Law of Large Numbers, explained.
The law of large numbers explains why a sample average or proportion becomes more likely to be close to its expected value as suitable repeated observations accumulate.
Why it happens
An early surprise has a large effect on a small sample. Later it becomes a smaller part of the total. Independent trials need not compensate for earlier results; their accumulated relative fluctuations become smaller.
For independent tosses with the same success probability, the observed proportion becomes increasingly likely to lie near that probability as the sample grows.
Read the result
Watch the running proportion rather than only the difference in counts. A curve may move away from its target temporarily, even as a larger sample makes large proportional errors less likely.
A worked example
Five heads arrive in a row
After five fair tosses that all show heads, the observed heads share is 100%.
The next toss still has a 50% heads probability. Additional ordinary tosses dilute the influence of the first five.
Long-run stabilization occurs through accumulation, not because tails becomes due.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Each seeded uniform draw is counted as heads when it is below p. The curve shows cumulative heads divided by toss count, with a horizontal reference at p. Absolute counts can drift even as proportions stabilize.
Where this idea is useful
A practical use
Estimate a product defect rate from a larger representative sample. More data reduces sampling noise, but it cannot repair a biased sampling process.
A common misconception
“A long sample must balance itself exactly.”
Convergence concerns proportions or averages under assumptions. It is neither exact balance at a finite point nor a promise about the next trial.
What this explanation leaves out
- The next toss never compensates for previous tosses. The displayed path need not approach its target monotonically. Dependence and changing probabilities require different assumptions.
Does more data always make an estimate better?
A larger representative sample can reduce sampling noise. It does not fix selection bias, incorrect measurement, dependence, or a changing process. Those require attention to how the data were collected.
Which is shrinking in relative importance: an early result or the uncertainty of the next trial?
Associated thinkers
Further reading
Explore the original research or the teaching reference behind this experiment.