"There are three kinds of lies: 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:
— Benjamin Disraeli
- The hit/miss belief: "A hit ratio below 25% is hopeless and a hit ratio above 75% is guaranteed. Everything else is a crapshoot." This stems from our evolved propensity to and preference for quick decisions, so we often times round everything up and down, in this case to 100%, 50%, and 0%. Obviously 75% does not equal to 100% nor does 25% to 0%, but you'd be surprised how often people forget those were just crude approximations.
- 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." In situations where extreme good/bad luck streaks are undesirable, the Gambler fallacy can be invoked, chiefly in the form of pseudo-random distribution (or PRD). Under PRD, consistent misses will slowly increase the chance of a hit, and vice versa. Many video games uses this variant of "random" without being noticed, because, "pseudo-random" feels more random than real random.
- Naïve 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, suppose you roll a die twice. The probability of a six is 1/6 each time, so the probability of at least one six in two rolls must be 1/3, right?^{note }
- 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: When a reasonable decision was made according to the odds, where the odds are calculatable, a measured risk is always taken. If this somehow backfired, calling this a wrong decision is fallacious because that would imply clairvoyance is presumed; Conversely, when all data and signs are ignored, thus a silly decision was made. Even if that worked out in the end that doesn't make the decision any less ill, as that's just literal dumb luck. For example, if two gamblers agree to roll a fair die, betting 1:1 where Gambler A wins on a 1, and Gambler B wins on a everything else, Gambler A is making an irrational decision to bet — the outcome of the dice roll has no bearing on said irrationality. If it's stupid and it works, it's still stupid and you're lucky
- 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. 50% is not a bad guess when you have nothing else to go on, but only if you have nothing else to go on.
- 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. Although there is no real "minimum sample size" to speak of, most accept the arbitrary magic number of anywhere between 20-50 (most notably 30). Obvious a higher sample size will grant better results, even though it might not be cost effective. In academia this isn't a problem as long as the sample size is shown, so the readers can decide for themselves whether the sample size is good enough. But your everyday layman will not notice this detail, and thus are prone to erroneous conclusion from scientific statistical research.
Examples of how this plays out in storytelling:
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Card and Dice Games
- It's common for a naïve poker players to (correctly) fold a bad hand, such as flush draw with no other outs, to a large bet, only to lament their "mistake" when it turns out their hand would've improved on the turn or river. Especially naïve players may try to "correct" their mistake by calling the next time they find themselves in a similar situation, which more savvy players can exploit.
- 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.^{note } 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.
- 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).
- Similarily, German card game Skat has rules in place, that if one player in a tournament has to use a shuffling machine (e.g. due to a disability), everybody has to, because hand shuffled decks are less random and favor the player after the dealer compared to the better randomization of a machine.
- Many players of the online version of Magic: The Gathering are convinced that the algorithm used to shuffle players' decks is flawed and 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.
- Go to any online poker forum and look in the General Discussion forum. More often than 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 the 'rigged' play screwing the loser.
- One popular roulette strategy (called Martingale) 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 } 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 be guaranteed a win, you'd need an infinite amount of money and time. If you had infinite money, you wouldn't need to bet. :)
- Real casinos always have betting limits, which remove the option to double up at some point. This will generally completely wipe out your winnings thus far that session if it happens.
- If your original lost bet is one dollar and you follow this stratagem, an unlucky streak of ten consecutive losses has you betting $1024 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.
- The presence of 0 and 00 on the Roulette Wheel, which is a loss regardless of which of the above options you bet on, means that there's not a 50% chance of winning. The odds are actually closer to 47%. With the exception of the quint bet, all of the bets on the table have the same expected outcome, trading win amount for likeliness.
- 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").
Comic Books
- In the Astérix album The Soothsayer, the Roman Empire tasks a centurion to round up all prophets and soothsayers in order to suppress pagan 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 villagers believe anything he tells them, which makes the centurion offer to let them go if he convinces the villagers to leave.
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.^{note }
- 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 } 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's success as a whole. At the successful completion of each of the early steps of the plan, Manny asks him for an update on their odds from that point, and Mike keeps giving odds lower than they were before, saying that the later steps in the plan have more opportunities to go wrong. However, these late opportunities for failure (which he was always aware of) should have been factored into every single odds calculation from the beginning. Each successful step should only improve the plan's chance of success from that point.
- 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." - Discussed, subverted, and justified in The Wheel of Time book Towers of Midnight, when Noal Charin points out to Mat Cauthon that rolling dice to determine which direction to go won't work, due to the higher likeliness of rolling odds than evens, and Mat counters that his magic luck means that it wouldn't be random, regardless.
- Just before Princess Diana gave birth to Prince Harry, one British newspaper article claimed that if the unborn child was a boy, that would complete a sequence with a "1 in 60,000" probability, and therefore it was overwhelmingly likely that the child would be a girl. This ignored the fact that a sequence of 15 boys followed by a girl has exactly the same before-the-fact likelihood of 1 in 65,536 as a sequence of 16 boys (or of any other specific sequence of 16 births), and that this is because the likelihood of any given baby being a boy is about 50%, regardless of previous births (especially if not births by the same mother, as in this case).
- One book on recreational mathematics described "the college bet", so-called because its inventor supposedly made enough money from it to put his son through college. If you select three card ranks randomly, and then draw a card from a fair 52-card deck, the chance that it will be one of your ranks is 3 in 13, right? Thus (the book claimed) if you draw three cards, the chance of at least one being one of your ranks is 9/13 (0.6923), and with four cards this rises to 12/13 (0.9231). If the author of the book had followed this "logic" a bit further, he would have seen that this gives the chance of one in five cards being one of yours as 15/13, or more than certain — which is clearly nonsense. (In truth, if each card draw is independent of the others (that is, the card is replaced and the deck shuffled after each draw), the three-card probability is 2170/2197 (0.9877 — substantially better than the naïve calculation), the four-card probability is 28480/28561 (0.9972), and the five-card probability is 371050/371293 (0.9993).)
