliver insisted it would either happen or not happen, a one-in-two chance!
▲* In the ''[[Corner Gas]]'' episode "Security Cam", Karen figures that there's a 50% chance of a riot breaking out in downtown Dog River, using exactly the same reasoning.
▲* On ''[[The O'Reilly Factor]]'', Bill O'Reilly argued that life expectancy was lower in the US than in Canada because the US has ten times as many people, and therefore has ten times the [http://mediamatters.org/mmtv/200907270052 number of accidents.]
▲* On the second episode of ''[[Burn Notice]]'', Michael guessed that a conman's former cellmate didn't drink, which made some sense in the context if he was genre savvy to those sorts of questions, but his explanation didn't: that he just guessed because the man either drank or he didn't, a fifty percent chance. So either Michael's estimate of teetotalers among the male prisoner population is extremely optimistic, or he needs to take a stats class.
▲** A bit more complicated: from the way the conversation was going, it was clear that the conman was testing Michael on whether he really knew the former cellmate. So the odds weren't whether the cellmate was a drinker, but whether the comment about the drinking was a "test question" or not.
▲* A common mention on the show ''[[Hell's Kitchen|Hells Kitchen]],'' as well as a number of other reality shows, is that at any given time a given contestant has a 1 in X chance of winning the grand prize, where X is the number of remaining contestants. Not only does this suggest that the winner is chosen at random (which is not the intent of the statement), but also that every contestant is equally likely to win. This is untrue, especially on shows which have a number of obvious dud contestants (such as ''Hell's Kitchen'').
▲** Also commonly used by wrestling commentators when discussing multi-person matches. They frequently claim that the champion in a 4-way match only has a 25% chance of retaining his title, with no regard to comparative skill levels or possible alliances between the participants. Of course, since the outcome is predetermined, it tends to be much more common for the champion to retain his belt. These sort of statistical predictions are even more stupid in matches like the elimination chamber where the final competitor to be released would clearly have a huge advantage if other other factors were equal.
▲* In the ''[[Law and Order]]'' episode "Coma", McCoy tries to ease Kincaid's conscience about subjecting a comatose victim to a high-risk surgery in order to remove a possibly trial-winning bullet. Subverted in that he's perfectly aware that it's bad statistics.
▲{{quote|''"Well, I see one of three things happening: she gets better, she gets worse, she stays the same, and we get strong evidence. Two out of three ain't bad."''}}
▲* Invoked in ''[[Survivor]]'' - As the players in the game dwindle, Probst tells them that they have a "one in ''x'' shot at winning the million dollars." The way he mentions this, it sounds like the winner of challenges (and at the game ''period'') is chosen at random, when it actually isn't. (You can argue that if you're in the final six with [[The Load]] and someone who the jury ''hates'', you would have a one in ''four'' shot since the jurors would not vote for them.) Justified in that he [[Fridge Brilliance|does this to motivate the players]], and it's part of his "character".
▲* The [[Fox News Channel]]'s fondness for flashy graphics to engage the viewer's attention occasionally lends itself to a few mistakes. Such as [http://pics.blameitonthevoices.com/112009/fox_news_math_fail.jpg a pie chart where the total breakdowns add up to 193%], or [http://www.mathfail.com/scientists-poll.jpg this poll with a breakdown that adds up to 120%]. Either with the pressure of the rush to get on-screen information ready by showtime, those responsible have little time to double-check their work; or, [[They Just Didn't Care|they care more about making a quick impression on the viewer than ensuring accurate information]].
==
* MMO players, almost without fail, will adhere to mindset two - they will notice the streak of resists/misses/landed enemy attacks/what have you that killed or almost killed them, but never notice the long, long, long string of hits that precede it. Any and all MMO forums will have a topic pop up fairly regularly asking whether (or sometimes screaming loudly even with no evidence to that effect other than they had a string of bad luck) the RNG is broken.
** ...which can lead to [[Final Fantasy XI|no small confusion]] at times.
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** Starting with the 10th, if a character does not get at least one stats increased during a level up, the game rerolls (unless a character has hit the cap on everything). Starting with the 11th game, a characters growth rates will be boosted or dropped if they are behind or ahead of the "average" stats. Like the main example, this helps deal with the ''very'' annoying chance that a character gets "RNG Screwed", except this is enough to force a restart on an entire ''file'' in some cases.
* [[Word of God]] to the contrary, most players of ''[[Puzzle Quest]]: Challenge Of The Warlords'' believe that the game "nudges" all sort of random stats in its own favor. As many people complain about the computer's habit of chaining together 4/5 gem combos and extra turns, it's even more blatant in Spell Resistance, where an opponent with 2% resistance across the board will block approximately 15% of spells. The player, with the same stats, will be lucky to block one spell in hundreds.
* This trope is often brought up in [[Massively Multiplayer Online Role Playing Game|MMORPG]]s, where many players believe that item drop rates can be mathematically calculated to determine how many monsters you must kill until you "should" find said item, by assuming that a 1% drop rate means that after you've killed a hundred, something's wrong if you haven't gotten one.
** Because of players complaining about this, the drop rate formula in ''[[World of Warcraft]]'' was changed to ''increase'' the drop percentage every time the quest item required ''doesn't'' drop and reset it after one ''does'' drop. Of course, this is also to avoid the wild variation in time a quest can take when it's truly random.
