You Fail Statistics Forever - Television Tropes & Idioms

By the third trimester, there will be hundreds of babies inside you.

"There are three kinds of lies: lies, damned lies, and statistics."

—Benjamin Disraeli

Teacher: I'd say there is a 50/50 chance. The Large Haldron Collider will either destroy us or it won't.
John Oliver: I'm not sure statistics work that way.
later
John Oliver: After the apocalypse, we should try breeding. (the teacher is male)
Teacher: I don't think that will work.
John Oliver: It either could happen or it won't.

—The Daily Show

Combat is chaotic, with a billion things going on at once, and the relatively simple task of "take this and stick it in that guy" will get bogged down with thousands of factors. In many RPGs, particularly tactical ones (Final Fantasy Tactics, the Fire Emblem series, etc.), there is an inherent equation for likeliness to hit and likeliness for a critical hit, and it is often shown to the player beforehand. Unfortunately, those who don't fully grasp concepts of probability can believe some apocryphal things that can lead to slanderous nerd rage directed towards the cards that probability hands them. These apocrypha come in two forms:

The hit/miss belief: a hit ratio below 30% is hopeless and a hit ratio above 80% is guaranteed while anything else is a crapshoot. However, a ratio of 25% is a 1 in 4 chance, meaning that there is still a good chance to hit; a chance of 75%, likewise, still has a 1 in 4 chance of failure.
The Gambler's fallacy (see You Fail Logic Forever): 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 after, then The Computer Is A Cheating Bastard. In truth, it just happened to be the way the "dice" fell. As it's often stated, "dice have no memory." The false belief that a random event will actively work toward its probability is the Law of Averages, while the fact that a random event will regress toward its probability given enough repetitions is the Law of Large Numbers.

It has been generally demonstrated that, because human brains are wired toward pattern detection, we are lousy at intuitively interpreting statistics; this is mostly responsible for the casino always winning. Trying to do anything to curb this problem often results in the worship of the Random Number God. Actively exploiting this problem to convince people leads to Lies Damned Lies And Statistics.

Sometimes, though, the random number generator (abbreviated RNG) IS screwy, which demonstrates that even programmers are not immune. Also, real dice rolls do not really have an unbiased distribution; this does not help.

In addition, sometimes The Computer Is A Cheating Bastard when it comes to random numbers, but this is difficult to prove.

Examples:

Card and Dice Games

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 froma 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.
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, 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 actual problem is a combination of You Fail Statistics Forever and the fact that shuffling in the physical game, especially when playing casual games with friends, is significantly less random than the online shuffler. Shuffling a physical deck in such a way as to achieve the most random results takes significant time and effort (seven good riffle shuffles is the least), so many players don't bother. Thus, players of the physical game are used to their decks being less than random. Suddenly encountering decks that are so much closer to truly random gives them a swift lesson in probability and statistics.
- Related to this is the practice of separating out, then 'weaving' land and nonland cards together after a physical game in hopes of ensuring a more even distribution. Which is a clear example of this trope — if you shuffle enough to truly randomize your deck, the starting distribution shouldn't matter, and if you don't you're effectively cheating.
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 and explanation of why it may seem that it is. Of course, 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 shuffle every new deck starting with 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,and is best suited to )
While most casino games are set up to take money from statistics failers over the long haul, 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.

Fanfic

In Tilman Stieve's Tales of The Twilight Menshivik stories, Mystique and Val Cooper have an affair, both as women and when Mystique has shapeshifted into a man. Val ends up getting pregnant(much as Word Of God has said did not happen with Mystique and Destiny in the 616-verse) and when she informs the rest of X-Factor that her conceiving was "something like a chance in a million"; Strong Guy(Guido Carrosella) replies that for X-teams, "Million-to-one chances crop up nine times out of ten."
- ...which is a line stolen from Terry Pratchett's Discworld series.
- While that line is amusing, it's not inherently wrong. When randomly choosing a number from zero to a million, all of the possibilities have million to one odds against. In that case, million-to-one will chances crop up ten times out of ten. You just can't predict which million-to-one chance will actually occur. Thus, the actual implication of the line is that there are many possibilities, none of which are likely.

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. You Fail Nuclear Physics Forever is also involved.
- The interviewer also suggested that he and the teacher try to breed after the end. Both were male, but 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.

RPGs, MMORPGs, and other Video Games

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.
- 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 (i.e. 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.
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 87.75%. This system is likely in place to make dodging-type units evade more and high-accuracy characters strike more.
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 egregious 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 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.
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 insure that a key attack doesn't miss.
The Tetris Guideline has mandated that all Tetris games since around late 2005 have an implementation to make fallacy #2 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 fallacy #2 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.
  - The Tetris piece-picking algorithms are many and varied. One, the appropriately named Bastet system, picks the worst piece possible for your current situation with an 85% accuracy.
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, unless the odds are 1% or worse, in which case victory is surprisingly possible (see Spearman v. Tank).
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").
This trope is hugely responsible for the Pokemon 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.
- 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.
Final Fantasy Legend, with the infamous Saw Game Breaker weapon. On the enemies it would work on at all, (see Game Breaker about the flaw making it work on too many enemies), it had a 50% chance of getting a One Hit KO. In practice, this meant that it would alternate between hitting and missing. Thus, if you wanted to use them in battle, just equip two different characters with them and have them both use it in a given round. If the first misses, the second would be guaranteed to hit.
Final Fantasy Tactics A 2: 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) 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.
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.
Warhammer 40000: 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.
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.

