What is going on?
For independent flips with a fixed chance of heads, the observed fraction of heads tends toward that chance as the number of flips grows. Early proportions can move dramatically: one new flip changes a two-flip sample much more than it changes a ten-thousand-flip sample.
Work through it
- Set the true chance of heads. It does not need to be a fair coin.
- Run a sample and follow the line showing the observed fraction.
- Generate a new sample to see a different path under the same underlying probability.
Put numbers to the idea
After 10 fair flips, 7 heads means an observed rate of 70%. After 1,000 flips, 530 heads means 53%. A larger sample can have a smaller percentage error even while its absolute excess of heads grows.
A little more clarity
Is tails due after a heads streak?
No. For independent flips, the next probability stays the same. The idea that an opposite result is due is the gambler’s fallacy.
Will heads and tails eventually become equal?
There is no guarantee of equality at any selected finite point. The theorem concerns the proportion approaching the probability, not a fixed count difference becoming zero.
How is this different from the central limit theorem?
The law of large numbers describes convergence toward a mean. The central limit theorem describes the approximate shape and scale of the distribution of sample means.