How Chance Works

Base rates, multiple opportunities, conditional probability, and the arithmetic behind why coincidences happen more often than intuition expects.

See also: Science & Skepticism covers how these probability tools fit into the scientific method more broadly.

• Multiple opportunities and selection effects — Why running many chances at a pattern makes finding one almost guaranteed, even when every individual chance was unlikely on its own.
• Probability is a model, not a verdict — Why a probability is always a statement about a chosen model and reference class, not a fact about a unique event — worked through the birthday problem.
• Base rates: how common is this, really? — The most under-used number in any coincidence story is how often the 'surprising' thing happens anyway, to someone, without any special explanation.
• The birthday problem, explained — How a room of just 23 people crosses even odds for a shared birthday — and why that surprises almost everyone who hasn't seen the calculation.
• Clustering and runs: randomness looks lumpy — Random sequences produce longer streaks and tighter clusters than most people guess, so clustering alone is weak evidence of a real pattern.
• Conditional probability: given what, exactly? — "How likely is A?" and "how likely is A, given B?" can have very different answers — and coincidence stories often quietly swap one for the other.
• Independence and dependence: what tells you what — Most 'multiply the probabilities together' coincidence math secretly assumes independence that real events, sharing culture, geography, and cause, rarely have.
• The law of large numbers: chance evens out slowly — Over many repeats, outcomes converge toward their true probability, but the law says nothing about any single short run.
• The law of truly large numbers: it happens to someone — Given a large enough number of opportunities, even wildly improbable events become almost certain to occur to somebody, somewhere.
• Networks and small worlds — Real social networks mix tight local clustering with a few long-range shortcuts, making unlikely-feeling encounters mathematically unsurprising.
• Post-hoc probability: painting the target after — Choosing which pattern counts as "the match" after seeing the outcome inflates apparent improbability, in coincidence lists and in research alike.
• Reference classes: compared with what, exactly? — Every probability judgment implicitly compares an event to a class of similar events — and changing that class can change the answer without any new evidence.
• Regression to the mean: extremes don't repeat — An unusually good or bad result tends to be followed by a more average one, for purely statistical reasons that have nothing to do with cause and effect.
• Selection effects: what you see isn't random — When data has already been filtered by its own outcome, conclusions drawn from it can be exactly backwards, as WWII aircraft-armor analysis shows.