The Law of Large Numbers, Practically
What 'it evens out' actually promises, on what timescale, and what it never promises.
"It evens out in the long run" is true, and it means something much narrower than most people take it to mean. Averages converge. Totals do not.
Flip a fair coin ten thousand times and the proportion of heads will be very close to half. The absolute difference between the number of heads and the number of tails will typically be around fifty, and it grows as you keep flipping.
Both of those facts are consequences of the same theorem, and holding them at once is what separates using the law of large numbers from misquoting it.
What the law actually says
The law of large numbers says that as the number of independent trials grows, the sample average converges on the expected value. Flip enough coins and the proportion of heads approaches 0.5.
It says nothing at all about the difference between counts. The proportion converging on a half is entirely compatible with the gap between heads and tails growing without limit, and that is exactly what happens.
The reason is that the gap grows with the square root of the number of flips while the count grows linearly. A hundred flips has a typical gap of about 5 — a 5% imbalance. Ten thousand flips has a typical gap of about 50 — a 0.5% imbalance. The gap grew tenfold and the proportion improved tenfold.
Proportion versus gap, as flips increase
- 100 flips: typical gap about 5, imbalance 5%.
- 10,000 flips: typical gap about 50, imbalance 0.5%.
- 1,000,000 flips: typical gap about 500, imbalance 0.05%.
- The gap grows; the proportion converges. Both are the same theorem.
How slow the convergence is
Square-root scaling is slow, and this surprises people every time. To halve the typical error in an average you need four times the data; to reduce it tenfold you need a hundred times.
That is the arithmetic behind survey sample sizes. A sample of a thousand gives a margin of error around three percentage points; getting to one point requires around nine thousand, and getting to a third of a point requires around ninety thousand. The cost grows quadratically for a linear improvement.
It also explains why small samples are so untrustworthy. Twenty observations have a typical error of around eleven percentage points on a proportion near a half, which is wide enough to be consistent with almost any hypothesis you might want to test.
The everyday misuse
The commonest misapplication is the gambler's version: a run of losses means a win is due, because it has to even out. The law says nothing of the kind. Future trials are independent of past ones, and the convergence happens by dilution rather than by correction.
That word is the important one. A deficit of thirty heads after a thousand flips is not repaid by an excess later; it becomes proportionally smaller as more flips accumulate around it. After ten thousand more flips the deficit is still about thirty and now represents 0.3% rather than 3%.
The second common misuse is applying it where the trials are not independent or identically distributed. A run of results from a process that is drifting does not converge on anything, and the law provides no comfort at all.
- The law does not correct past deviations — it dilutes them.
- It requires independent trials from an unchanging distribution.
- It says nothing about totals or gaps, only about averages.
- Convergence is at square-root speed, which is slower than intuition suggests.
Where it genuinely applies
Insurance is the canonical case. An insurer cannot predict any individual claim and can predict the aggregate across a hundred thousand policies with considerable accuracy, because the average of many independent claims converges tightly.
Casino margins work the same way, and it is worth noting the asymmetry: the house plays millions of hands and the player plays a few hundred, so the law applies to one side of the table and not the other. A player is entirely in the variance regime.
Randomised assignment in trials and experiments relies on it too. Randomising treatment across a large sample makes the groups similar on average in every respect, including ones nobody measured — which is why randomisation is so powerful and why it works poorly with small samples.
Splitting a bill, revisited
A concrete version of all this: a group of four takes turns paying for dinner by random draw. Expected shares are equal, and the law of large numbers guarantees the proportions converge.
After ten dinners the typical gap between the most and least frequent payer is around three occasions — a substantial imbalance that people notice and resent. After a hundred dinners the typical gap has grown to around ten occasions, while the proportional imbalance has fallen from 30% to 10%.
So the honest description is: it becomes proportionally fairer and absolutely more uneven, and the group experiences the second one. Which is why a method that is fair in expectation can still be a bad idea for people who will remember.
Frequently asked questions
Does the law of large numbers mean things even out?
Averages even out. Totals do not — the gap between heads and tails typically grows as you keep flipping, even as the proportion approaches a half.
How can both be true?
Because the gap grows with the square root of the trials while the count grows linearly. Ten thousand flips has a bigger gap and a much smaller imbalance than a hundred.
Does a deficit get repaid later?
No. It gets diluted. A deficit of thirty after a thousand flips is still about thirty after eleven thousand, and now represents a much smaller fraction.
How fast does convergence happen?
Slowly. Halving the typical error needs four times the data; reducing it tenfold needs a hundred times.
Why are small samples so unreliable?
Twenty observations have a typical error of around eleven percentage points on a proportion near a half, which is consistent with almost any hypothesis.
Does it apply if the process is changing?
No. The law needs independent trials from an unchanging distribution. A drifting process converges on nothing.
Why does the house always win?
Because the casino is in the long run and the player is not. The law applies to one side of the table and not the other.
Does it justify random bill-splitting?
In expectation, yes. In practice a group of four sees a gap of about three occasions over ten dinners, which is what people actually notice.
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Related guides
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