Gambling terms, explained straight

Odds, Probability & Bankroll

Law of large numbers

The law of large numbers is a probability theorem stating that the sample average of repeated independent trials converges to the true expected value as the number of trials grows.

Weak form
The probability that the sample mean differs from the true mean by more than any small amount tends to zero as trials increase.2
Strong form
The sample mean converges to the true mean almost surely as trials go to infinity.1

Key points

  • Random fluctuations cancel out over many trials, pulling the average toward the underlying mean.1
  • The law does not guarantee short-run correction — it describes long-run convergence, not individual streaks.

Common questions

What is the law of large numbers in probability?

In probability, the law of large numbers says that as you repeat an independent experiment many times, the average of your results gets closer to the true expected value. For example, the more times you flip a fair coin, the closer the proportion of heads gets to 50%.

Where this term is used

Odds & probability Odds Explains how odds express the ratio of probability between an event happening and not happening. Odds & probability Roulette payouts Lists the payout odds for standard roulette bets and how they relate to the house edge. Odds & probability Betting odds Covers the three main formats for displaying betting odds: fractional, decimal, and moneyline. Odds & probability Kelly criterion A formula for bet sizing that maximises long-term growth given an edge and the odds. Odds & probability Kelly betting Practical application of the Kelly criterion to sports betting and other gambling. Odds & probability Martingale strategy A betting system that doubles after a loss, aiming to recover previous losses with one win.

Sources

  1. Ley de los grandes números en.wikipedia.org Provides the standard definition and explanation of the law of large numbers.
  2. Probability Models - Notes 10 webspace.maths.qmul.ac.uk Formalises the weak law of large numbers with the epsilon-delta definition.

Sources are drawn from regulators, universities and published research, and each one is labelled with what it actually is — a preprint is not called a paper. Bookmaker and affiliate pages are never cited here, because a page that sells betting is not a neutral authority on it.

Written and maintained by Ellis Warrender, writer and maintainer.

Last updated 28 July 2026. Spotted an error? Tell me and it gets fixed and logged on the corrections page.