In Neon Genesis Evangelion, Dr. Akagi and her subordinates were fond of proclaiming that the chances of something necessary/spectacular would happen were zero-point-(Eleventy Zillion zeroes)-one percent; naturally, these things always happened.
Played straight in Mermaid Saga. However, in this case, the million to one chance of mermaid's flesh granting immortality probably is really a million to one. Over the last eight hundred years, only four appear to have gained complete and uninhibited immortality.
GaoGaiGar actually lampshaded it with its famous catchphrase: "With courage, 1% becomes 100%!".
TTGL may warrant its own subtrope of this; the Million to None Chance.
A smattering of All There in the Manual makes this happen in Death Note: Tsugumi Ohba claims that any time L throws out a percentage chance of Light being Kira (normally 4 to 10 percent), it's really over 90%.
The premise of Detective Conan lies in the Million To One Chance that a poisonous pill would shrink him into a 6-year-old body instead of killing him. Fortunately for him the creator of the drug was the other person to shrink, so that she could help him.
Exploited in every episode of Yu-Gi-Oh!, and all animes based on card games, for that matter: The Magic Poker Equation is in effect. Although technically, in most cases the chances of a player "miraculously" drawing the card he needs at the right moment (a "topdeck" as some players call it) is one in 40, much better odds than a million to one.
Also happens near the end of Code Geass. For Lelouch and Suzaku to break through the shield of and thus board the Damocles, they have to fly through the opening in the shield made to fire F.L.E.I.J.A.s. In order to do that, they essentially have to disarm a fired F.L.E.I.J.A. using Nina's F.L.E.I.J.A. Eliminator, but that requires real-time environment conditions to be input by Lelouch for 19 seconds before detonation and the Eliminator has to hit within .04 seconds of detonation. Odds of success: outrageously low, but the definition of this trope can tell you what happened...
This was probably not used to show how miraculous the move was, but to show how skilled Lelouch was and how disciplined Suzaku was. The Million to One Chance bit probably failed to calculate that Suzaku would magically know the exact moment to throw the eliminator thanks to the Geass effect.
At the end of Rocket Girls, the main characters, falling from 3000 km above the Earth, are expected to disintegrate and burn during reentry. However, they figure they can skip over the top of the atmosphere to shed the excess momentum that threatens them, but that doesn't work when Akane faints from the G-forces; since she can't provide the timing for rocket burns they need, Yukari has to fire and time the burns by instinct. The result is this charred body of the Mangosteen's pod... but somehow their luck held up and they survived.
Averted with Kaiji; whenever something extremely unlikely happens, it's always due to cheating of some type. When not Kaiji doing so, he tends to blame himself for not having good enough luck!
Subverted in a chapter of Franken Fran. A Born Lucky serial killer dies to the astronomically high odds of being struck by lightning.
Subverted in Puella Magi Madoka Magica; Kyubey implies to Kyouko that it's theoretically possible to make witch!Sayaka human again, but extremely unlikely that it would work. Kyouko tries it anyway...and fails, because there's no indication that it is possible either. There's just no evidence that it isn't.
In Superhero comics, one-in-a-million chance is code for 100% certainty.
In the Disney Ducks Comic Universe, Gladstone Gander's luck usually beats all odds. Huey, Dewey and Louie even tried to use it against him in the Don Rosa story "Oolated Luck". Donald and Gladstone entered a raffle and Donald's nephews had one entry ticket with Donald's name and the others for Gladstone. Donald won the first prize (a trip in a cruise ship) and Gladstone won the second prize (a lifetime supply of oolated squigs). It turned out Gladstone was lucky for not winning the trip since the ship got stuck in an iceberg. (The Lucky Bastard didn't seem to be concerned for his cousin) And one of the squigs had swallowed a diamond.
In the story Yet Again With A Little Extra Help, during a swordsman tournament in the Land of Iron, Sasuke and Tenten are facing each other, and Naruto makes his bet. The two fight out to a tie, and Naruto won all the money on that fight.
Scabbard: You flipped a coin when you placed bets on this fight didn't you.
Naruto: Yeah, you got me.
Scabbard: And it landed on its side... didn't it?