** Indeed, while the mean number of kills is 100, the actual number will be greater than 100 37% of the time. Of course, this also means that 50% of the time it will require fewer than 70 kills.
* Speaking of ''[[World of Warcraft]]'', during the famous "[[Leeroy Jenkins]]" video, someone is asked to do a number crunch to calculate their odds of finishing an encounter. [[Poe's Law|It's actually not as simple as that - it was done to make fun of guilds]] as well as [[Straw Vulcan]]s who may often rely on statistical fallacies.
* ''[[City of Heroes]]'' actually
* ''[[Dungeon Fighter Online]]'' has a dice roller that is perfectly random for the first instance of every sequence (first upgrade attempt, or random item pickup, or something similar), but then often produces identical results for the next several sets (Failing an identical upgrade five times in a row, the same player getting every single item in a dungeon). It often "corrects" itself and skews the other way until results are even. The hit/miss ratio is the same, either producing a lot of hits or a lot of misses in a row, only rarely looking like the actual statistic.
* The ''[[Tetris]]'' Guideline has mandated that all Tetris games since around late 2005 have an implementation to make the gambler's fallacy actually happen (and make players complain less of being screwed by the [[Random Number God|RNG]]): Instead of rolling a D7 to select a piece, newer Tetris games take a sequence of all seven pieces and deals random permutations of it. Thus, after every 7th piece, all seven have appeared with equal frequency. This also makes every 7th piece completely predictable.
** Prior to that, the ''[[Tetris the Grand Master]]'' series also had an algorithm to make the gambler's fallacy come true: The game rolls 6 times (4 in the first TGM) and takes the first result that isn't identical to any of the four most recent pieces dealt. It's still possible for this to "fail" and give you the same pieces over and over again since the game only rolls a fixed number of times; it's just much less likely than with a simple RNG approach.
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** A more direct example are the year-in, year-out complaints that either the stats or the on-field experience are unrealistic, by pointing to the raw numbers. Since ''[[Madden NFL]]'' is a video game, the developers have to shorten the quarters because most gamers aren't willing to invest multiple hours on a single game. So ultimately this means that gamers are running between 50-70% as many plays as a real NFL contest. Yet many expect to produce as many points or exciting moments, while somehow maintaining realistic results on a per-play basis. This is mathematically impossible. EA chooses the former, heavily slanting the game in favor of the offense, which has caused somewhat of a [[Broken Base]] amongst fans of the series.
** Regression to the Mean overall is fueled by a misunderstanding of statistics, and has many (sometimes serious) consequences. "You say Homeopathy/Acupuncture/pseudoscience of your choice worked for your arthritis pain? Wow. When did you take it? When you felt at your worst. Did it ever occur to you that short of ''trying'' to make things worse, you would almost certainly feel better a while after hitting rock bottom? Does "nowhere to go but up" mean anything?"
* ''[[Warhammer
* ''[[Blood Bowl]]'': There's always a 1 in 6 chance of succeeding or failing because ones always fail and sixes always succeed. Players hate this because you tend to fail at the worst possible time. Failing also ends your turn in most cases, so superstition abounds.
** This is also the "rebuttal" of any claims of the AI cheating in the computer game based on it, not taking into account that the exact sequence of rolls is predetermined at the start of any given game (which they mention IN their rebuttal) and thus CAN BE LOOKED AT before they happen, thereby giving the AI an opportunity to cheat. [[Save Scumming|Of course, it also provides an avenue for players to cheat where save games are available...]]
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** [[Crapsack World|In that example, the results are very true to the setting.]]
* In a strange twist, ''Final Fantasy VII: Crisis Core'' had the DMR, a slot-machine of various character faces that spins during combat, creating different effects. The only way to level-up is for three "7"s to align. Isn't that awful?!? Leveling based on total randomness?!? Except. . .it isn't. The manual ''lies''. The DMR is actually controlled by an ''insanely complicated'' mathematical formula that, in-game, manifests itself as the strange impression that chance always ''just so happens'' to work out exactly the way natural progression should. In essence, one in a million chances succeed nine times out of ten.
* The programmers of ''[[Sid
▲== [[Theater]] ==
▲* ''[[Cox and Box]]'': In the (sometimes cut) gambling number, the titular characters roll nothing but sixes on their dice, leading them to suspect the other is cheating. Although they both ''are'', no dice-weighting is quite ''that'' good.
▲* Deconstructed in [[Rosencrantz and Guildenstern Are Dead]] by [[Tom Stoppard]]. A coin flipped nearly a hundred times comes up head each time, and they try to figure out how it's happening. Two explanations Guildenstern develops are divine intervention and random chance.
▲** Hamlet himself (though in his [[Hamlet|own play]]) provides the in-universe explanation: "The time is out of joint". Presumably this affects the law of probability somehow.
▲** Of course it is [[Meta Fiction|that kind]] of play so the canonical explanation might well be that Stoppard is doing it.
▲== [[Tabletop Games]] ==
▲* Among its [[So Bad It's Horrible|many flaws]], ''[[FATAL]]'' says that to determine the probability of an event, you roll two percentile dice,<ref>That's a 100-sided dice, or more commonly two 10-siders with one representing the tens digit, for non-gamers</ref> and if the second one is equal to or greater than the first, you succeed. That means that everything has a flat 50.5% chance of happening. And yes, you're supposed to do this for '''anything'''.
== [[Web Comics]] ==
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