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."
Not so much. There are four groups of 25% in 100%. Go ahead and count them. We'll wait.
There is a 1 in 4 chance of hitting any one of them.
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.
The Definition of Probability: There's two ways probability can be defined. The first is what should happen in a random process in the long run. The second is the degree of certainty with which a belief is held. The first definition applies to statements like, "The probability of rolling a six on a fair die are one in six." The second applies to statements like, "My favorite team will win this game." This difference can be very important. These two views are called frequentist and Bayesian, respectively.
Decision-Making and Probability: Many make a mistake related to all these fallacies. When a decision-maker makes a decision to play the odds in a situation where he can calculate the odds, he's taking a measured risk based on what he knows at the time. This is his best decision based on what he could possibly know at the time. If this fails, calling it a wrong decision is fallacious because it would require knowing the less-likely alternative would happen. For example, if two gamblers agree to roll a fair die, betting 1:1 where Gambler A wins on a 1-5, and Gambler B wins on a 6, Gambler B is making an idiotic decision to bet - and the decision remains idiotic even in hindsight if he wins.
Even Probability Distribution Fallacy: When someone assigns equal probabilities to a list of possible events where an even probability distribution cannot be presumed: "The suspect either shot the victim or he didn't shoot the victim, therefore there's a 50% chance he shot him." Ad Absurdum, one would have to claim that you have a 50% chance of winning the national lottery because you either win it or you don't.
Projecting From a Small Sample Size: Even a fair coin has a significant probability of coming up heads seven times in ten flips, so you can't project from such a trial that the coin has a 70% probability of coming up heads.
Note that while this often makes fun of the developers messing up at statistics or the author having no clue how it works, this can actually be invoked or justified.
Also see 20% More Awesome, which involves quantifying the unquantifiable.
Examples of how this plays out in storytelling:
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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)
Players of any RPG, when a character fails at a good plan due to some really off-the-wall lousy rolls or succeeds at an absurd plan through sheer dumb luck
Board game players who fall into the type of thinking in the Monopoly example
Wargamers who misinterpret why some of their opponents quickly calculate odds, then make their decisions based on what's likely to occur from a particular gambit
Many Bridge players feel that the computer-generated hands used for many duplicate games are more unusual (i.e. favor more unlikely distribution of cards) than human-shuffled hands. They're right, but in a backwards way: The computer-generated hands are more likely to be completely random than hands dealt from a human-shuffled deck. Even the best human-shuffled deck will retain a few cards in the same relative order as they were played in the last hand; computer-generated hands don't (except at the frequency you'd expect from random chance).
Many players of the online version of Magic: The Gathering are convinced that the algorithm used to shuffle players' decks is flawed and is biased. (Some say the bias is towards "mana flood", where you get too many mana-producing cards (and not enough spells to actually use that mana with), while others say towards "mana screw", which is the exact opposite — not getting enough.) In reality, the algorithm is completely incapable of either, since it does not consider what type any given card is when performing the shuffle. The reason for the dissonance between physical and online play (when there is one at all — mana screw and mana flood are common on cardboard too) is that having to physically shuffle a deck enough to provide a truly random distribution every time would be incredibly annoying, particularly given the number of times some decks end up being shuffled in a single game. Most people just take their land cards, which end up all in one pile at the end of a game and put them into the deck at fairly even intervals to avoid there being giant clumps of nothing but land. For practical reasons, even in tournaments it's accepted that the deck doesn't have to be truly randomly distributed — it just needs to be random enough that a player can't predict what comes next.
This one-land-per-two-cards sorting prior to the deck shuffling is accepted practice in tournaments, provided that the decks are properly shuffled after said stacking - which then leads you to wonder why you would even bother, given that it would destroy any stacking you might have done. If it didn't, after all, then it would be considered cheating and would lead to disqualification.
Go to any online poker forum and look in the General Discussion forum. More often then not, you'll find a sticky about the game not being rigged, and an explanation of why it may seem that it is. Most forums will also have a 'Bad Beats' section for whining about said 'rigged' play screwing the loser... (never mind that they were chasing a flush draw and getting really poor pot odds on the call...)