The Powers Of Harmony: One of the abilities of the Element of Laughter is to invoke this trope to the bearer's advantage or that of their allies. Inversely, it can also be used to cause your enemies bad luck.
The odds might be better than a million to one in this example, but in the tournament finals of ''Yu-Gi-Oh The Thousand Year Door Redux, both Drake and Kyle are in a no-win situation. It became obvious days ago that this was Not Just A Tournament, and whoever wins the duel between them will duel one of the Queen's three Designated Victims, a situation which, judging by what happened in the first three duels of the semi-finals, will be bad for whoever wins their duel, and worse for whoever the winner duels next. But Kyle does have an idea, although he doesn't have much time to explain it to Drake:
Drake: This better be a good plan, Kyle.
Kyle: Actually, it's a stupid plan, As in, the type you try when you're desperate, and people tell you 'it would never work', but it's the only thing you can think of because you're in a pretty bad situation?
Drake: Like the kind we're in now?
Kyle:: Exactly. Still, I seem to remember that when Kaiba gave Yugi that Fiend's Sanctuary card before his duel with Marik, he claimed it would improve Yugi's chances from 3% to 20%. Not the best improvement, but it was enough. So start hoping that slightly improving the odds in our favor will help me pull it off…
In Titan A.E., the main character space-floats around a docking bay, when his friend warns him against the chance of an actual docking ship. He responds with, "Oh, please, the chances of a ship docking here are a thousand to one." He then turns around and utters the line, "...and that would be the one."
In the first Ghostbusters movie, our heroes manage to stop Gozer by crossing the streams of their proton packs. Egon had earlier warned his comrades against "crossing the streams" on pain of being vaporized, but here states that there's "a slim chance" they'll survive. Since they're the heroes, they do.
Played totally and completely straight in Baby Mama, where Tina Fey's chances of conceiving are a million to one. She goes out and gets a surrogate etc, but by the end of the movie, guess who's pregnant.
Subverted in Dumb and Dumber. When the woman Lloyd is in love with tells him their chances of getting together are one in a million, his response is "So you're telling me there's a chance? YES!" Later he's outraged to find she's married: "What was all that one in a million talk?"
In Run, Lola, Run, Lola wins back-to-back spins at roulette on the number 20. Not exactly one in a million, but one in 1,369 — A 0.073% chance of happening. This film is an interesting case. Because multiple outcomes of the same situation are shown, they probably all happened. The other possibilities are just not shown.
Also works as a meta-example. When the scene was shot they just shot her playing and intended to do a staged shot showing the ball landing on the 20. When shooting the shot of her playing, the ball landed on 20. They threw it in.
The Gamers gives us a twenty-to-one chance - mainly on account of there being no one-million-sided dice.
At some point in Animorphs the reader may sit back and think, "Hey, wait a second. Yeerks are a race with insanely superior weapons. Not only that, but anyone can be a Controller. And this is a worldwide invasion. The heroes are six teenagers who live in a small town in California that can turn into animals? How can they stop the invasion? A bunch of animals couldn't beat the U.S. Army, never mind the Yeerks." This is lampshaded many, many times throughout the series, as the kids admit that at best all they do is slow down the Yeerks from time to time. They mostly lose battles and they agree that they'll never really be able to beat the Yeerks. They do eventually win, due in large part to the morphing technology being so dangerous and versatile. Rachel sums it up pretty well during David's betrayal when the kids are reflecting on how hard it is to kill an Animorph:
Rachel: "Just us. Just us against an enemy that could become any living thing. An enemy that could be anywhere, at any time. An owl in a tree, a spider in your house, a cat in the night, and then... Then, when you were unprepared, when you were vulnerable, a lion or a tiger or a bear.
I was starting to see why Visser Three hated us so much."
Memorably invoked in the novel Guards! Guards!, where the Genre Savvy Watch try to make the odds of shooting a Dragon in its "voolnerables" exactly a million to one through various means such as blindfolding the archer, putting soot in his face and making him stand facing the wrong way on one leg while singing the Hedgehog Song — but end up with some other, non-specified, incredibly low odds instead, which isn't improbable enough and thus predictably fails. Then played straight (and lampshaded by the narration) immediately after when the annoyed dragon retaliates: Their chances of surviving that turns out to be exactly a million to one.
