# Coin flipping

Coin flipping or coin tossing is the practice of throwing a coin in the air to resolve a dispute between two parties.

Coin flipping is a method that trusts the decision to pure luck, since there is no possibility for strategy, and any attempt to alter the odds (such as, most obviously, using a fake coin with both sides the same) is considered cheating. It is generally assumed that the outcome is unpredictable, with equal probabilities for the two outcomes (the fair coin), although careful analysis has shown that is not quite the case.

Coin tossing is popular because it is fast, more or less fair, and requires nothing but a little spare change.

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## History of coin flipping

The historical origin of coin flipping is the interpretation of a chance outcome as the expression of divine will. A well-known example of such divination (although not involving a coin) is the episode in which the prophet Jonah was chosen by lot to be cast out of the boat, only to be swallowed by a giant fish (Book of Jonah, Chapter 1).

Coin flipping as a game was known to the Romans as "navia aut caput" (ship or head), as some coins had a ship on one side and the head of the emperor on the other. In England, this game was referred to as cross and pile.

## The process of coin flipping

During coin flipping the coin is "flipped into the air", i.e., caused to both rise and rotate about an axis parallel to its flat surfaces. Typically, agreement is reached that one person will explicitly assign the action that will ensue from one positioning of the coin, and another, presumed to have the opposite interest or to be impartial, performs the following steps:

• resting the coin on the sides of several segments of the bent index finger of the dominant hand,
• pressing the tip of the bent thumb of the same hand against the palm-side of the index finger, so that friction there holds the thumb back from extending further,
• tensing the muscles that extend the thumb, thereby storing energy in the form of tension in those muscles,
• further extending the thumb and/or slightly uncurling the index finger, thereby overcoming the finger's frictional grip against the thumb-tip so it slips, and freely and rapidly extends, with it or its nail
• hitting the bottom face of the coin, centered within the half of the coin that is less in contact with the bend index finger, and thus
• simultaneously pushing it more or less upward and setting it rotating around an axis parallel to the circular faces of the coin;
• optionally, suddenly raising and quickly stopping the hand involved, in coordination with the releasing of the thumb, thus imparting extra vertical momentum (but little additional rotary momentum) to the coin. (Depending on the skill of the coin-tosser, and any resulting horizontal motion, the optional upward jerk of the tossing hand may be needed to ensure the coin stays aloft long enough to get the catching hand into position, or for the tosser and observers to move out of its path.);
• saying "Call it", to alert the party so designated to say (while the coin is in motion) either "Heads" or "Tails", designating the outcome that will correspond to the previously agreed upon outcome;
• once it falls back to a convenient height, either
• catching the coin in an open palm, or
• bringing one hand down over it, to prevent its bouncing away, as it lands on the other hand or arm, and quickly removing the upper hand from it, or
• avoiding interfering with it as it falls onto a sufficiently smooth and uncluttered point on the ground;
• all those involved jointly observing whether it has landed "showing heads" --with the side bearing the portrait or profile uppermost -- or "showing tails".

There may be several rounds in a single game of coin flipping if the participants agree to this ahead of time, but typically there is only one; this keeps the contest quick and prevents the losing side from asking for more rounds after the toss.

The coin may be any type, as long as it has two distinct sides, with a portrait on one side. The most popular coin to flip in Canada and the United States is the quarter because of its size; in the UK a 2p, 10p or 50p piece is favoured. However, participants will use any coin that is handy.

## Coin flipping in dispute resolution

Coin flipping is used to decide which team gets the kickoff, which way the teams will play, or similar questions in soccer matches, American football games, and almost any other sport requiring such decisions. The most famous case of this in the U.S. is the use of coin flipping in National Football League games, especially the Super Bowl. A special mint coin, which later goes to the Pro Football Hall of Fame, is used for this purpose, and other coins in that edition are sold as collectors items. The actual NFL rule is that the team winning the coin toss elects whether to choose which team kicks off, or whether to choose which team defends which end, in the first quarter; the other team makes the other one of the two choices, and then makes the same election at the start of the third quarter. In cricket, the toss is often of critical importance, as the decision of the winning captain to bat or bowl first has a heavy influence on the outcome of the game; in other sports the result of the toss is less crucial and merely a way to fairly choose between two more or less equal options.

