From Tardis Wiki, the free Doctor Who reference

According to Romana II, coincidences were a statistical inevitability, and they were bound to happen sometimes. (AUDIO: Subterranea) The Fifteenth Doctor described a coincidence as a form of accident, with two things bumping together unexpectedly. (TV: The Church on Ruby Road [+]Loading...["The Church on Ruby Road (TV story)"]) The Eleventh Doctor claimed to believe that coincidences were what the universe did for fun. (TV: Closing Time [+]Loading...["Closing Time (TV story)"]) On Traken, it was said that the universe was made up of coincidences all coming together to make one happy accident. (PROSE: Divided Loyalties)

Statistically, events on their own would be considered potential coincidences or anomalies until checked against similar examples. (TV: Adrift) Greater degrees of recurrence, or "bigger" coincidences, (TV: Invasion of the Dinosaurs) tended to suggest that apparent anomalies might be connected, and hence were not coincidences. (AUDIO: The Thirteenth Stone)

Some people did not believe in coincidences at all. (TV: The End of Time [+]Loading...["The End of Time (TV story)"]) Tula Chenka was among them. (AUDIO: The Mystery of Sector 13, A Matter of Conscience)

The Eleventh Doctor was known to make discoveries or deductions by paying great attention to apparent coincidences. He reminded himself, "Never ignore a coincidence, unless you're busy. In which case, always ignore a coincidence." (TV: The Pandorica Opens [+]Loading...["The Pandorica Opens (TV story)"]) The Twelfth Doctor followed a similar line of thinking, offering that two variables could either be causally linked or constitute an unlikely coincidence. (TV: The Pilot [+]Loading...["The Pilot (TV story)"])

Claims were sometimes made that patterns were mere coincidences in a deliberate attempt to cover up a lie or secret, (TV: Father's Day [+]Loading...["Father's Day (TV story)"]) even a conspiracy. (TV: Adrift)

The Thirteenth Doctor claimed that coincidence was one of two things she didn't believe in, the other being goblins. (PROSE: The Good Doctor)

After the Fourteenth Doctor cast a superstition at the edge of the universe, (TV: Wild Blue Yonder [+]Loading...["Wild Blue Yonder (TV story)"]) he inadvertently gave goblins the power to use coincidence to "bumble" through time and steal baby humans to feed the Goblin King. The Fifteenth Doctor intervened to stop them stealing Ruby Sunday in December, 2004, undoing their cracked timeline and defeating the goblins using his intelligent gloves. (TV: The Church on Ruby Road [+]Loading...["The Church on Ruby Road (TV story)"])