Long-Run Behavior

Probability · Axiom Academy

What happens when we repeat something over and over... and over? Let's flip a fair coin many times. What fraction of flips will be heads? Let's watch how the running average approaches the true mean over time. Play roulette at a casino. Bet 1 on red each spin. See what happens over many spins... The sample mean approaches the population mean. Random fluctuations average out. This is the Law of Large Numbers . This theorem explains why casinos are profitable, why polling works, why insurance companies stay in business, and why experiments with large sample sizes are more reliable. The Law of Large Numbers tells us about averages , but it doesn't tell us about the distribution of those averages. That's where the Central Limit Theorem comes in...

This is the written version of the interactive lesson above. See the full Probability course.