In professional (off-line) poker tournaments, the dealer start to shuffle every new deck by simply scattering the cards on the table and mixing them around (similar to how one would shuffle dominoes). Then the cards are loaded into whatever shuffling device is used. (This type of shuffle is commonly called a Beginner's or Corgi shuffle.)
If you're interested, in the UK this shuffle is known as a 'chemmy' (pronounced shemmy), named after the game 'chemin-de-fer' made popular in French casinos, but known to most as the game Baccarat seen in a number of James Bond movies.
Casino games are set up so that over a long period of time, the statistical average favours the house (house advantage or house edge). In Poker, the bets are fair, since you're playing against other players rather than against the house, but instead the casino makes money through rakes of the pot and fees.
The side bets on Craps tables are particularly blatant, because the fair odds are so simple to calculate. For example, the odds of rolling two sixes are 1/6 * 1/6 = 1/36 (1:35 odds), but the payoff on that side bet is 30:1.
The one fair bet in Craps, the free-odds wager, pays a fair, proportional amount should you win. That said, there is no space on the Craps table for it (the player has to "know" to place it at the right time), and it can only be placed as a supplement to your original bet (which is subject to the house percentage.)
Technically, there are single player casino games which sometimes offer a theoretical gain to the player with the right strategy. However, since the gain is very small, any mistake will set you back a lot.
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 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.^{note }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. The belief is that you will eventually win, and thus win the initial bet. There are a few reasons why this doesn't work:
To always win, you need an infinite amount of money and time.
This is known as the Gambler's Ruin (which is different from the Gambler's Fallacy mentioned above). Essentially, the way it works is that if you enter a game with a finite amount of money, and never stop placing bets, you will at some point lose all your money, at which point you can no longer gamble. Technically this requires the person you're betting against to have infinite money, but when you consider how much more money a casino has than you do, the casino for all practical purposes does have infinite money.
In real casinos there is always a betting limit, which removes the option to double up at some point.
The green fields on a roulette table, a 0 and sometimes also a 00, reduces the chance to win to 18/37 or 18/38, which is less than 50%. Because the expected profit is negative, the sum of many such bets is also negative.
If your original lost bet is 1 dollar and you follow this strategem, an unlucky streak of 10 consecutive losses has you betting 1024 dollars to chase after your original loss of a single dollar. Even if you do win, all you get is your original dollar back. You were better off just betting one dollar at a time and hoping for a winning streak.
There is an optional "Event Deck" for the board game Settlers Of Catan. Using it instead of the dice makes probabilities "even out" somewhat (going through most of the deck before reshuffling guarantees that each number will come up about as often as it "should").
In the Astérix 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.
The dice hate you.
Literature
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:
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^{note }Illinois, not Cairo, Egypt 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'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.
The best way for her to deceive the examiner would be to not even bother to use her ability during the session, but merely pretend to concentrate while drawing the cards blindly.
The Moon Is a Harsh Mistress: The supercomputer Mike calculates the odds of a plan for a Lunar revolution. At the successful completion of the early steps of the plan, however, he decreases the odds of success, saying that the further along in the plan they go, the more opportunities there are for something to go wrong. However, if Mike had known about these opportunities for failure from the beginning, they would have already factored into the odds for success. Each successful step should only improve the plan's chance of success.
Brian Aldiss wrote several stories where humans are near extinction, and where society thus makes heavy use of robots; one characteristic of such robots is that their logic often leads them to mistake clear reasoning for correct reasoning. One such story has this exchange:
"There is more country than city." "Therefore there is more danger in the country."
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 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.[^{note }Michael doesn't specifically say it was a fifty-fifty chance, but rather that he had two choices and one was correct and one wasn't. Without knowing anything about person he's talking about, and only knowing that it's likely a test question, he's technically correct in that he basically flipped a coin and guesses from one of the options available. He guesses right, but not because of statistics: it's because of luck, which he points out.
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.
