A specific case of Million to One Chance: the laws of probability are nothing compared to the power of Narrative Causality.
Often seen in franchises involving space travel, possibly because Sci-Fi Writers Have No Sense of Scale and can't figure out how big a single planet is, let alone an entire galaxy. It often occurs in real-life settings as well, albeit on a smaller scale.
As a corollary:
If you're instructed to find something on a planet, and don't know where on the planet that something is, just land at a random spot. Most likely, you will land not far from your destination. If you were told to seek a person, they may even find you themselves.
If one of your friends has a long-lost relative they last saw on some distant planet, be assured that you will bump into said relative shortly after landing on the next planet, whatever it happens to be.
Every planet will have one capital city or another "spot of activity", and controlling that spot means instantly conquering the entire planet. These planets are also almost invariably 100% monocultural, with only one government that needs to be factored in to any plan.
Kanon. Although this may be due in some small part by miracles, it's still damn unlikely for Yuuichi to run into people whenever he steps outside.
In InuYasha, Feudal Japan appears to be populated by a total of about twenty people, all of whom are at any given time within convenient brawling distance of one another.
In Star Blazers, the man that Queen Starsha rescued from the Gamilon ship that crash-landed on Iscandar just happens to be Alex Wildstar, Derek's long-lost brother, presumed dead.
Magical Girl Lyrical Nanoha has some pretty unlikely meetings, such as Arf and Alisa Bannings, or Hayate, Fate, and Nanoha in season two. Vivio also happens to run into Riot Force Six rather than, say, a police officer.
An extremely mundane example appears in episode 11 of Servant × Service. When Saya goes out to town on her off day, she muses about the possibility of meeting people she knows and ends up meeting practically the entire cast (with the exception of Touko, who's studying at home) in various places.
Inverted in Flash Gordon. Mongo was a big, extremely multivaried place. Then again — at least in the early days of the comic and most TV & film adaptations — the main characters are stranded on Mongo and can't visit other planets, so it makes sense for Mongo itself to be portrayed as a richly diverse world.
In NYX it turns out that the gang banger who gunned down protagonist Kiden Nixon's father right in front of her as a child, is none other than X-23's pimp and now major criminal and gang leader in his own right, Zebra Daddy, who has been chasing them for almost the entire second half of the story.
Star Wars, of course. In The Empire Strikes Back, Luke is told to look for Yoda in the Dagobah System. That's all he's told about where to find Yoda. And not only does he get the right planet, but he even lands within a mile or so of Yoda's hut.
The EU justifies this particular example on both counts. The Essential Atlas states that Dagobah is the only Earthlike planet in its star systemnote the other four are respectively volcanic, airless, iced-over, and a gas giant, and the vast majority of intelligent species in the Star Wars galaxy are carbon-based and oxygen-breathing. And in Heir to the Empire, Luke theorizes that Yoda used the Force to scramble his X-Wing's sensors and guide him to the correct location.
In another Star Wars example: In A New Hope, despite the surface area of the Death Star being millions of miles, Luke and company end up within jogging distance of everything and everyone of importance to their errand - when they had no control over what part of the space station they were going to end up inside.
In Star Trek V: The Final Frontier, after passing through the Great Barrier, the heroes land in a random spot on a random planet. After wandering around for a bit, they find the exact spot where the temple-thing comes out of the ground.
In Star Trek, when Kirk just happens to run into Spock Prime while marooned on Delta Vega. The latter was sent to a location where they could observe a certain unexpected astronomical event, while the former was, presumably, dropped within walking distance (or maybe getting eaten distance) of a Starfleet base, with no reason whatsoever for the two locations to be anywhere near each other — apart from the Theory of Narrative Causality, of course. Lampshaded in the novel adaptation.
In Enemy Mine the Human and the Drac both manage to not only crash on the same planet, but within walking distance of each others spaceships.
In Pitch Black the ship crashes on the planet, conveniently within walking range of the settlement, though it was intended as an aversion. Ken Wheat, the original writer of the film with his brother, Jim, explained that in the first draft of their script The Ship had detected the Settlement and tried to land near there so as to be near an area where there might be supplies.
In The Windup Girl by Paolo Bacigalupi, Emiko is running for her life and looks certain to be killed when Anderson Lake just happens to be riding past in his rickshaw and rescues her.
Sheckley's Mindswap. That method of looking for Ze Kraggash actually pays off. Somewhat.