Later on, they succeed in another million to one goal (well technically their pet swamp dragon does).
In the very next book Eric, Death informs the mages that Rincewind has exactly one chance in a million of returning from the Dungeon Dimensions. At this stage, the book doesn't even bother to remind the reader of the rule.
Mentioning the odds here is more meant to refresh and emphasize Rincewind's Action Survivor nature rather than allude to the fact he's due to return.
Equal Rites, the third book in the series, is the first one to state that "one-in-a-million chances crop up nine times out of ten", because of narrativium
Viciously questioned in The Last Hero by Cohen, one of the last surviving barbarian heroes. When he meets the Lady, he tears into her about how many of his fellow heroes died due to taking million-to-one chances that failed them. As he points out, she's the million-to-one chance, and also all the chances of dying. As such, Cohen has no respect whatsoever for her.
Not just Million to One, infinity to one, hence the name of the drive. Million to One Chance events are generated by much simpler (finite) improbability generators, and it is by using one of these that the Infinite Improbability Drive was brought into existence; since such a device is a "virtual impossibility" but not completely impossible, its existence was a finite, calculable probability. The man who programmed the computers to calculate the precise probability of the Drive's potential existence wound up discovering the exact scientific principles behind the Drive in his results.
In the Hitchhiker's guide universe, most probabilities are distinct (ie rarely do two highly improbable events have the same probability) which is why it's possible to specify the event you want to occur using a finite/infinite improbability drive. You induce all events with that probability, but usually there's only one or the side-effects of the others are trivial or at least benign. What's even more mindboggling is that the probability that a particularly notable improbable event will have a probability whose digits are particularly notable such as Ford and Arthur being rescued / 1 over 2 to the phone number of the party where Arthur first met Trillian is itself a number which can be plugged into the drive.
In addition, any event that is infinitely improbable is most likely to instantly happen as soon as the field is turned on.
Subverted viciously in the Doctor WhoExpanded Universe novel The Book of the War, where the Remote, a race whose hat is being a living receiver for mass media, have been primed for battle by being exposed to endless transmissions glorifying battle in an effort to make them spontaneous and unpredictable. What the architects of this plan didn't realize was that the transmissions were so formulaic and full of cliches that it had the exact opposite effect. As a consequence, the Remote's first assault had the entire army take the Million to One Shot against the Evil Tower of Ominousness.... and get mowed down by sniper fire. Also an interesting case of Death by Genre Savviness.
The gods will be forbidden to help you, and none but a mortal can restore you to heaven, and at a conservative estimate the odds against somebody pulling off a trick like that are one in ten thousand billion trillion.
Of course, it happens by the end of the book.
In Jacqueline Carey's Kushiel's Legacy books, heroine Phèdre nó Delaunay is very frequently told (or admits herself) that her plans are madness and suicide. Yet, with the exception of a few Unwitting Pawn moments, they always work.
Played straight as an arrow in "Day of the Ants". The one ant that was carrying a message from the humans trapped under the colony in the mad scientist's lair finds its way to the protagonist, who happens to be the cousin of one of the trapped human. The friend who was with her lampshades this by saying that even if it's a one to a million possibility, there's still one chance of it happening.
Lucifer's Hammer: Odds of the Hamner-Brown comet hitting the earth start out at a zillion to one, then they're a million to one, then they're a thousand to one, and then Oh Crap!
Death Lands. In "Crater Lake" Finnigan faces off with a sec man holding a laser pistol that's so unreliable it only fires one time out of every hundred. This makes Finnigan cocky. And rather gruesomely dead.
In Charlie and the Chocolate Factory, there are only five Golden Tickets in the entire world to be found amongst the millions upon millions of Wonka Bars on store shelves. People with the means to buy many candy bars per day still don't have a guaranteed shot at finding one, so Charlie Bucket, whose family is so poor that a bar of chocolate is a once-a-year birthday treat for him, seems to have no chance at all. Ultimately, he gets four chances at finding a ticket, and the fourth time is the charm — on the day before the tour to be held, no less. This is such a good example of this trope that more recent adaptations tend to lampshade/play with it, as in the 2013 musical — Willy Wonka, while getting acquainted with the tour group, asks Charlie "Aren't you the boy who got his ticket at the very last moment? Don't leave it so late next time." Taken even further in the first movie adaptation since Charlie thinks the last Golden Ticket had already been found when he bought his two bars. The moment he starts unwrapping his second bar, a news broadcast reveals that the last ticket was a forgery. The only reason he could even afford the chocolate bar with the ticket was because he just happened to find a discarded fifty-pence/dollar bill (practically a fortune to Charlie due to his family's poverty) on the street.
