Flip a coin ten times and you might get seven heads and three tails. That looks unfair, like the coin is broken. It is not — short runs of chance are just streaky and lumpy by nature. The magic happens when you keep going. Flip the coin a hundred times, then a thousand, then more, and the share of heads slides closer and closer to a perfect half. The early lopsided streaks do not get cancelled out; they simply get drowned by the flood of flips that come after. In the simulator you can flip one coin at a time or a thousand at once and watch the running fraction of heads wobble at first, then home in on 50%.
Most people think a run of heads must be balanced out by coming tails. In fact the coin has no memory and the raw gap between heads and tails often grows; only the fraction settles, because later flips dilute early streaks.
What's actually happening
Genuine randomness is far streakier than people believe. Flip a fair coin ten times and getting seven or eight heads is completely ordinary — it happens roughly a sixth of the time. So a short run looks suspicious: clumps, runs of the same result, a tally that leans hard one way. Our intuition expects chance to be tidy and even, alternating obligingly between heads and tails, and it is offended when reality serves up HHHHTHHHTH instead. Nothing is wrong with the coin. Small samples are just lumpy.
Now keep flipping. As the count climbs into the hundreds and thousands, something steady emerges from the noise: the proportion of heads creeps toward one-half and stays there. This is the law of large numbers, proved by Jacob Bernoulli around 1700, and it is one of the load-bearing walls of probability. It guarantees that the average of many independent tries converges on the true underlying chance. The simulator shows it as a wandering line that thrashes about on the left, where flips are few, then flattens and presses itself against the 50% mark as the flips pile up.
There is a subtlety that trips up nearly everyone, and the simulator quietly exposes it. The law does not say that a run of heads will be "balanced out" by a coming run of tails — the coin has no memory, and after five heads the next flip is still a clean fifty-fifty. In fact the raw difference between the number of heads and tails often grows larger as you flip more, not smaller. What shrinks is the difference as a fraction. Ten extra heads matter enormously out of twenty flips and barely register out of twenty thousand. The ratio settles not because luck corrects itself, but because the mountain of later flips makes the early lopsidedness too small to see. This single fact is the entire business model of casinos and insurance companies: they cannot predict your next flip, and they never need to. Over millions of flips, the average is a near-certainty, and they price it to their advantage.
The average of many independent trials homes in on the true chance not by self-correcting but by dilution, which is the entire business model of casinos and insurers.
- 1Flip a real coin and keep a running tally: after each flip, write down the total heads divided by the total flips so far.
- 2For the first ten or twenty flips the number jumps around wildly — 1.0, then 0.5, then 0.67, lurching with every flip.
- 3Keep going to fifty or a hundred and watch the jumps get smaller and the number settle near 0.5. The more you flip, the harder it is to budge — that is the law of large numbers in your own hand.
Common questions
No, the coin has no memory and the next flip is still a clean fifty-fifty. Believing otherwise is the gambler's fallacy, dramatised in 1913 when a Monte Carlo wheel landed black 26 times in a row.
No, the raw difference between heads and tails counts often grows larger as you flip more. Only the difference as a fraction shrinks, because each extra flip shifts the ratio by an ever-smaller amount.
They cannot predict your next flip and never need to. Over millions of trials the average is a near-certainty, so a roulette wheel's small edge is invisible to one gambler yet inevitable across all of them.