A physical coin flip is very slightly biased toward the face that was up when it left your thumb — about 51 per cent of the time, according to work by Persi Diaconis, Susan Holmes and Richard Montgomery published in 2007. The cause is not weight distribution or minting. It is that a real flip precesses, wobbling around its axis rather than rotating cleanly, so the coin spends marginally more time with its starting face upward.
It is a small effect and it is a genuine one. A digital flip has no such bias, which makes it the fairer of the two even though it feels less trustworthy to watch.
Why does the wobble matter?
Because a coin tossed by a human is a deterministic physical system, not a random one. Given the exact launch velocity, spin rate and axis, the outcome is fully determined — the reason it feels random is that people cannot control those inputs precisely enough to exploit it.
The precession is what breaks the symmetry. A coin flipped about a perfectly perpendicular axis would spend equal time on each face; a real one tilts, and the tilt has a preferred direction set by how it was launched. Averaged over many flips, that shows up as roughly one extra same-side result per hundred.
The same paper showed the effect disappears if the coin is spun on a table instead of flipped, where a different and larger bias appears from the rim being slightly asymmetric.
What does 51 per cent actually cost?
Almost nothing on one flip and something measurable on many. Over a hundred flips the expected imbalance is one result; over ten thousand it is a hundred, which is well outside what chance alone would produce.
For settling an argument, the bias is irrelevant — it is far smaller than the effect of whoever is calling being able to see which face went up. For anything where a run of results is aggregated, it is the kind of thing that is worth removing rather than reasoning about.
How does a digital flip avoid it?
By not simulating a coin at all. There is no physics being modelled; the browser generates a cryptographically secure random value and the result is read from it directly, so the two outcomes are equally likely by construction rather than by approximation.
The relevant call is the browser crypto source, which is designed for key generation and is held to a much higher standard than the general-purpose pseudo-random function used for animations. It is seeded from the operating system entropy pool, not from the clock, and its output cannot be predicted from previous values.
The animation, where there is one, is presentation. The result is drawn before the coin starts moving, and nothing about the motion influences it.
Ceremony is the honest thing being given up. A physical flip has a witness, a sound and an object everyone can inspect, and that is worth something in a room even when the arithmetic is worse. Where the decision is being made in front of people who need to accept it, the coin in the pocket is often still the better instrument.
Why is my tally never exactly even?
Because an even tally would be evidence of a rigged draw rather than a fair one. In a hundred fair flips, landing on exactly fifty heads happens about 8 per cent of the time; anything between 40 and 60 is entirely ordinary.
The imbalance shrinks as a proportion and grows in absolute terms. A thousand flips will typically sit within a couple of per cent of even while being a dozen or more away from an exact split. That is what convergence looks like, and expecting the raw counts to level out is the gambler’s fallacy wearing a lab coat.
What is the seed for?
Reproducibility. Leaving the seed empty draws fresh randomness every time, which is what you want for a decision. Entering a seed makes the sequence deterministic, so the same seed produces the same run of results — useful when you need to demonstrate afterwards that a draw was not rerun until it gave the answer you wanted.
A seeded draw is not less random in appearance; it is less private. Anyone with the seed can reproduce the sequence, which is the point when the sequence has to be auditable and a problem when it has to be secret.
Questions people ask
Does the previous flip affect the next one? No. Each draw is independent, and a run of six heads does not make tails more likely on the seventh. The coin has no memory and neither does the generator.
Is a spun coin better than a flipped one? Worse. Spinning on edge amplifies the effect of the rim profile, and spun coins can deviate from even far more than one per cent.
Could I exploit the 51 per cent? In principle, by always calling the face that was up before the flip. In practice you rarely see it, the effect is one in a hundred, and anyone doing this at scale would be noticed.
Is the browser random source good enough for a prize draw? For fairness, yes. For a draw that has to be shown to be fair afterwards, the seed and the visible history matter more than the generator does.
The digital version wins on fairness and loses on ceremony, which is a reasonable trade for most decisions. The coin flip keeps a history of what it has said, the dice roller covers the cases where a coin is too blunt, and the random number generator handles any range you like.