Live Action TV
Spoofed in Red Dwarf with the luck virus. Lister, Cat, and Kryten travel to a certain planet to investigate the death of a bacteriologist. While there, Lister is injected with the virus, which allows him to pick all the aces out of a deck of cards. He also tries to hit a dartboard's bullseye while throwing over his shoulder, but the virus wears off and he instead hits the center of Kryten's forehead. In the climax of the episode, Rimmer goes insane and locks the three in quarantine, slowly draining out the oxygen. Due to the virus however, Lister successfully guesses the code to unlock the door, then manages to trip over the necessary components to cure the insanity while running backwards down a hallway.
Lister, in another episode: "The chances of it happening are one in..." [big explosion] "...one."
In the Doctor Who story "The Creature from the Pit", the Doctor's companion Romana tells him that the odds are 74,384,338 to 1 against his Crazy Enough to Work plan actually working. He tells her that 74,384,338 is his lucky number. (The plan works, of course.)
A Seinfeld episode has Kramer marvelling at stories told by proctologists, which all end with the statement "It was a million to one shot, doc. Million to one," since none of their patients want to admit how whatever object is stuck really got there. At the end of the episode, Frank is the victim of such a million to one shot, falling on a pasta sculpture Kramer had made. Naturally, he leads with the same statement.
In "The Naked Time", they have to implode the anti-matter engines to prevent themselves from crashing into a planet that is itself imploding. The procedure has never been done before: Spock and Scotty have only minutes to prep the engines, and they only have one shot to get it right. Kirk insists, "We've got to take that 10,000 to 1 chance!" They do. It works.
A rare aversion is in "Errand of Mercy". Kirk decides to break into the Klingon headquarters and kidnap the commander. Spock calculates very long odds against success. They get as far as the commander's office—and are captured.note Because the Klingon Empire is a Police State, everyone—including the commander—is under constant surveillance. The commander's Monologuing is actually playing for time until Klingon security arrives. Kirk and Spock are only saved by the Organian Deus ex Machina.
In Covert Affairs at the end of season 1, Annie is undercover in London and needs to lose a lot of money, in a non-suspicious way, so that she will not be allowed to the leave the country. She loses all her money playing craps, then takes out a loan for a few thousand dollars, and bets it all on snake eyes -which is very risky. She acts confident that she'll win, but is secretly hoping she'll lose. The roll comes up snake eyes. She cheers, but internally she's freaking out because the whole point was to rack up massive debts. She bets it all on snake eyes again, and his time, she loses.
This sometimes makes the difference on MythBusters when a myth has to be busted. While the possibility that an occurrence could happen, the myth's conditions are so difficult to replicate beyond pure fluke that the myth has to be busted (usually when dealing with myths that question the practicality of some far-fetched idea).
In The Hitchhiker's Guide to the Galaxy, Arthur and Ford are rescued by the Heart of Gold seconds before they would have died after being ejected from a Vogon ship. Arthur states that the chances against it were astronomical after Ford tries to act as if he was counting on it as a certainty. Because Arthur had lived his entire life on Earth and Ford had been stuck there for over a decade, the ensuing weirdness such as an infinite number of penguins with a revised script for Hamlet didn't tip them off to the fact that they had an advantage in the form of the Infinite Improbability Drive.
Not at all rare in the Stargate Verse, but a particularly well-lampshaded version happens in Stargate Atlantis when Rodney and Jeannie almost destroy an alternate universe. The goal is to draw energy from a parallel universe but the process would create dangerous exotic particles in the alternate reality. Carter assure them this won't be an issue, though, since "the odds of us choosing at random one that’s inhabited are astronomically slim." Surprise, surprise: they choose an inhabited one. And not just any inhabited universe, but one similar enough to their own that everyone in the main cast has counterparts there.
