What is the law of large numbers?

It is a theorem stating that as the number of independent, identically distributed trials increases, the sample mean converges in probability to the population mean (expected value). For a fair die, the expected value is (1+2+3+4+5+6)/6 = 3.5.

Why does the line fluctuate a lot at first?

With few trials, each individual outcome has a large relative effect on the average. As more trials accumulate, the influence of any single outcome diminishes and the average stabilizes near the expected value.

Does this prove the law of large numbers?

The simulation illustrates the law visually but does not constitute a mathematical proof. The formal proof relies on Chebyshev's inequality or the Borel-Cantelli lemma depending on the variant (weak vs. strong LLN).

What is the expected value for each experiment?

For a six-sided die the expected value is 3.5. For a coin flip where heads = 1 and tails = 0 the expected value is 0.5 (i.e., 50% probability of heads).

Is the random number generator truly random?

The simulation uses the browser's built-in Math.random(), which is a pseudorandom number generator. It is not cryptographically random but is more than sufficient for illustrating statistical convergence.

Law Of Large Numbers Demo

Simulate dice rolls or coin flips and watch the running average converge to the expected value as trials increase.

Simulation Settings

Experiment

Number of Trials 500

10 1,000 2,000

Animation Speed

Trials

Running Avg

—

Expected

3.50

Deviation

—

Running Average vs. Expected Value

Running avg Expected

Press "Run Simulation" or "Step (+1)" to begin.

Summary

The law of large numbers is a fundamental theorem in probability that states as the number of trials increases, the sample average converges to the expected (theoretical) value. This interactive demo lets you roll dice or flip coins and watch the running average plotted in real time against the expected value, making the concept tangible and intuitive.

How it works

Choose an experiment: a six-sided die (expected average 3.5) or a fair coin (heads = 1, tails = 0, expected average 0.5).
Set the number of trials per run using the slider (up to 2,000).
Click "Run Simulation" to instantly generate all trials and plot the running average.
The chart shows each trial on the x-axis and the cumulative average on the y-axis.
A dashed line marks the theoretical expected value so you can see convergence visually.
Use "Step" to add one trial at a time, or "Reset" to start over.

Use cases

Teaching probability and statistics to students.
Demonstrating why casinos always win in the long run.
Illustrating why sample size matters in research.
Showing how insurance risk pools become predictable at scale.
Understanding why poll averages become more accurate with more respondents.
Exploring the difference between short-run randomness and long-run predictability.

Frequently Asked Questions

Last updated: 2026-05-28 · Reviewed by Nham Vu