The law of large numbers, shown with playing cards
· 6 min read
What the law says
As you repeat an independent random experiment more and more times, the observed frequency of an event approaches its true probability. Short runs wobble; long runs settle.
What it does not say
It does not say that results “even out” by compensating for earlier ones. A fair process has no memory. After ten heads in a row a fair coin is still just as likely to land tails as it was at the start; the early imbalance is simply diluted by later flips. (Drawing cards from a deck without putting them back is different: the deck does have memory, and that is exactly what the hypergeometric distribution describes.)
How fast does it settle?
The standard error of an observed proportion is √(p(1 − p) ÷ n). For the chance of at least one ace in five cards (p ≈ 0.341) that is about 4.7 percentage points at 100 samples, 1.5 at 1,000, 0.47 at 10,000 and 0.15 at 100,000. Each tenfold increase shrinks the error by about 3.2 times.
SE = √( p (1 − p) / n )Watch it happen
The convergence chart in the simulators plots the running estimate against the number of samples on a logarithmic axis. The line jumps around at first and then hugs the exact value, and the sampling variability tool shows the spread directly by repeating the whole experiment hundreds of times.