The Law of Large Numbers

What You'll Learn

The Law of Large Numbers states that as the number of trials increases, the empirical (observed) probability of an event converges to its theoretical probability. Individual outcomes are unpredictable, but patterns emerge over many repetitions.

The Three Phases

• Phase 1 — Single Event: One flip/roll/birth. The outcome is unpredictable — could be success or failure. Watch the animation and see the result displayed for 3 seconds.
• Phase 2 — Few Trials: With 20 trials, empirical probability fluctuates around the theoretical value. Each outcome is shown clearly so you can see the variability.
• Phase 3 — Many Trials: With 200+ rapid trials, watch the empirical probability line converge toward the theoretical probability line on the graph.

Quantities Displayed

• P(theory) — Theoretical probability: P(Heads)=0.500, P(Roll 6)=0.167, P(Male birth)=0.512 (all dimensionless)
• n — Number of trials performed
• k — Number of successes observed
• P(empirical) = k/n — Observed frequency of success
• |P(emp)−P(th)| — Absolute deviation from theory (decreases on average with more trials)

Sources

Ross, S. M. (2014). A First Course in Probability (9th ed.). Pearson. ISBN: 978-0321794772

World Bank. (2021). Sex ratio at birth. data.worldbank.org/indicator/SP.POP.BRTH.MF.ZS