In an episode of Gilligan's Island, a radioative meteorite falls on the island, and the crew is in danger of the radiation killing them. They learn of an incoming storm, and the Professor gets the idea of trying to put a lightning rod on it and hope lightning strikes it, but admits the chance is a million to one. When the Skipper asks what could be worse than that, Gilligan suggests, "A million to none?" Realizing that he has a point (slim chance is better than no chance) they try it, and it works.
Invoked in Castle when Beckett was trapped on a pressure-sensitive bomb;
Beckett: Castle, please, you have to leave me, there's no reason for both of us to die. Castle: Oh, I didn't come here to die, I came here to diffuse the bomb. There's still a chance. Beckett: Yes, a one-in-one-hundred-thousand chance! Castle: Great, while there's still a chance, I'm not giving up.
This trope is very common among the top babyfaces in Professonal Wrestling. So common, in fact, that the fans became very Genre Savvy when WWE attempted to invoke this trope on John Cena. Thanks to this, his already simmering and vocal Hatedom just got louder, citing his status as an Invincible Hero, and each pay per view ended up as a monthly "pay to watch John Cena overcome impossible odds once again" event to them. The culmination was at 2006's New Year's Revolution pay per view, where Cena was the first in the Elimination Chamber, surviving through 5 men while having about 90% of the fans in attendance against him. The crowd seemed ready to riot. Then Edge, one of the most hated men in the company at that time, backdoored his way into a match right after this and subverted this trope, defeating Cena to win his title despite being fresh and Cena having just fought through five men in a steel cage. The crowd was so happy to see this subverted that Edge became one of the biggest stars in the company almost overnight. Cena's hatedom has since cooled down when WWE downplayed him being a one-man odds beating machine.
Played with in Penny Arcade Adventures Episode 1, where the chance of a summon working is stated as 1 in millions (if not billions), but it's really more like 1 in 4.
The final battle of Star Fox Assault has the final boss mention a zero chance of victory for the protagonists. It is entirely possible to win the battle; however, the announcement of the zero chance is in fact an attempted Mind Screw by the Big Bad.
In Portal 2, GLaDOS gives this evaluation of the player character's chances of winning the final battle:
GLaDOS: I'll be honest, the odds are a million to one, and that's with some generous rounding. But if we're going to explode, let's at least explode with some dignity!
In Discworld, Rincewind has to collect a number of Plot Coupons (tattoo, sword that goes ting, secret identity, camouflage...), but not before determining, with the aid of Nobby, which ones would land his chance of success at exactly a million to one.
Actually a recurring theme in the When They Cryverse. In Umineko No Naku Koro Ni, Kinzo relies on a huge gamble involving the epitaph to give Yasu the headship and get them to forgive him as evidenced by the quote on top of the chapel: "You will only be blessed at a probability of a quadrillion to one." In the same EP, Bernkastel tells Lion and Will that the chances of Lion existing in a fragment is about 1 out of 2,578,917. And later on, cruelly reveals that in all those fragments, Beatrice/Lion suffers the same dead-end fate.Not that it stopped either Will or Lion.
Abner: There's a million reasons why that is not going to work.
Dimo [referring to himself, Maxim, and Oggie]:Dun vorry. Dere's three reasonz it iz.
In Homestuck, Vriska's entire plan to take out Jack Noir relies on rolling the best possible result on eight magical eight-sided dice (a 16,777,216 to 1 chance), unleashing their ultimate magical attack. It would have worked — because she'd been "stealing ALL of the luck" from her entire team.
Haley: Exactly! It was totally dramatic! How did I miss?
Spoofed and lampshaded again in #584. Being even more Genre Savvy than the rest of the group (Durkon's response to the above is "I think maybe ye been spendin' too much time wit Elan"), Elan realizes that a 10% chance of an imp summoning a demonic ally isn't anything to worry about, but a million-to-one chance of the imp summoning a monster that could actually kill them is a sure thing. This results in one of the page quotes.
Vaarsuvius: (sigh) And once again, Probability proves itself willing to sneak into a back alley and service Drama as would a copper-piece harlot.
Justified in Schlock Mercenary when Gav goes through a wormgate and immediately dies at the hands of a hostile guard that he had sent through the wormgate before. And meets up with his scientist buddies out of harm's way. Turns out that Gav went through the wormgate when it was set to send duplicates to all the other wormgates - meaning that Gav hit both those million-to-one shots by simply playing the odds 950 million times.