These things come up several times during the show. Someone is infiltrating or bluffing or undercover and they just improvise along. In the specific example of the drinking cellmate question, he had to respond immediately without showing any signs of thinking it over. He was probably more concerned with having a back up strategy in case he got it wrong then the actual question. It's more like chess then gambling. Being able to adapt to the opponents moves and work around it or force his options is arguably more important then perfectly predicting what move will be made. All that said, it's still a fallacy.
A common mention on the show 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. 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 & 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.
"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 does this to motivate the players, and it's part of his "character".
If the people polled can pick more than one option, poll results can easily add up to more than 100%. The first poll might be such a case, but a pie chart is a poor choice for showing that kind of data. The second poll suggests someone was stupid, whether the people who made the graphic, the people who calculated the numbers, or the people who voted for more than one mutually exclusive option.
Music
This is how synchronisation (most notoriously of The Wizard of Oz with Dark Side of the Moon) "works"; the brain notices the few dozen coincidences and goes "oh wow" at them, whilst ignoring the thousands of non-coincidences which surround them.
RPGs, MMORPGs, and other Videogames
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.
This kind of fallacious reasoning reached memetic proportions in the pre-expansion World of Warcraft encounter "Onyxia's Lair." A player who'd run into a string of bad luck assumed that the developers had made the encounter harder, and loudly complained: "She deep breaths more!". (Not breathes, breaths.) Today, "She deep breaths more" is the official name of an in-game Achievement you can earn in Onyxia's Lair.
To further complicate things, some MMOs actually do use a skewed RNG, precisely because true randomness could, in theory, result in a string of misses one real day long, or the opposite. Since MMOs rely on a very predictable form of randomness (e.g., no plucky level 1 can be able to beat a level 20 monster because the monster miraculously rolls no hits, but if attacking a level 4 he must be able to win through pure luck some of the time), various measures can be put in place to make sure the game generates the good, reliable sort of random.
The biggest place this tends to show up is in loot drops, in order to ensure that a player gets loot AT LEAST so often. This is actually very important for helping to maintain addiction.
All Fire Emblem games after the fifth display inaccurate hit/miss percentages. The game actually uses the average of two random numbers to determine a hit, so a 75% chance to hit is really 87.5%. This system is likely in place to make dodging-type units evade more (and thus more viable) and high-accuracy characters strike more and lessens the chance that such a character dies (Due to permanent death and limited saving, this means restarting the entire level in most games) against all 3 of the random mooks that has a 2% chance to hit each.
To prevent Save Scumming abuse, the 10th game's (unlike the 9th's, which was completely random) bonus experience system ^{note }EXP that may be freely given to any unit between levels, helpful for raising Magikarp Power characters, getting that precious extra level of stats ups or helping the guy lagging behind will always increases a character's 3 stats with the highest growth rate (Has an x percent chance to raise this stats on every level up). This wound up making it more broken, as some units quickly hit the Cap on their main stats (Aran), causing stats that would other almost never grow to suddenly increase at insane rates.
It is less broken and more "different". Bonus experience in the game should never be spent levelling characters without maxed stats, but instead should be used to level characters who have already hit the caps on their main stats, resulting in hoarding and struggling a bit to get through the levels, followed by a sudden rise in level to the maximum amount when they finally hit that point at which using bonus experience is optimal.
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 a 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 MMORPGs, 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 for quest items 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. This is also to avoid the wild variation in time a quest can take when it's truly random. The formula for non-quest-related gear drops, however, is still truly random and doesn't vary from one attempt to the next.
Indeed, while the mean number of kills is 100, the actual number will be greater than 100 37% of the time. This also means that 50% of the time it will require fewer than 70 kills.
City of Heroes actually has a mechanic that behaves like the second part, called the "streakbreaker". For a given base percentage chance to hit, if a player or mob misses a certain number of times in a row, the next hit is guaranteed. For a hit chance below 20% you have to miss something like 100 times in a row, but for hit chances above 90%, it only takes one miss to get a guaranteed hit on the next attack. If you were paying REALLY close attention, you could use this to ensure that a key attack doesn't miss.
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 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.
