What 'fair' means in probability
A fair coin is one whose two outcomes are equally likely: the probability of heads equals the probability of tails, at exactly 1/2 each. Fairness has a second, equally important component — independence — meaning no flip is influenced by any previous flip. Together these properties make the toss unpredictable in the individual case and perfectly regular in aggregate, which is precisely what a tiebreaker needs.
Real coins approximate this ideal. Modern minting produces coins symmetric enough that no deliberate bias is detectable in circulation, and analysis of large flip experiments consistently finds splits within a fraction of a percent of even — well within the noise any honest sample produces anyway.
The physics (and the tiny asterisk)
In 2007, statistician Persi Diaconis and colleagues modeled the physics of a vigorously flipped coin and found a subtle effect: the coin precesses as it spins, and the side that starts facing up has roughly a 51% chance of ending up — not a loaded coin, just mechanics. The bias is small enough that no human flipper could notice it, but it is real.
Other real-world quirks are even smaller: a coin landing on its edge (estimated at roughly 1 in 6,000 for a nickel), spinning coins drifting slightly toward the lighter face, and surface bounce effects. None of these survive into the digital world: this page's flips come from cryptographic randomness, which has no mass, no thumb and no asterisk.
How to test a coin for fairness
The classic test is also the plot of this site: flip it many times and watch the percentage. A thousand flips of a fair coin will almost always land within a few points of 50%, and the 10-times and 100-times pages let you feel how wide the scatter is at smaller samples. One ten-flip batch proving nothing is the most important lesson in the curriculum.
Statisticians formalize this with a binomial test, but the intuition is enough for everyday life: in small samples, wild swings are normal; in large samples, only genuine bias survives. If you want a coin with mathematically perfect fairness, you are already holding it — it is the one on this page.