As any student of cause and effect can tell you, a coincidence is when two or more events happen either simultaneously or in sequence, without any sort of obvious (and in most cases, inobvious) causal connection. Generally coincidences surprise us because, naturally enough given their nature, we just weren't expecting them. Most people know that sometimes things just happen at the same time. But some people just refuse to believe it. There are, after all, reasons people say "where there's smoke there's fire." They also often say "There Are No Coincidences."
Interestingly, this line can be made by both the Agent Scully and the Agent Mulder. One believes that the improbable has a simpler explanation and one believes that it has a fantastic one. Both are Genre Savvy in that Contrived Coincidence is something that should usually be avoided in serious plots.
Usually used either
to motivate investigation into possible reasons why an apparent coincidence actually sprung from a common cause — the Conspiracy Theorist has an advanced case of it, and no willingness to stop the investigation
Yuuko from Xxx HO Li C. This idea is her whole entire shtick and a major plot point, if not the major point of the series; though she also comments that the ultimate cause of an event may be an incident so minor that it's next to impossible to recognize the connection, even with supernatural help. She has a point — for example, the only reason that we, the readers, know that Doumeki's destroying one specific spiderweb while housecleaning is at all relevant to the story is that the manga devotes an entire page to depicting it, as opposed to the many other spiderwebs that have presumably been destroyed at some point or another without being shown. (Watanuki's job is being Yuuko's live-in slave, after all.) Unlike real life, fictional works have limited space, so only the significant events are depicted.
The phrase came before in an older work, Cardcaptor Sakura and even before the idea of fate had come before in most of clamp works, since the begining.
"See what you have to ask yourself is what kind of person are you? Are you the kind that sees signs, that sees miracles? Or do you believe that people just get lucky? Or, look at the question this way: Is it possible that there are no coincidences?"
In The Matrix, Morpheus believes in fate and prophecy, and thus does not believe in coincidence.
In Obi Wan Kenobi's experience, there is no such thing as luck, since The Force pervades everything.
In the Expanded Universe, this is acually a Force User's philosophy in general. If something really improbable happens, it's because The Force wanted it to go that way. The more coincidences pile up, the more the Force is at work.
V For Vendetta: "I, like God, do not play with dice and I don't believe in coincidences."
Later in that movie, V tells another character, "There are no coincidences, Delia. Only the illusion of coincidence."
In Kung Fu Panda, Master Oogway tells Master Shifu, "There are no accidents," in relation to Po being chosen as the Dragon Warrior. Master Shifu later repeats this to Po.
In The Mummy Returns, Rick tries to shrug everything off as a coincidence and Ardeth tells him "There's a fine line between coincidence and fate."
Elmo McElroy: No, Shit don't just Happen. Shit takes time. Shit takes effort. Twenty million dollars worth of effort.
In other words, your car breaking down just in time to strand you in the middle of nowhere as a snowstorm starts is a coincidence. A twenty-million-dollar drug deal being busted up by dozens of skinheads with automatic weapons just before you close escrow? That's somebody with more cash than he knows what to do with wanting to make your life hell.
In The Dark Knight Rises, Gordon tells Blake that, as a detective, "[he's] not allowed to believe in coincidences anymore".
Literature
Robyn from The Tomorrow Series believes that "coincidences are God's way of telling you to wake up".
This is repeated several times in the Young Wizards series.
In David Brin's The Uplift War, an alien is annoyed with the human word "accident," which is muddy in meaning, and the humans even say, "There are no accidents." At the end, contemplating the events that led to their defeat and thinking that if some of them hadn't happened, victory might have been possible — but, the alien realizes, "There are no accidents."
Tony Hillerman's detective Joe Leaphorn explicitly gives "There are no coincidences" as his philosophy in solving mysteries, as stated in The First Eagle for instance.
In Goldfinger, James Bond tries to convince Goldfinger that their third meeting is a coincidence. He fails.
In Graham McNeill's Warhammer 40000Ultramarines novel Dead Sky Black Sun, when Uriel meets Colonel Leonid, who can tell him what is in the Chaos fortress, Uriel tells him that it was not chance that brought him to meet Leonid.
In C. S. Goto's Blood Ravens trilogy, "Coincidences are for the weak-minded and the ignorant."
In Kate Seredy's The Singing Tree, when arguing that they should take Marton Nagy home despite his lack of papers, one argument is that it was obviously Destiny that brought them there to recognize him and jog his memory loose — they had only stopped there because a cat had stowed away in the cart and started to have kittens — and who are they to argue with destiny?
In John C. Wright's The Golden Age, Phaethon ponders whether a meeting is coincidence or arranged by the Earthmind, an AI with a trillion times the brain power of a human such as himself.
In Jasper Fforde's The Eyre Affair, when Thursday meets the activists who want her to speak against the war, they declare it can't be a coincidence.
