Plain-language answer
Statisticians Persi Diaconis and Frederick Mosteller coined the phrase “the law of truly large numbers” to describe an informal but genuinely useful observation: with a sample size large enough, any outrageous thing is likely to happen to somebody.(Diaconis & Mosteller, 1989) It is not a formal theorem the way the law of large numbers is — it is a memorable label for the combined effect of a huge number of independent opportunities, each individually unlikely, adding up to near-certainty that a few of them will hit.
Why it matters
Confusing this principle with the formal law of large numbers is one of the most common sources of “impossible odds” claims. A specific person winning the lottery twice is astronomically unlikely. But given how many people play lotteries, how many lotteries exist worldwide, and how many years lotteries have run, it becomes not just plausible but close to expected that someone, somewhere will win a lottery twice within a few decades. The event feels miraculous only if the reference class is wrongly narrowed to “this exact person” instead of correctly widened to “anyone who has ever played.”
Worked example: applying the phrase carefully
Diaconis and Mosteller use the phrase specifically to caution against treating genuinely rare-per-person events as inexplicable once the total population and time span considered are large enough to make them likely in aggregate.(Diaconis & Mosteller, 1989) With roughly eight billion people alive, each living tens of thousands of days, the number of “person-days” in which something could coincide is enormous — commonly estimated in the tens of trillions. Against a pool that large, events with a per-instance probability of one in a million can be expected to occur many thousands of times somewhere in the world, every single day.
This is precisely the same underlying arithmetic as multiple opportunities and selection effects: with enough independent tries, the probability that at least one succeeds climbs toward certainty, even when each individual try remains unlikely. The “law of truly large numbers” is best understood as a memorable name for that effect applied to the scale of the whole world’s population, rather than a separate mathematical principle.
Common misconception
“This happened to me, and the odds of it happening to any one specific person are a billion to one, so it must mean something” skips past the fact that the reference class implicitly being used (“did this happen to someone, anywhere, in a large enough population and long enough time span”) was never that narrow. The law of truly large numbers is a caution against exactly this substitution — a tiny individual probability does not stay tiny once summed across a truly large number of opportunities.
Limits and open questions
The phrase is deliberately informal, and Diaconis and Mosteller intended it as a rule of thumb rather than a precise formula: turning it into an actual number still requires estimating the real population of opportunities, which is often uncertain and easy to get wrong in either direction. It is also not a license to treat every implausible-sounding event as automatically explained by population size — some events remain genuinely rare even across a global population, and the phrase is only a caution to check the population size before assuming otherwise.
Related
- The law of large numbers is the distinct, formal theorem about convergence of long-run averages — not the same idea, despite the similar name.
- Multiple opportunities and selection effects works through the underlying “probability of at least one success” formula in detail.
Key takeaways
- “The law of truly large numbers” is Diaconis and Mosteller’s informal name for why rare-per-instance events become near-certain in aggregate, given a large enough population and enough time.
- It is not the same as the formal law of large numbers, despite the similar name, and should not be cited as if it were a rigorous theorem.
- Before treating an event as inexplicably rare, check how large the true population of opportunities for something similar to happen actually was.