Live-Action TV
- A high school science teacher on The Daily Show 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. Artistic License – Nuclear Physics 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 number of accidents.
- A common mention on the show Hell's 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).
- Wrestling commentators discussing multi-person matches 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. Given the fact that the outcomes are always predetermined, however, neither skill nor chance really has anything to do with it.
- 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."
- The Fox News Channel's fondness for flashy graphics to engage the viewer's attention occasionally lends itself to a few mistakes. Such as a pie chart where the total breakdowns add up to 193%◊, or 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 care more about making a quick impression on the viewer than ensuring accurate information.^{note }
- In the "Captain jerk" episode of Henry Danger they set up a contest where people have to guess a random number to meet Captain Man and Kid Danger. The random number is four digits, but it takes a very long time (a few days) for the final winner to guess the number, especially as they show the entire city obsessively trying to guess the correct number.
Music
- This is how synchronization (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.
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 85 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. The ultimate reason is that they're characters in a play, so the author is making it happen.
Tabletop Games
- Players themselves sometimes fall afoul of the gambler's fallacy.
- FATAL says that to determine the probability of an event, you roll two percentile dice^{note } , 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.
- World Of Synnibarr also does this.
- 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 almost always at least a 1 in 6 chance of succeeding or failing because when rolling a d6 (i.e. most dice rolls), 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.
- In the 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 three 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. The results are very true to the setting.
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.
- ...which can lead to no small confusion at times.
- 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.
- Because of players complaining about this, the drop rate formula for quest items 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.
- The players' failure to get how probability works was joked about in the 2015 April Fool's Day patch notes, which presented a large number of fictional changes to the game, one of which was:
The S.E.L.F.I.E. camera mission "Field Photography" is not appearing for all players due to its random nature. We didn't change anything; we just wanted to point out what random means. - 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 tenth game's (unlike the ninth's, which was completely random) bonus experience system ^{note } 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^{note } , 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 tenth, 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 eleventh game, a character's 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.
- To prevent Save Scumming abuse, the tenth game's (unlike the ninth's, which was completely random) bonus experience system ^{note } 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.
- 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.^{note }
- City of Heroes actually has a mechanic called the "streakbreaker" that behaves like the second part. 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 a hundred 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 seventh piece, all seven have appeared with equal frequency. This also makes every seventh 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 six times (four 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.
- The Tetris piece-picking algorithms are many and varied. One, the appropriately named Bastet^{note } system, picks the worst piece possible for your current situation with an 85% accuracy.
- Prior to that, the Tetris: The Grand Master series also had an algorithm to make the gambler's fallacy come true: The game rolls six times (four 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.
- Civilization:
- 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 } , unless the odds are 1% or worse, in which case victory is surprisingly possible (see Spearman v. Tank).
- Alleviated somewhat in V, 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.
- Not helping matters is that whoever wrote the code for battle predictions fell into this trope themselves. Due to the calculation failing to take several important factors into account, the result varies from "close enough to make a judgment call" to "no connection to what will happen whatsoever." See the Civilization folder under The Computer Is a Cheating Bastard for the details.
- This is really a matter of game theory, not statistics. In general, you can't afford a policy of entering battles with an 80% chance of winning, because the cost to the player of losing each battle is so much greater than the benefit of winning.
- This trope is hugely responsible for the Pokémon entries on The Computer Is a Cheating Bastard, and is the number one 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 Pokémon, 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 Pokérus (a "virus" that doubles a Pokémon's stat growth) has a 3/65,536 chance to be on a Pokémon, or 1/3 the probability of finding a "shiny" Pokémon. Many players have never seen the Pokérus, while a few have been lucky enough to get it more than once. Once you have a Pokémon with the Pokérus, 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:
- 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 insist that the game cheats in favor of aliens on higher difficulties and Classic mode. 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 computer 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.
- John Madden Football:
- 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 six on a normal die is likely to be lower than that. The technical name for this is regression toward the mean.
- 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.
- 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.
- Discussed and defied in one ending of Stories The Path Of Destinies. After discovering that the completed Skyripper could potentially destroy the universe if fired, he discusses the matter further with the sage Calaveras, who narrows it down to a 1 in 128 chance. Reynardo decides this is no big deal; when confronting Zenobia, he even explains the 1 in 128 chance in terms of the Gambler's Fallacy (i.e. that the risk will increase the more he fires the weapon, but the first shot should be perfectly safe), which an exasperated Zenobia points out is not how odds work at all. Needless to say, he fires it, and it destroys the universe.
"One in 128. What are the odds?"
Visual Novels
- World End Economica has the protagonist, who is a little full of himself, apply the Gambler's Fallacy to trading on the stock market.
Webcomics
- In Darths & Droids, Pete (R2-D2) likes to "pre-roll the ones out" of his twenty-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.
- Awkward Zombie addresses the matter here.
"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."
- Sam Starfall uses statistics in artistic fashion to establish that riding the top of a moving train is safer than climbing a staircase.
- Larry Leadhead applies this sort of reasoning to the wargaming tables.
Web Original
- Sometimes websites run stories about so-called "winning systems", such as this one. The most cursory read of the article reveals that the system this guy has "discovered" is merely the martingale, described further up this page.
- This Jalopnik article advocates bringing more diesels to the US market by showing how big of the VW Jetta sales chunk is diesels. This means next to nothing, since most Jetta buyers buy them for the TDI engine.