The Xanth novels define "coincidence" as a word used by Mundanes to explain away something magical happening.
Good Omens. At first glance it seems that a bunch of random stuff (starting with the babies getting mixed up, and escelating from there) came together to interfere with the Divine Plan. It's not until it's over that the protagonists start to wonder if the Divine Plan they've been told about is the actualDivine Plan.
In Jack Campbell's The Lost Fleet, some characters declare it can't be chance that they found Black Jack Geary's survival pod.
In Michael Flynn's Up Jim River, when she hears of a murder of a woman she spoke with, the harper thinks it can't be a coincidence. She tells Donovan, and he agrees.
In the Liaden Universe series by Sharon Lee and Steve Miller, fate often seems to be an active agent with a will of its own. Characters refer to it as "event" or "the Luck," and it is said to move in strange ways around those of Clan Korval and Line yos'Phelium in particular, making them the ultimate Weirdness Magnet. Individuals who are Genre Savvy enough to recognize it (including those of Korval themselves) make allowances for it in their plans—Bechimo's builders who warned him to steer clear of Clan yos'Phelium in Ghost Ship, and Zaneth Katrina who wishes nothing to do with Clan Korval at the present time because the Luck is too unsettled in Dragon Ship. Those who don't believe in the Luck or the powers of the dramliza end up puzzled by the way Korval is always at the nexus of extraordinary happenstance and often assign human intentions to utter coincidence. (Or what would be utter coincidence if it weren't for the Luck causing it.)
For example: what are the odds that a half-brainwashed Agent of Change disengaging from a mission would meet, become companions with, and eventually lifemate a woman who happens to be the granddaughter of a missing member of a long-lost clan with whom his is allied—a woman who grew up on the planet that his cousin is shortly going to civilize so Clan Korval can move to? And that this woman's own clan's world is about to be invaded by Yxtrangi, bringing with them the very member of that race he had encountered ten years before? The entire series is one long chain of increasingly unlikely "coincidences". (It drives the Department of the Interior, and more than a few people who are actually in the know about Clan Korval's history with the Luck, right up the wall.)
This is a major plot point in Bridge Of Birds, with Master Li slowly realizing that all the unlikely events and reappearances of characters and side stories that keep happening to him and Number Ten Ox are the work of a god doing his best to clue them in on their Fetch Quest being part of a much bigger quest without breaking the rule of gods not being allowed to intervene directly in mortals' affairs. The chapter where he explains all this to Ten Ox is even titled "There Are No Coincidences in the Great Way of Tao".
This line is said repeatedly in Beautiful Creatures.
Just after Honor Harrington has made her grand escape from the prison planet Hades with over half a million escapees, apparently coming back from the dead to do so, Rob Pierre and Oscar St. Just discuss her apparent ability to have been in just the right spots to screw with Haven's war planning for a decade or more. Pierre admits that half his strategic analysts believe it's just coincidence. The other half think she's in league with Satan.
In Dorothy Gilman's The Clairvoyant Countess, Madame Karitska discusses the possibility with Mr. Faber-Jones, that his meeting with a criminal, enabling his capture, was no coincidence.
Comes up several times in The Wheel of Time, in particular thanks to three incredibly powerful ta'veren traipsing around, distorting probability with their presence. Invoked memorably when Verin tried for days to leave a particular town but was repeatedly flummoxed by random delays, until finally she decided that she was supposed to be there and settled down to see who would show up. In broader terms, resident pragmatist Gareth Bryne sums it up:
Once is happenstance, twice coincidence, three times a conspiracy.
Live-Action TV
The Romulans of Star Trek, as a species, do not believe in luck. If something goes wrong, it is because someone either intentionally made things go wrong, or else someone messed up and unintentionally made things go wrong.
Bashir: "Don't tell me you don't believe in coincidences."
Garak: "I believe in coincidences. Coincidences happen every day. But I don't trust coincidences."
On CSI: Crime Scene Investigation Gil Grissom has repeatedly said in a handful of episodes that he does not believe in coincidences. He even quoted Goldfinger at one point:
"Once is happenstance. Twice is coincidence. Three times is enemy's action."
However, in the episode "Chaos Theory", Grissom is forced, at the end of the episode's investigation, to accept that the death and disappearance of college student Paige Rycoff was the result of a series of unfortunate random events that, when taken in sequence, were only connected because, together, they directly led to the girl dying in a tragic accident that merely appeared at first blush, to be murder.
At the end of the aforementioned episode, Gil is forced to look into the mirror when the parents of Paige refuse to believe the explanation even with all the evidence, the one thing that Gil absolutely believes in an investigation, due to grief.
Doctor Tony Hill, from Wire in the Blood, has a psychological justification for not believing in coincidences:
"Jung's theory of synchronicity says there are links between events beyond cause and effect. Patterns the conscious mind can't perceive. [pause] There is no such thing as coincidence or accident."