Ask anyone who's played Civilization IV (especially those who play mods like Fall from Heaven) and they will tell you that any combat with less than 80% odds is suicidal and should be avoided at all costs ^{note }though this isn't purely for the chance weighting, as it affects how much your unit gets damaged, which combined with the AI favoring large stacks of weak units, means your unit will likely die next turn, unless the odds are 1% or worse, in which case victory is surprisingly possible (see Spearman v. Tank).
Alleviated somewhat in the sequel, which is kind enough to give you all of the information BEFORE you attack and provides a rough estimate of where the forces will end up in strength after the round of combat. It was actually criticized heavily for its near-perfect accuracy in prediction! Later patches actually made it a bit more random.
This may just be an example of "risk aversion". A success rate of 75%, for example, is still a loss rate of one in four, and replacing those lost units costs resources. (Conversely, at the stage of the game where the enemy have tanks, keeping your spearmen alive drains more resources than they're worth, so why not throw them into battle for that 1% chance of victory?)
This trope is hugely responsible for the Pokémon entries on The Computer Is a Cheating Bastard, and is the number 1 thing the game's professional players complain about to similar levels of usage.
In a more topical instance, players have a random 1/8192 chance of finding an alternately colored Pokemon, similar to albinism and what not. Many players only encounter one or two in several years of playing, others never find one, and some find them with surprising regularity.
In a similar way, the Pokerus (a "virus" that doubles a Pokemon's stat growth) has a 3/65,536 chance to be on a Pokemon, or 1/3 the probability of finding a "shiny" Pokemon. Many players have never seen the Pokerus, while a few have been lucky enough to get it more than once. Once you have a Pokemon with the Pokerus, though, it's very easy to spread it around the party.
The first generation did have statistical errors due to bugs, such as attacks that should never miss actually having a 1/256 chance of missing due to the code using "less than" checks instead of "less than or equal to" checks.
Similarly, the "absolute guarantee" of the Master Ball catching a wild Pokémon is not; there is an incredibly small chance it will fail when used, but luckily hasn't happened yet.
Fortunately, this only applied to the first generation of games. After that, it bypasses the catch calculator entirely.
Final Fantasy Tactics A2: many people report that attacks that give a 95% success rate fail often. It seems likely that this is the case given the number of complaints (especially since the previous game didn't have these problems - then again, the previous game had an actual flaw in its RNG where success rates tend to be universally higher than the shown numbers suggest) but obviously it's impossible to say for sure.
X-COM's accuracy reports during combat aren't exactly blatant lies, but they're not exactly accurate, either. What X-COM does for a hit check is up to two rolls. The first is done against the accuracy check, and if it passes, you automatically get a dead-on shot. The other roll, if the first fails, is the deviation from where you're aiming, which may also end up being nil, resulting in a dead-on shot. So that 75% Accuracy the game reports? More like 77% to hit the target you're aiming at, and up to around 20% to hit someone else, resulting in somewhere around a 86% (on average) chance of someone getting hit by any given shot in a heated battle. Oh, and 100% accuracy reportedly doesn't exist.
XCOM: Enemy Unknown ditches this in favor of telling player straight how likely he is to hit. However, many players insisted that the game cheated in favor of aliens on higher difficulties, more specifically on Classic which many felt was supposed to be the correct difficult. This eventually got so bad, that the RNG was taken apart for testing by the community... and found to be 100% correct. As it turns out, on the lower difficulties, the does actually cheat for the player. Thus once the aid is removed on the higher difficulties, players feel cheated when their "sure" shots no longer hit all the time.
The Madden Curse works this way. Generally, the cover is awarded to some athlete who just had a phenomenal season. The next season, the player is often beset by the sorts of bad luck that befall all athletes (injuries, bad games, etc) except they receive more attention. In some cases, it may also be a Self-Fulfilling Prophecy if the player gets a big ego and skimps on workouts, or if other players are more motivated to play hard against him. But mostly it's just that any season of a player is likely to be average (for his or her capabilities) and any season which leads to feature a player in games or magazine stories is likely to be way above average, so it's just a good chance of a "dice roll" showing a lower number, just like the next number after you rolled a 6 on a normal die is likely to be lower than that.