Theater

Averted in Rosencrantz and Guildenstern Are Dead. "Heads." This is perceived as a failure of statistics, but it's more likely the the coin was simply biased. Since it happens, and is correctly noted as being very unlikely, no one is failing statistics.
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.

Webcomics

In Darths And 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!"
- And, of course, assuming the dice can be unbalanced, doing that with enough dice enough times will get you the ones that are prone to rolling ones.
Math being one of it's primary subjects, XKCD has a few statistics-failure comics, like the one at the top of the page, and this one about guessing numbers (read the alt-text).

Film

There is an unintentional lampshade in Maverick - Maverick tells the audience that he has believed since he was a child that he has a special power - that if he thought hard enough about a card, he could cut right to it. Then he tells us that "it never works." He must have a special power, as, if it's never worked, and he's been trying nearly daily since he was a child, he would have cut to the right card simply by chance at least once.
- Perhaps he exploited this fact in the climactic hand, by intentionally trying to draw every card except the one he needed?
GI Jane's premise: in an attempt to see if women can serve in combat duty, the DoD lets one woman try SEAL training. The course, as stated in the movie, has a 60% failure rate, meaning you'd need a sample size of at least three to even pretend you've got a realistic bearing. Of course, a Jackie Robinson Story is better drama.

Literature

In Brandon Sanderson's Mistborn, at one point the heroes notice that precisely 16% of people exposed to a certain substance are affected. More than that, in every sample, precisely 16% are affected. This is taken as evidence of an underlying condition in the people being affected, when in fact it's precisely the opposite. When taking samples from a population, you expect deviations.
[[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 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.

Real Life

People are prone to claim that some statistical study is flawed because the sample is a very small proportion of the total population. However, the margin of error of an unbiased sample depends primarily on its absolute size, and not on its ratio to the population.
- The Nielsen TV rating in the US use just 5000 families and are generally accurate. Not completely, but generally accurate.
In this forum post, somebody asserts that all the great advances of history have been made by 1 in 100 billion people. Considering that 100 billion is a fair estimate of the total number of human beings who have ever been born, the assumption is probably unsound.
The perspective of eight-year-olds on this is interesting. I am studying probability with my class, and I asked them what result they would expect if I rolled a dice (D-6), and why. Answers included: "You will get a one because it is the lowest number."; "you will get a six because it is the highest number/the best number (six wins an extra turn in many children's board games)."; "you will get a three/four because it is the middle number."; and "You will get (insert number here) because it is my favourite number."
Behold, sir! This apparently ordinary coin, when asked a simple yes/no question, will give you a correct answer fifty percent of the time!, unless, of course, it happens to land on its edge...
Speaking of D&D, we have Patricia Pulling, who rode the Satanic Panic hard by claiming that Dungeons And Dragons (among other things) led to Satanic activities. She once claimed that 8% of the population of Richmond, Virginia indulged in Satanic activities... then explained that she reached the number by estimating that 4% of the teenage population and 4% of the adult population (which would add to, oh, 4% (or less) of the total population) were involved in Satanic activities.
People who play the lottery usually have a favorite number, and they always have a rationale for why they're being clever about it. "Well, of course 123456 isn't going to win. That would be crazy. But 425387 — that's plausible!" No. That's not how it works. Every ticket has the same chance. The fact that the odds of 111111 coming up seem completely nil ought to put it in perspective.
- An interesting take on this: it's better to pick high numbers (40 000+) because many people play favorite numbers, which often happen to be dates. Picking higher numbers doesn't increase the chances of winning, but it may increase the chances of being the only winner, or one of only a few.
  - Related: in New Scientist's Do Polar Bears Get Lonely?; if the combination 1,2,3,4,5,6 was drawn in the UK Lotto, the jackpot would be shared by 10,000 people who thought they were being clever!
  - In New Zealand one Lotto draw picked 31, 32, 33, 35, 36, 38. The 38 winners got a cool $36,000 each; the average is around three winners. (Just imagine if the 38 drawn had been a 34.) And apparently 700 people pick 1, 2, 3, 4, 5, 6 each week.
The entire series of The Bible Code is an epic failure in statistics; you can find similar codes in any sufficiently large text.
There's an urban myth that during pregnancy, doing something complicated (and embarassing/potentially dangerous) with a certain brand of drain cleaner will give a 50% accurate prediction of the sex of the unborn child. This persists despite statistically-savvy expectant mums pointing out that there's another procedure which has the same 50% accuracy, and is far quicker, cheaper and safer — toss a coin.
This troper once saw a TV programme (Brit!) in which a professor illustrated this trope by showing members of the public an ordinary three-tumbler combination lock and asking what the chance of hitting upon the combination first time out was. People came up with all sorts of wild guesses, up into millions to one. What makes this even more bizarre is that recognising that a three-tumbler lock has 1000 combinations doesn't really need any understanding of statistics at all, just a basic knowledge of counting.
A popular set of statistics used in abstinence-only Sex-ed classes will claim that condoms "only have a 50% chance of even having the ability to prevent [insert pregnancy/STD here]", which is possibly the most confusing way to state a statistic possible. What are they really saying? Simply that a condom that isn't used won't work at all.