Law of Large Numbers

Key idea

Sample means converge to the expected value.

In depth

As the number of independent trials grows, the sample mean converges to the true expected value. It explains why casinos are profitable in the long run despite random short-term results.

Worked example

The proportion of heads in n flips approaches 0.5 as n grows.

Real-world uses

• Explaining why casinos win on average and why a poll gets accurate with a bigger sample

Common mistake

Believing the law means deviations will be balanced out by opposite ones (gambler’s fallacy).

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Probability & Statistics

• Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
• Residuals and R² – Error of a fit and proportion of variance explained.
• Simpson’s Paradox – A trend reverses when groups are combined.
• Law of Total Probability – Split an event over a partition.
• Bayes’ Theorem – Update beliefs with evidence.
• Medical Tests and Base Rates – Why positive results can still be probably false.