Aylee Bot: "There are now 950 million of you. According to the latest available galactic census data, blue-haired, Caucasian human males are now the largest single sapient ethnicity in the galaxy. You outnumber several entire sapient species. You are no longer merely human. You have become your own, weird demographic."
Minty Fool: Finally gamers will be able to calculate even the slimmest odds.
Miss Guided: Like their chances of ever finding dates?
In Tabletop, The odds of Ryan Higa's feat of rolling ten brains on his first turn in Zombie Dice approach infinity. Wil Wheaton speculates that, in order to balance his luck out, somewhere in the universe a planet spontaneously imploded.
When the Game Grumps played the Playstation 2 Wheel of Fortune game, Danny landed on "Lose a Turn" four times in a row. As one comment points out:
Supermutant6112: Alright, there are 24 spaces on the board. Every single turn, there is a 1 in 24 chance of landing on the "Lose a turn" spot. This means that Danny had a (1/24)^4 chance of getting FOUR turns lost. That is one in three-hundred-thirty-three-thousand-seven-hundred-seventy-six, or, to put it in simpler terms, three in a million.
Inverted in an episode of Dynomutt Dog Wonder. Dynomutt is reprogrammed for evil and given the task of killing his master, Blue Falcon. Falcon, knowing Dynomutt'stendencies all too well, announces that the odds are a million to one... in his favor. (And yes, the odds do hold up this time.)
Played straight in Thunderbirds, of course. "That thing [aircraft heading towards the tower they're in] is never going to hit us! It's a million to one chance!" Guess what happens. Go on.
Subverted in the first episode when the authorities try a desperate plan to board a plane in mid-air and the chances are explicitly described as a one in a million chance. As it happens, it doesn't work.
Futurama hangs a lampshade on it when Fry tries to use the Earth's most powerful radio telescope to find Bender somewhere in the universe. After spending a few days searching, Leela notes that even if he spends his whole life looking for Bender, he'll have searched less than a ten-thousandth of the universe. As he gives up, he spins the viewfinder aimlessly and says "I wish I had Bender back!" - which just happens to reach the ears of a God-like being that is conversing with Bender, and promptly sends him back to Earth. Leela's comment on this is priceless:
Leela: "This is, by a wide margin, the least likely thing that has ever happened!"
Futurama has another brush with odds when Bender becomes an Ultimate Robot Fighter. After his manager notes that his antics are getting stale, he is pitted against 500-ton Destructor while wearing a tutu. The announcers note that the odds on Bender are infinitely low:
Rich Little: "The Vegas odds tonight stand at an unprecedented 1000:0. A bet of zero dollars on Bender pays $1000 if he wins. Still, very few takers."
George Foreman: "It's not- not a smart bet."
It ends up being subverted as Bender loses when Destructor falls on him.
This man—it's a "45 billion to one chance", rather!
It turns out, quite a lot of things in real life exhibit this trope. Many events that are assumed to have their chances controlled by the normal bell curve distribution, have actually been found to obey Mandelbrot's (yes, he of the pretty fractal patterns) fat-tailed distribution. Which means that improbable events actually happen more often than statisticians expect.
The managers of the hedge fund, Long Term Capital Management, said that there was only a one in a billion chance that their investment strategy would falter. Four years after it was founded, it collapsed, nearly taking the world economy with it.note One major reason for the collapse was that LTCM was no longer following that strategy, but rather a variant of it. Another reason is that they gamed their strategies using computer models, forgetting that actual investing is done by human beings, who do not always act with complete rationality.
An exception to the rule is perhaps worth mention here. One scientist has said that the Large Hadron Collider has less than a one in 10^billion*billion chance of destroying the world. Far more energetic events (meteoric impacts, volcanic eruptions, earthquakes, etc.) take place on Earth on a regular basis. Yet the invocation of thistrope to real life makes that chance seem far scarier than it should. (The same scientist said, likely tongue-in-cheek, that one has a one in a hundred billion chance of spontaneously evaporating while shaving. It's probably less than that.)
