Coin Flipper: How It Works
A coin flip is the simplest random event there is, and it is where almost every intuition about probability goes wrong. This page covers what a fair sequence actually looks like, and why it does not look the way people expect.
The gambler's fallacy
After five heads in a row, the next flip is still 50/50. The coin has no memory, no mechanism to balance itself, and no awareness of what came before. The belief that a run makes the opposite outcome 'due' is the gambler's fallacy, and it is the single most persistent error in probability.
What is true is that before flipping, a run of six heads is unlikely — one chance in 64. Once five have happened, that improbability is spent. Only the sixth flip remains, and it is an even chance like every other.
Streaks are normal
| In 100 flips | Probability of occurring |
|---|---|
| A run of 5 or more | Over 97% |
| A run of 6 or more | About 80% |
| A run of 7 or more | About 55% |
| A run of 10 or more | About 5% |
A run of six identical results is more likely than not in a hundred flips. People asked to write down a 'random' sequence produce far fewer and shorter runs than genuine randomness does, which is why hand-invented random data is easy to identify statistically.
Regression, not correction
The law of large numbers says the proportion of heads approaches 50% as flips accumulate. It does not say the counts converge. After 100 flips you might be 10 ahead; after 10,000 you might be 40 ahead — a larger absolute gap and a much smaller proportional one. The imbalance is diluted, not corrected.
Real coins are not perfectly fair
A physical coin flip is close to fair but not exactly. Research has found a small bias toward the face that was up when the coin was launched — roughly 51 to 49 — because the coin spends marginally more time on that side during its precession. Spinning a coin on a table is considerably more biased, sometimes heavily, because of the rim's weight distribution.
A digital flip using a cryptographic random source is fairer than any physical coin.
Making an unfair coin fair
Von Neumann's method removes any consistent bias. Flip twice: HT counts as heads, TH counts as tails, and HH or TT are discarded and the pair repeated. Since HT and TH have identical probability whatever the coin's bias, the result is exactly fair. It costs flips — a badly biased coin needs many — but the correction is perfect and requires knowing nothing about the bias.
When a coin flip is a good decision tool
Two situations. When the options are genuinely equivalent, flipping saves the time you would spend deliberating over a choice that does not matter. And when you cannot decide, watching the result often reveals a preference — the disappointment or relief on seeing the outcome is information you did not have access to before.
It is also the fairest tie-breaker available, which is why it opens sporting fixtures. Nobody can argue with a mechanism that gives neither side an advantage.