Michael Garibaldi, from Babylon 5 prefers to make his own luck rather than just relying on chance.
Anthony DiNozzo: "In the immortal words of Leroy Jethro Gibbs [puts on a deep voice] 'I don't believe in coincidences'."
Gibbs: "You know how I feel about coincidences, Abs."
Abby: "Equatorial pygmies know how you feel about coincidences, Gibbs."
However, after Gibbs rules a group of chop-shop employees out of the investigation of a dead body found in the trunk of a car they stole:
Anthony DiNozzo: "I thought you didn't believe in coincidences, Boss."
Gibbs: "I don't. But I do believe in bad luck.''
Laverne in Scrubs. "Everything happens for a reason."
Then JD's narration comments on how this is subverted at the end of the same episode when Laverne is left in a coma by a random car accident
Then played straight in the next episode, Laverne holds on to life just long enough for her best friend Carla to get past denial of the situation and say goodbye
LOST's John Locke is of the "everything happens for a reason" sort. As things unfold in Season 6, he just might be right.
Mr. Eko also seemed to feel this way, advising John not to "mistake coincidence for fate."
Farscape references Albert Einstein just to hammer in, "Yeah, Scorpius has your number, Crichton, and he will never stop screwing with you!"
Harvey: Scorpius... Iz... Like Gohd! He doez not play dize vit ze univerze!
And Harvey, the invisible friend in Crichton's head is dressed like Einstein. Complete with Einstein Hair. Farscape's that kind of show.
The Doctor has a tendency to say this, usually while figuring out the plot. It's almost like he knows he's in a universe that has people writing the script. Or he's just had nine hundred years of experience in how these things work.
He didn't used to. The First Doctor was more likely to call things that were obviously enemy action coincidence than to call a coincidence fate. So the experience explanation seems likely.
One episode of El Chapulín Colorado featured the hero at an area where it's usual to have falling rocks from space. Not believing it, Chapulin sarcastically announced the arrival of one that made stops in Jupiter and Saturn. Then a rock falls and hits someone nearby. Chapulin dismissed it as a coincidence that could happen anywhere on the world and challenged another one to appear. It did, hitting the same someone. "Two coincidences", the hero said and then challenged again. A third rock fell and Chapulin decided to hide just in case a "fourth coincidence" appeared. Several rocks fell later, and the hero weakly dismissed them as coincidences.
Once Upon A Time: Other characters have this, too, but Baelfire explicitly states that it's one of the few things that he and his father, Rumplestiltskin, agree on.
Radio
Mr. Whittaker from Adventures in Odyssey believes this. (Although in a very early episode he comments on something being a coincidence.)
Videogames
In Legacy of Kain: Soul Reaver II, Vorador refuses to believe that Raziel's sudden appearance and the imminent destruction of the Pillars is a coincidence.
The quote is associated with a technology called "Probability Mechanics". One suspects that the nonsensicalness is rather the point.
In Record Of Agarest War, Duran thinks that it's just a coincidence that they found a hot spring on their camping site, but Winfield thinks the other way.
Winfield: Ha! Coincidence, you say? Divine providence, say I! The gods love us!
A repeated theme in The Elder Scrolls, especially Morrowind and Oblivion, but also present in the resolution to Daggerfall. The Septim dynasty, in particular, tends to find this out the hard way; Uriel Septim drops the phrase just seconds before taking a dagger to the back, while Martin Septim slowly goes from feeling in control of his destiny to believing things are pre-planned.
While not a complete rejection of coincidences, Kreia in Knights of the Old Republic II believed that true coincidences were rare and that events usually chalked up to coincidence actually happened because of the Force.
Bastila also lampshades it in the first game, saying such things are so commonplace to those who are active Force Users that they simply "get used to" it.
The reason Einstein did not like the quantum mechanics relating to chance was because of this. "Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He does not throw dice."
To which Niels Bohr responded: "Albert, stop telling God what to do."
Stephen Hawking's take on the matter is "Not only does He play dice, He sometimes throws them where they cannot be seen!"
Technically, he didn't believe in randomness beyond the laws of physics and the boundary conditions of the universe. Those two things can still cause some pretty coincidental stuff to happen, just like pseudo-random number generators can cause coincidences in computer games.
The Yoruba tribe in Africa believe there is only one reason people die: witchcraft. They refuse to accept such things as "natural causes" or "horrible accident" as causes of death.
A Soviet-era phrase, "This was no accident, comrade," referred to both the Marxist ideological notion that powerful and inexorable historical forces drive human progress, and that more mundane human agents (i.e., the KGB or other parts of the totalitarian state) were secretly behind events that seemed to occur by happenstance.
It was also said that every major accident has a first, second and last name.
