Law of large numbers, in full
Jacob Bernoulli proved an early form of it, published in Ars Conjectandi in 1713, and the statement is about averages. For independent trials drawn from the same distribution, the sample mean converges to the expected value as the number of trials grows. Flip a fair coin often enough and the proportion of heads approaches 0.5. Roll a fair die often enough and the average pip count approaches 3.5. Convergence is claimed for a ratio, and the ratio is the only thing claimed to settle down.
The usual misreading is that counts even out, that a surplus of heads gets repaid by a later surplus of tails. The opposite is typical: the absolute gap between heads and tails tends to grow with the number of flips, not shrink. For a fair coin the expected size of that gap scales with the square root of n. The proportion still converges, because a gap growing like the square root of n, divided by n, heads to zero.
Put numbers on it. After 100 flips a gap of about 8 is unremarkable, say 54 heads to 46 tails, four percentage points off even. After 1,000,000 flips a gap of about 800 is equally unremarkable, say 500,400 to 499,600, which is 0.04 percentage points off. The gap grew a hundredfold while the proportion moved a hundred times closer to half. Nothing pulled anything back. The denominator simply outgrew the drift.
- Probability
- LLN, Bernoulli's theorem
A worked example
Roll a fair d6. Its expected value is (1+2+3+4+5+6)/6 = 3.5. After 60 rolls you might see six come up 6 times against an expectation of 10, a shortfall of 4, which is 10 percent of rolls against an expected 16.67 percent. After 6,000 rolls the shortfall might be 40, ten times larger, yet 960 sixes out of 6,000 is 16.0 percent, ten times closer to target.
Sources
Jacob Bernoulli's law of large numbers appeared in Ars Conjectandi, published in Basel in 1713, eight years after his death.
MacTutor History of Mathematics — Jacob Bernoulli
See also
Where it comes up on this site
Law of large numbers: frequently asked
Only as a proportion. The share of heads approaches 0.5, while the raw difference between the two counts typically drifts further from zero the longer you flip. Both hold at once because the gap grows roughly with the square root of n and the total grows with n, so dividing one by the other sends the imbalance to nothing even as the imbalance itself gets bigger. Nothing corrects anything. The denominator just wins.
There is no threshold. Cutting the wandering in the proportion by a factor of ten takes a hundred times as many trials.
It says the long-run average return per ticket approaches the game's expected value, which is negative. About any single ticket it says nothing.