The Law of Large Numbers
Jacob Bernoulli / Andrey Kolmogorov · 1713
"As you take more independent draws from the same process, the average of what you see converges to the true underlying expectation — which is why small samples routinely mislead and large samples quietly correct."
Bernoulli proved in 1713 why your gut misreads streaks: flip a fair coin ten times and 70% heads is common; flip it ten thousand times and 51% heads is rare. Large numbers pull the average back to its true center.
The weak law says sample averages converge in probability to the expected value; the strong law strengthens this to almost-sure convergence. The mechanism is not that the coin 'compensates' — each flip remains 50/50 — but that deviations get diluted as the denominator grows. This is why early startup metrics, with n=20, are almost always noise, and why A/B tests, polls, and investment track records cannot be read without asking 'what is n?'
What does the Law of Large Numbers actually guarantee about sample averages?
Read more about the topic
The explanation above is written with AI assistance. These are the originals — go to them to check it.
- Law of Large Numbers — Bernoulli's Ars Conjectandi (1713) overviewWikipedia
Bayes' Theorem
"Belief should update in proportion to evidence — Bayes' theorem is the exact math for how much a new piece of evidence should move your confidence."