Coin flips

The game is older than the theory. Romans called it navia aut caput — "ship or head" — after the ship's prow and the portrait that appeared on opposite faces of their coinage. Medieval English sources call the same game "cross and pile", after the cross stamped on one face and the pile, the name for the lower of the two dies used to strike a coin. For most of that history a toss was not a model of chance so much as a way of handing a decision to something outside human argument.

Treating it as a mathematical object came later. Girolamo Cardano's Liber de ludo aleae, written in the 1560s but not printed until 1663, was the first systematic attempt to compute odds in games of chance. Jacob Bernoulli's Ars Conjectandi, published posthumously in 1713, supplied the result that makes a coin useful: as the number of trials grows, the observed proportion of heads converges on the true probability.

The most quoted empirical check was made under duress. John Kerrich, a South African mathematician, was interned in Denmark after the German invasion in 1940, and passed some of that time flipping a coin 10,000 times. He recorded 5,067 heads — 50.67% — and published the running tallies in 1946 as An Experimental Introduction to the Theory of Probability. The graph of that proportion settling towards one half is still reproduced in statistics textbooks.

Physics has since complicated the picture. Persi Diaconis, Susan Holmes and Richard Montgomery argued in 2007 that a hand-flipped coin wobbles as it spins, spending slightly longer with its starting face upward, and predicted it should land the same way up as it started about 51% of the time. A team led by František Bartoš later tested this with 350,757 recorded flips and reported 50.8%, with a credible interval excluding an even split. A real coin is fair about which face shows; it is very slightly unfair about changing.

Key properties

• Each flip here is an independent Bernoulli trial, so the number of heads in n flips follows the binomial distribution B(n, p).
• For a fair coin the expected number of heads in n flips is n/2 with standard deviation √n / 2 — about 50 either side of 5,000 in 10,000 flips.
• Exactly 500 heads in 1,000 fair flips has probability only about 2.5%, even though it is the single most likely count.
• The longest run of identical outcomes grows roughly with the logarithm of the number of flips, so a streak of nine or ten somewhere in a thousand flips is ordinary rather than remarkable.
• Past flips carry no information about the next one; the belief that a run of heads makes tails "due" is the gambler’s fallacy.
• A coin with any fixed, unknown bias can still yield perfectly fair bits by von Neumann’s trick: flip twice, read heads-then-tails as 0 and tails-then-heads as 1, and discard the two matching pairs.
• The law of large numbers guarantees the observed proportion converges on p, but says nothing about the absolute gap between heads and tails, which typically grows like √n.
• These flips come from a deterministic PRNG seeded by the text you supply, so they are reproducible by design and unsuitable for anything that needs unpredictability.

Where they turn up

• In 1845, in Francis Ermatinger’s house in Oregon City, Asa Lovejoy of Boston and Francis Pettygrove of Portland, Maine settled the name of their new town with a best-of-three penny toss. Pettygrove won; the 1835 coin is held by the Oregon Historical Society.
• On 14 December 1903 the Wright brothers tossed a coin at Kitty Hawk to decide who would fly first. Wilbur won and his attempt stalled seconds after leaving the rail; Orville made the successful flight three days later.
• The UEFA Euro 1968 semi-final between Italy and the Soviet Union finished 0–0 in Naples and, under the rules of the day, was decided by a coin toss. Italy went through and won the tournament.
• Cryptography has a formal version of the problem: Manuel Blum’s "coin flipping by telephone", presented at CRYPTO ’81 and published in SIGACT News in 1983, lets two parties who do not trust each other agree on a fair bit using a commitment scheme.
• Sports use the toss as a procedural tie-breaker — the NFL overtime toss, the cricket toss for choice of innings — which treats it as fair by convention rather than relying on any measurement of the coin itself.