Coin flip guide
The Law of Large Numbers in Coin Flips
As independent fair flips accumulate, their heads proportion tends toward one half. This concerns relative frequency, not a requirement for the next flip to correct the past.
Counts versus proportions
Expected heads is n/2. The standard deviation of the heads count is √n/2, while the standard deviation of its proportion is 1/(2√n). Count fluctuations can grow while percentage fluctuations shrink.
A concrete comparison
At 100 flips, a deviation of 10 heads from 50 is 10 percentage points. At 10,000 flips, a deviation of 100 heads from 5,000 is one percentage point. A larger absolute gap can accompany a smaller relative gap.
Try the experiment
Run 100, 1,000, and 10,000 flips. Save each CSV and compare observed heads, expected heads, and the percentage-point difference from 50%. Repeat each size several times so one lucky batch does not determine your conclusion.
Read the path honestly
The cumulative chart can approach and retreat from 50%. The law does not promise a smooth path, an exact split, or a smaller deviation in every larger batch. The table gives theoretical one-standard-deviation scales, not guaranteed intervals.
| Flips | Expected heads | SD of heads proportion |
|---|---|---|
| 100 | 50 | 5 percentage points |
| 1,000 | 500 | About 1.58 percentage points |
| 10,000 | 5,000 | 0.5 percentage points |
Sources and further reading
Worked examples use the independent fair-coin model unless a section explicitly discusses physical tossing.