Harry Potter and the Deathly Hallows: Whilst jumping around the British countryside entirely at random, our heroes land within a few hundred meters of a group of people they know, and Harry just happens to wander by them while they discuss plot-relevant events. Britain, remember, covers some 210,000 square kilometers.
And in the film, they just happen to Apparate right in the middle of of a group of snatchers.
This happens in favor of the villains in the background of Goblet of Fire. Pettigrew decides to stop at an inn on the way to meeting Voldemort, and runs into a Ministry official who happens to know the location of a loyal Death Eater, secretly being held under house arrest by his father and assumed dead by the rest of society.
In Dan Simmons' Illium and Olympos this is justified and deconstructed. Everyone lives close to the teleporters all across the planet because there is no need to go very far from them. The unfortunate result is that they've managed to forget about the entire rest of the planet.
Older Than Radio: In Henry Fielding's Tom Jones (1749), characters who are travelling separately are forever running into each other at inns along the road. Critics have tried to justify these remarkably convenient coincidences by making learned references to the average speed of a stagecoach and the density of coaching inns along the major roads in Georgian England.
The Hitchhiker's Guide to the Galaxy: The first humanoid alien Arthur meets after he goes into space is a guy who crashed a party he'd been to. The second is the Earth woman that guy hit on at said party after Arthur had been chatting her up himself. They acknowledge that this is weird, although this is the Improbability Drive in action, so... A Sci-Fi Scientist Did It?
Some critics have pointed out that a good number of plot points in A Clockwork Orange are driven by Alex running into an old friend.
It happens on smaller scale in Feliks, Net & Nika. What are the chances that in city of more than million people children of two man working in the same top secret facility will turn out to go to the same (regular) school and to the same class?
In The Vor Game, Miles Vorkosigan is on a space station several wormhole jumps from home, and just happens to get tossed into a holding cell that contains his old friend Emperor Gregor. And later, on an entirely different station, he runs into General Metzov, the man whose career he had ended in the opening chapters of the book.
In The Warrior's Apprentice, one of the first people Miles meets after arriving on Beta Colony is the freighter captain that his mother had conned into giving her a lift off the planet eighteen years earlier. And then, several planets away from both their home worlds, he runs into Elena's mother.
Jaden Korr (of Jedi Academy fame) happens to be in precisely correct spot in all of space to intercept an Old Republican Jedi master who was flung into the future due to a hyperdrive malfunction.
Live Action TV
In Sliders, they would always randomly appear in the precise place and time where four strangers could, over the course of a few hours, completely alter the way of life on the planet. (We did briefly see the Sliders in universes where they had no particular impact, usually at the very start of an episode. Presumably, there were any number of such banal slides and the network was only showing us the interesting ones.)
Justified in Stargate SG-1. Very few civilizations see much advantage in venturing more than a few kilometers from the Stargate, it usually being the only way on or off the planet (or, for primitive cultures, being integrated in the local religion since the "gods" come through it).
Also justified in that they tend to mention towns and villages elsewhere on the planet. If you're just looking for civilization, and the civilization happens to cover the entire planet, "landing randomly" works just fine.
Stargate Atlantis, where those who live in deadly fear of the human-eating Wraith never move away from (or block) the stargate the Wraith ships emerge from - generally making it easy for the ships to fill their human quota in about half an hour.
Well, considering that when they do do those things, the Wraith come in from space and bomb the place to hell, it might make more sense.
Very obvious in Doctor Who, where the TARDIS never seems to land on the opposite side of the planet from wherever the local intrigue is going on. The episode The Doctor's Wife tells us that the TARDIS is doing it on purpose, even in the early seasons when the ship's flights were entirely random.
For example, despite having an entire planet to argue over, the Thals and the Kaleds apparently live within walking (or gliding) distance of each other in Genesis of the Daleks.
In one episode of Farscape, Zhaan searches for her missing crewmembers by asking a bartender on a random planet nearby. Because clearly there is only one bar on the entire planet which they could have visited if they had been there, which, thankfully, they did not.
A Pa'u Did It?
The first season of Heroes suffered badly from this. Characters just seemed to run into each other all the time, even when they came from distant places like Tokyo. Perhaps the most blatant example was when Hiro met Nathan at a roadside diner, and shortly afterwards, Sylar also happens to show up on it (in time to kill Hiro's new love interest.)
Actually two different diners. Nathan and Hiro were outside Las Vegas. Hiro, Charlie, and Sylar were outside Odessa, Texas. Hiro just loves waffles.