The Russian mathematicians Irina Aref'eva and Igor Volovich did the numbers and it turns out its rather more likely that someone in the future will lock onto one of the incidental wormholes the LHC's collisions might make for passing moments, and use them as a 'year zero' for a future time machine to come visit us.
Richard Dawkins mentions in The God Delusion that if the chances of life arising on any given planet was one in a billion, then life would still have arisen on a billion planets in the universe, using one of the lower estimates of a billion billion planets in the universe (the actual figure is probably much higher). This astonishing figure relies on the fact that while a one in a billion chance is much too small for humans to comprehend comfortably, with a large enough sample a billion to one becomes a certainty. And the existence of life on this planet is an example of this certainty.
The chances of life arising (and forming a technological civilization) are collected in the Drake Equation, which predicts how many technologically advanced civilizations should be in existence. The problem is that the vast majority of terms in the equation are probabilities which require broad speculation. As we learn more about the nature of the universe, the uncertainty of terms in the equation goes down, which is not the same thing as the probability of other sapient, technologically advanced life existing.
There's also a discussion on the definition of probability here, where frequentists will claim the true probability is the same - it's just calculated it incorrectly. Another group disagrees and has a different definition. But that's a separate discussion.
Rare diseases deserve a mention here. It's true that there are diseases that affect only one in every 10,000 or 100,000 people, but it's also true that hundreds of these conditions have been identified, and there are almost certainly hundreds more that are as yet unidentified. So don't be shocked if you or someone you know is diagnosed with a rare disease - perhaps that disease is rare, but rare diseases in total affect a lot of people. Odds are you know or will know someone with one.
And so, Television Is Trying to Kill Us. The real-life lesson taught to all medical providers is summed up as, "When you hear hoofbeats, think horses, not zebras." Unless you're in Africa. In other words, when a patient comes in with a set of signs and symptoms, start looking for the common sources of those signs and symptoms before you start thinking about weird diseases. From a patient care viewpoint, it makes sense; if someone comes into the ER with low blood sugar and altered mental status, starting immediate treatment for diabetic shock is almost always better than figuring out all the other strange ways the patient might have low blood sugar and AMS. The problem is that thanks to Hollywood and the news, "millions of diabetics averting catastrophe and remaining fairly healthy thanks to managed diet, exercise, insulin," isn't news, but "man dies of rare illness doctors mistook for diabetes," is. The public winds up Wrong Genre Savvy, often thinking they have weird diseases rather than simple ones, or else expecting whole battery of tests when the answer is obvious from just a couple of blood tests and a quick examination. Dr. House hasn't helped matters much, and countless medical practitioners subtly curse that show (even if they're some of the biggest fans). So (should you be an American), yeah, you will likely know one person in your life with Tay-Sachs, Gauchier's, Huntington's, or something equally rare. Almost everyone else you know will die to cardiovascular disease, malignant neoplasms, kidney or mental diseases, liver failure, the flu, and the occasional accident, murder, or suicide.
This is pretty much how national/international lotteries can be successful. The chances of winning might be one in 75 million, i.e. all but impossible. But with hundreds of millions of entrants, there is in all probability going to be a few winners of the vast jackpot, which sells the dream to the next hundreds of millions of entrants.
When dealing with continuous random variables (almost anything with an infinite number of possible outcomes), each individual outcome will occur with probability of one over infinity. They're not talking one-in-a-million, but so close to zero-in-a-million it can't be comprehended. However, one outcome always happens despite technically being impossible. Hence the word almost.
Consider a square with sides one meter long. Now consider the diagonal of that square as a line. The probability of a randomly chosen point within that square - remembering that that square has an infinite number of points within - being within a certain region is P = (area of region) / (area of square). Now, a line has one dimension. Hence, the line has a length, but zero area. Therefore, the P of a randomly chosen point landing on the line segment = 0/1 = 0. Yet it's possible. This is where mathematicians split hairs between ''almost sure'', happening with P = 1, and sure, happening always, no matter what. As a matter of fact, whatever point does wind up chosen had a P = 0 chance of being chosen, so it had an assigned probability of 0, for no-way-no-how, but still happened.
Whether the probability of such an event should actually be considered zero is subject to argument. There are number systems (for example, the hyperreal numbers) which permit "infinitesimal" values. This solves the apparent contradiction because then the probability of the events described above is not zero and it makes sense that they can occur. Such number systems bring in problems of their own, though, and currently suffer from a lack of popularity.