A great example of this would be Brett Lorenzo Favre's appearance on Madden. It appeared that Favre had subverted, nay, broken the Madden curse while playing for the New York Jets. Then the Jets lost four straight games and a bid for the playoffs. "What went wrong?" you ask. Brett Favre played the last month of the season with a torn bicep in his throwing arm and no one did anything about it.
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 40,000: Fears of "bad dice" abound. The previously mentioned lack of even distribution and the tendency of rolling methods to influence the result only adds fuel to the fire.
Bad dice can be a real thing. Las Vegas maintains a very strict policy for evaluating the balance of dice BEFORE they go into a game, and replacing them at regular intervals. As most 40k players play with cheaper, mass produced dice, and will rarely replace them unless they have visually obvious defects, it is perfectly plausible for a player to be using biased dice.
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. It also provides an avenue for players to cheat where save games are available...
In A Song of Ice and Fire roleplaying system, there is a table for rolling random events in your family's history. This would be perfectly fine, except that you roll 3 dice for the events (thus making the events in the middle more likely), and the table is in ALPHABETICAL ORDER. Thus Doom (the worst thing that can possibly happen to a family) is more likely than a mere Catastrophe (still bad, but not even half as bad), just because Doom is closer to the middle of the list than Catastrophe is.
In a strange twist, Final Fantasy VII: Crisis Core had the DMW, 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 DMW 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. Furthermore, while it's not shown anywhere in the game, enemies actually do give experience when killed, increasing the odds of hitting the combination that gives Zack a level up. As can be expected, this means that getting 2 or more level-ups in a row or shortly one after another can only happen if you've been under the effects of Curse status which disables DMW for an extended perioid of time and killed a ton of enemies during that time and if you kill enough enemies, you're eventuallly guaranteed to get the combination that gives you a level up.
The programmers of Sid Meier’s Alpha Centauri fell afoul of this trope when they wrote the code to estimate the battle odds displayed before a combat: they used an obvious-but-wrong method of working out chained probabilities, leading to the game tending to grossly underestimate the actual odds of victory. For example, a strength-8 unit with 30 hitpoints attacking a strength 8 unit with 10 hitpoints would be shown as having a 75% chance of victory; the actual odds of winning are 99.93%. Under the right circumstances, this could result in the game predicting a one-in-a-million chance of winning, when the actual odds are 90%.
In Achaea, you can flip a coin. Some players argued that the chance was approximately 60/40, rather than the expected 50/50. However, after careful study of the game code, one developer found out that the players were actually right, and corrected the programming error, while also claiming that it wasn't impossible to have the coin fall on edge.
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. 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 own play) provides the in-universe explanation: "The time is out of joint". Presumably this affects the law of probability somehow.
It is that kind of play so the canonical explanation might well be that Stoppard is doing it.
Tabletop Games
FATAL says that to determine the probability of an event, you roll two percentile dice^{note }That's a 100-sided dice, or more commonly two 10-siders with one representing the tens digit, for non-gamers, 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 that doesn't have a specific rule for it.
In Darths & Droids, Pete (R2-D2) likes to "pre-roll the ones out" of his 20-sided dice. He takes a huge number of dice and rolls them once each, and selects the dice that rolled a one. He rolls those dice again, and selects the dice that rolled one a second time. Since the odds of any given d20 rolling a one three times in a row is 1 in 8,000, another roll of any of these dice has only a 1 in 8,000 chance of rolling a one again, right? ... No.
When one of said dice does roll a 1... "Now it's even luckier!" ^{note }Assuming the dice can be unbalanced, doing that with enough dice enough times will get you the ones that are prone to rolling ones.
Even better - his reason for pre-rolling is the fact that he doesn't believe in "lucky dice".
This is not some goofy joke. There actually are people in Real Life who do stuff like this, and whole heartedly believe it works.
"Even after studying statistics, all my knowledge and understanding of it goes out the window when I play Fire Emblem and I get mad at math not doing what I think is best for it."