Plain-language answer
Two events are independent if knowing that one happened tells you nothing about whether the other happened. They are dependent if it does. Coin flips are (very close to) independent: knowing the first flip was heads tells you nothing about the second. Two people’s choice of baby name is usually dependent: both are drawn from the same era’s popular names, the same culture, sometimes the same family — so a “coincidental” name match is far less surprising than treating the two choices as independent random draws would suggest.
Why it matters
The most common way an “astronomical odds” calculation goes wrong is by multiplying together probabilities for details that are not actually independent. If two biographies “coincidentally” share a birth country, a profession, and a first initial, multiplying three seemingly rare probabilities together produces a headline-grabbing number — but if the profession and the era make certain first initials, birth countries, and career paths correlated with each other (shared educational systems, shared class background, name fashions specific to a period and region), the events were never independent, and the multiplication rule for independent events does not apply. A rigorous treatment of coincidences has to model these dependencies explicitly rather than assume independence for convenience.(Diaconis & Mosteller, 1989)
Worked example: why multiplying probabilities can mislead
For genuinely independent events A and B, P(A and B) = P(A) × P(B). This is the rule behind “a coincidence with odds of a million to one” style calculations: multiply a handful of small probabilities and the product shrinks fast. But the rule only holds under independence. If A and B share a common cause or context — both more likely in the same time period, the same profession, the same region — then P(A and B) is larger than the independence formula predicts, sometimes by orders of magnitude, because knowing A is true already makes B more likely than its raw, on-its-own probability would suggest.
Diaconis and Mosteller’s methodological treatment of coincidences makes this one of its central cautions: many “spectacular” published coincidence probabilities come from multiplying marginal probabilities for events that were correlated all along, and the resulting figure describes a scenario that was never actually the one in question.(Diaconis & Mosteller, 1989)
Common misconception
“I checked that each individual detail was unlikely on its own, so the combination must be astronomically unlikely” only follows if the details are independent. Shared context is the norm, not the exception, for almost everything about a human life — nationality, era, class, education, profession, and social network all correlate many superficially separate facts with each other.
Limits and open questions
Quantifying how dependent two real-world variables are is often much harder than establishing that they are dependent at all — it usually requires actual population data (how correlated are profession and first-letter-of-name in a given era and country?) that is rarely available for a specific anecdote. In practice, the honest response is often to flag that an independence assumption is doing unjustified work in a calculation, without being able to supply the precise corrected number.
Related
- Multiple opportunities and selection effects covers the companion assumption — that all the “opportunities” being counted are themselves independent tries.
- Conditional probability gives the formal language (P(A given B) ≠ P(A)) for what dependence means.
Related interactive
- Social network crossings (in development) will let readers see how shared social context breaks the independence assumption behind “small world” encounters.
Key takeaways
- Multiplying probabilities together to get a dramatic combined figure is only valid when the underlying events are independent — and real biographical, cultural, and social details usually are not.
- Shared cause, culture, era, or network is the default state for most facts about people’s lives, not a special exception to look for.
- A rigorous coincidence analysis states its independence assumptions explicitly and flags where dependence would change the answer, rather than quietly assuming independence because it makes the arithmetic simple.