Fans of Table Top Games are usually woefully unable to appreciate what the odds of something happening actually are. Consider a popular house rule for some games where an attack is rolled on a 20-sided die. If a twenty is rolled, it is rerolled with a chance for massive bonus damage. If a second twenty is rolled, the target is dramatically slain. Now, let's say that each character is subjected to only twenty attacks in a game. The odds of any one attack pulls this off is only one in 400, so nothing to worry about, right? Well, over the course of ten gaming sessions, it becomes almost a 40% chance of being killed just by this mechanic. In a six person group, over ten sessions, it's more than 95% probably a player is instantly killed like this. And if you're playing the granddaddy of TT games, considering it usually takes at least four-six or so sessions to go up a level...
Magic tricks abuse this all the time. The reason that a magician's ability to always cut to your card and produce it on command seems surprising is because you don't perceive the trick. Even if you know it's a trick, the visuals help maintain that what you saw happen was what actually happened. Therefore, the odds that the magician can successfully produce your card is apparently one in fifty two. Of course, it's really almost one in one, and if you know the secret to a given trick, it's obvious why.
Playing with the odds here leads to a fun trick, wherein the target picks a card, and then the "magician" finds all sorts of bizarre ways to ensure that the card cannot be tracked... then draws the top card of the deck. 51 times out of 52, it's a joke trick. The fifty-second, you blow someone's mind. Extra fun to use on magicians who are watching for the sleight, and will never find it.
You'll need to know quite a few magicians in order to have a good chance of pulling it off in front of even one. Also, most magicians are wise to such sophistry and if irked can really make the performer look like a fool. Just ask him to repeat it at the next gathering. And the one after that, if necessary. Then show them up with a few actual tricks.
Affirmed in any case of the Birthday Paradox, where a group of just 23 people will have a 50% chance of having at least one birthday in common. The reason is because for every additional person added to a group you add a number of comparisons equal to the number of persons already in the group. Birthday Paradoxes often result in counter-intuitive outcomes with Unfortunate Implications. For example a DNA test that can narrow down a match between two random people to a probability of 13 billion to one will on average yield 100 exact matches in a sample prison population of just 65,000. Can also be used to attack high strength cryptographic algorithms.
Steve Whiteley bet £2 on a horse that had lost the last 26 races. It won the race and as the only person who bet on this horse he won a jackpot of £1,445,671.71. 
The odds of a random person being struck by lightning and surviving in any given year are 1:280,000. The odds of three brothers all being struck by lightning and surviving is 1:345,000,000. The odds of three brothers all being struck by lightning in the order in which they were born and surviving is 1:2,700,000,000. This very thing happened to brothers Jack, Aaron, and Nathan Helms of Apopka, Florida.
This is related to the rare diseases example above. A probability of 1 in 3 billion sounds very low, but you have to consider what the "positive" outcomes are. What if three cousins were hit in the order on which they were born? Or maybe in alphabetical order? If you count all newsworthy events you can quickly get a very reasonable probability.
The Helms brothers have nothing on Roy Sullivan. He was struck by lightning and survived on seven different occasions, the odds of which being a whopping 1:10,000,000,000,000,000,000,000,000,000.
NASA's management claimed that the risk of catastrophic malfunction on the shuttle was 1 in 100,000. After the Space Shuttle Challenger disaster, Richard Feynman immediately realized that this claim was risible on its face; as he described, this assessment of risk would entail that NASA could expect to launch a shuttle every day for the next 274 years while suffering, on average, only one accident.
The chances of one meeting a frighteningly similar-looking Doppelgänger, or Distaff Counterpart, to oneself on any given day are quite low but over the course of a lifetime the chances are good that people encounter a few.
Chances are good that someone out there shares your name, even if it's a fairly unusual one. Not only might they have the same first name as you, they might even have the same last name! People tend to pick names for their kids that they like and/or find meaningful, and there's a good chance that what your parents thought was an Awesome Mc Cool Name, another parent will too. Every year, there's a list of top-ten trendy baby names. Many parents also name their kids after a family member, a famous person, a saint, deity, or Biblical figure, etc.