Cleverly lampshaded in the original Land of the Lost, where the artificial pocket dimension the Marshalls are trapped in is not only small, but warps over on itself, so that if you walk far enough in one direction, you will return to your starting point. The local "mountain range" is, in fact, just the endlessly repeated image of the same mountain, and if you stand on its peak and look at the neighboring beak with binoculars, you can see your own back.
Averted and then played straight in Battlestar Galactica (Reimagined). Starbuck crash lands on a barren moon and a big deal is made of how difficult it is to find one person on a planet when all you've got is "visual scanning". At one point they even show a map of the moon with the comparatively small area they've managed to search drawn on. They then throw this out the window by having Starbuck find a crashed Cylon Raider that apparently came down not far from where she crashed. Contrived Coincidence maybe, maybe not, as she personally shot it down before she crashed from damage it inflicted.
Later in the series, Starbuck makes various similar leaps with predictability especially during the final episode where she manages to make an FTL co-ordinate out of the song she and the Final Five kept hearing, just in time to avoid the collapse of the Cylon colony ship under nuclear attack, only for this to turn out to be a new habitable planet, precisely what the fleet had been looking for since the planet formerly known as Earth had turned out to have been nuked by humans attacking earlier Cylons. Not only do all of those ducks get lined in a row, but it turns out there are indigenous humans genetically compatible with the humans on the fleet, despite total biological isolation of the two populations prior to this episode. Lucky coincidence indeed.
Present in Angel season 2, in the Pylea arc. After lengthy discussion of how two people going through the portal might wind up halfway across the world from each other, -and- coming up with a plan to stop them from doing so, Angel, Wesley, Gunn, and Lorne get to Pylea and find they're a few miles from where Cordi ended up after her own trip through the portal. Cordi herself came through about the same place that Fred had, on yet another trip. Possibly justified in that it's mentioned that the portals need psychic energy to open. The Pyleans, and a few of the wild animals nearby, produce quite a bit of said energy.
There are over a dozen characters in Dino Attack RPG explicitly said to be ex-military, and somehow all of them were in the same platoon together.
Somewhat averted in Battletech. Most worlds sport populations in the hundreds of millions, and a few have them in the billions. Similarily, most worlds will have numerous cities and other settlements. That being said, there are remote worlds with almost ludicrously small populations (Some as low as a thousand) where there is only a single community on the whole world.
Both Averted and Justified in Traveller. From the point of view of your usual intrepid intersteller adventurer a starport is the only part worth thinking about on a given planet. However most planets are quite large and whole campaigns can be featured on just one of them.
Space 1889 sort of justified. Humans on Mars are not that common and tend to hang out among themselves, and Victorians usually socialize with and write letters to primarily adult people of their own gender and social status. Knowing a little bit about all humans on Mars of your gender and social class is doable.
Painfully obvious in Kingdom Hearts, where one "world" consists of only a single thoroughfare and one block to either side in Agrabah, and one spot in the desert with the Cave of Wonders, and another consists of just Captain Hook's pirate ship and the top end of Big Ben. Other worlds are similarly diminutive.
It's still bad, but gets better in Kingdom Hearts II, especially for the worlds that are redone. The new worlds, however...meh, could've been more expansive.
But wait! Arguably, Kingdom Hearts actually averts this. After all, being a giant Cross Over game, the worlds it contains are already fictional worlds mostly suffering from this trope, and therefore their meta-representations in KH are actual size!
In the Ratchet & Clank series, a great many worlds can be visited, and each one consists of a single action adventure zone, no larger than the levels in Sly Cooper, (which are all set in various parts of one single world). If it weren't for Chairman Drek'sEvil Plan and its importance to the plot, there would be no reason for space travel at all.
Averted with Metropolis, where the portion explored is completely different in the first game, third game, and Tools of Destruction.
Averted in Final Fantasy IV, where the heroes travel to the moon and find that, though it is indeed smaller than the normal world, as one might well expect of a moon, it nevertheless does have a fully detailed worldmap. It's just... rather sparsely inhabited. Again, as one might expect of a moon.
This is also because all of the humanoid inhabitants are sleeping below the surface, and the only other people living there, the Humingways, occupy one cave.
Averted in Haven: Call of the King, a game which goes to show exactly why it's played straight most of the time. In the later stages of the game, you're tasked with finding 12 hidden dungeons in order to get the best ending. You have a space ship, and have to check the game's several worlds for them. As these are full sized planets, it will literally take hours worth of flyovers in your space ship to find one, partly because your ship doesn't move nearly with the kind of speed you'd expect of an intergalactic vessel.
