The mathematics of one flip
A fair coin has two equally likely outcomes, so the probability of heads on any single flip is exactly 1/2 — 50%. That is the entire theory of the single flip, and it carries a crucial corollary: flips are independent. The coin has no memory, so after ten heads in a row the eleventh flip is still 50/50. Believing otherwise — that tails is somehow 'due' — is the gambler's fallacy, and casinos have been built on it.
Independence is also what makes the coin useful as a decision tool: there is no strategy, no pattern and no history to exploit, which is why the toss has survived millennia as humanity's favorite tiebreaker.
Many flips: where it gets interesting
Probabilities multiply. The chance of two heads in a row is 1/2 × 1/2 = 1/4; ten in a row is about 1 in 1024. Meanwhile the count of heads in a large number of flips follows the binomial distribution, which concentrates around half with a spread that shrinks as the sample grows: at 10 flips a 70% split is common, at 100 it is rare, at 1000 it is essentially unheard of.
The simulator on this page runs a hundred flips per click so you can check every claim above empirically. Run it ten times and you will see the batch-to-batch scatter; the 1000-times page shows the same experiment at a scale where the law of large numbers pins the result tightly to 50%.
Coin flips in the wild
Beyond the classroom, coin-flip probability underwrites real institutions: NFL games begin with a toss, some elections tie-broken by drawing lots follow equivalent fairness logic, and computer scientists use the same Bernoulli trials as building blocks for randomized algorithms and simulations.
Even physics has weighed in: work by Persi Diaconis and colleagues found that a vigorously flipped physical coin lands on the side that started facing up about 51% of the time — a tiny real-world bias your browser's cryptographic randomness is happy to be free of.