Mass Effect's planets generally consist of about a square kilometer of mountainous terrain. You can see areas beyond the tiny map, but you're not allowed to go there - and, at any rate, all the stuff on the planet worth exploring is within a short drive of everything else.
"You're leaving the bounds of the operational area, you're leaving our scopes, you need to turn around Commander" says Joker every time you try to go a little too far out. Although on one particular planet there's an annoying bit of ore that's JUUUUUUST outside the operational area and you have to very, VERY carefully inch over to it on foot or Joker picks you up and deposits you back at the "beginning of the level". Great scanners you got there, Normandy...
Everyone that you meet in the first game shows up in the sequel. EVERYONE. At least, everyone who isn't dead. Most of them are emailing you, though, and comment something like "Man, it was hard to get your address!" ...an address belonging to a terrorist organization wanted in all of Citadel space. Then again, given their penchant for plastering their logo on everything, it wouldn't be surprising if it were something like "email@example.com"
Played straight with searching for Liara. The smallest to which your superiors can narrow down her location is a sector with four navigable star systems. Although they do recommend starting the search on "the planet with the Prothean ruins", without even specifying its name. Likewise, Liara can only narrow down the Conduit's location to "somewhere on Ilos", and you only find it by locating Saren and airdropping right behind him.
The Knights of the Old Republic games take this trope to an even farther extreme than the Star Wars movies. Regardless of whether the protagonists land on a desert planet, an ocean planet, or a planet that is one big city (Coruscant-style), their destination is always just a few zones away, perfectly walkable on foot, even if they don't know its location. Also applies when their spaceship crash-lands in the middle of nowhere.
In the case of Tatooine(desert planet) and Manaan(water planet), there were only one setlement on each planet, so there really wasn't anywhere else to go. Also on Tatooine, when finding the Star Map there is really no way to know how far it was before you find the cave containing the map, especially given that you are unable to travel there without a map. Also on Taris(city planet), you and Bastila both ejected from the same ship at roughly the same time, meaning it would be highly unlikely for you to end up in different locations. Everything else that you encounter is largely related to Bastila's capture. Although the fact that you travel to Tatooine of all places is really an example of this.
The King of Fighters meta-series has several of the oldest fighters (Takuma, Saisyu, Chin, etc.) having either known each other superficially or being old friends. Specially, Takuma Sakazaki knew Jeff Bogard rather well, and he also was an acquintance of Kyo Kusanagi's father Saisyu; also, Chin Gentsai was an old friend of Tung Fu Rue.
Technically, though, he only knew Tedd's father, who is the head of the American The Men in Black.
Parodied in the Futurama episode, "A Taste of Freedom", where aliens manage to enslave the entire earth by winning a single battle and then leaving a single occupying base - once that is destroyed, the invasion disappears.
Anyway, when Zapp Brannigan handed the defense codes over to Hugh Man he apparently crippled Earth's entire military force (which often seems to consist entirely of the Nimbus, which was blown up in the battle).
In The Transformers, the Autobots and Decepticons fight all over the planet, not just the Rockies. Yet, it takes less than two hours for Optimus and the gang (a group of cars) to get from Colorado to Central Africa. The same goes with their adventure in India. And while a race from Paris to Istanbul sounds doable in a day, the map shown on the show looks deplorable!◊
In the episode "Around the Berry Big World," Strawberry Shortcake goes on an "Around the World in 80 Days" style trip and is deflected at every turn, yet she always manages to end up near one of her international friends (from the "World of Friends" line).
Top Cat plans an all-you-can-eat pizza binge for him and his pals in "Rafeefleas" but picks the wrong pizza shop to do it:
T.C.: A thousand pizza parlors in New York and we had to pick the one run by Officer Dibble's cousin!
It actually is a small world... when comparing urban centers to rural. The reason humans tend to run into each other - even across the earth from where they met - is they tend to hang around cities and other places where humans live.
If the entire human race of several billion people were put into a megacity at the same population density as New York City, it would be the size of the state of Texas. Seems big, but compared to the amount of available land on the earth is pretty tiny - .0046% of the total land area of the earth.
Though even if you landed at the central space port of that city the droids you were looking for would most likely not be within any kind of walking distance, let alone on the route you have randomly chosen.