In some jurisdictions, a coin is flipped to decide between two candidates who poll equal number of votes in an election, or two companies tendering equal prices for a project. (For example, a coin toss decided a City of Toronto tender in 2003 for painting lines on 1,605 km of city streets: the bids were \$161,110.00, \$146,584.65, and two equal bids of \$111,242.55. The numerical coincidence is less remarkable than it seems at first blush, because three of the four bids work out to an integral number of cents per kilometer.)

In more casual settings, coin flipping is used simply to resolve arguments between friends or family members. Unlike Rock, Paper, Scissors, coin tossing is almost never done purely for amusement.

## Physics of coin flipping

Experimental and theoretical analysis of coin tossing has shown that the outcome is predictable, to some degree at least, if the initial conditions of the toss (position, velocity and angular momentum) are known. Coin tossing may be modeled as a problem in Lagrangian mechanics. The important aspects are the tumbling motion of the coin, the precession (wobbling) of its axis, and whether the coin bounces at the end of its trajectory.

The outcome of coin flipping has been studied by Persi Diaconis and his collaborators. They have demonstrated that a mechanical coin flipper which imparts the same initial conditions for every toss has a highly predictable outcome.

Morover, they have demonstrated both mathematically and experimentally that the underlying physics of coin tosses appears to have a slight bias for a caught coin to be caught the same way up as it was thrown, with a probability of around 0.51. They also state that stage magicians and gamblers, with practice, may be able to greatly increase this bias, whilst still making throws which are visually indistinguishable from normal throws.

Since the images on the two sides of actual coins are made of raised metal, the toss is likely to slightly favor one face or the other. This is particularly true if the coin is allowed to roll on one edge upon landing; coin spinning is much more likely to be biased than flipping, and conjurers trim the edges of coins so that when spun they usually land on a particular face.

Although it is extremely rare, there is an extremely slight possibility that a coin will come to rest on its edge (estimated at roughly 1/6000 for a U.S. nickel.) If the coin remains on its edge, while it may cause temporary distraction, the only fair course of action would be to toss the coin again.

## Coin flipping in fiction

At the start of a famous 1939 movie, a state governor has to select an interim Senator and is being pressured by two sides to choose their respective candidate, Mr. Hill or Mr. Miller. Unable to choose, he flips a coin in the privacy of his office... but it falls against a book and lands on edge. And so he makes neither choice, and Mr. Smith Goes to Washington.

Conversely, the 1972 movie of Graham Greene's novel Travels with my Aunt ends with a coin toss that will decide the future of one of the characters. The movie ends with the coin in mid-air.

The comic-book villain, Two-Face, has a double-sided coin (both sides are "heads") with one side defaced--a parallel to his actual character, because one side of his face is deformed--which he relies on for all his decisions.

## Number-theoretic version of "flipping"

There is no fair way to use a coin flip to settle a dispute between two parties over distance-- for example, two parties on the phone. The flipping party could easily lie about the outcome of the toss. Instead, the following algorithm can be used:

1. Party A chooses two large primes, either both congruent to 1, or both congruent to 3, mod 4, called p and q, and produces N = p*q; N is communicated to party B but p and q are not; N will be a product of two primes congruent to 1 mod 4. The primes should be chosen large enough that factoring of N is not computationally feasible.
2. Party B calls either "1" or "3", a claim as to the mod 4 status of p and q. For example, if p and q are congruent to 1 mod 4, and A called "3", A loses the toss.
3. Party A produces the primes, making the outcome of the toss obvious; party B can easily multiply them to check that A is being truthful.

See bit commitment for examples of similar protocols.

## References

• Joseph Ford. "How random is a coin toss?" Physics Today, 36:40-47, 1983.
• Joseph B. Keller. "The probability of heads". American Mathematical Monthly, 93:191-197, 1986.
• Vladimir Z. Vulovic and Richard E. Prange. "Randomness of a true coin toss". Physical Review A, 33:576-582, 1986.

• Heads or Tails? (http://www.sciencenews.org/articles/20040228/mathtrek.asp) (A discussion of the predictability of a coin toss; with references)
• The Not So Random Coin Toss (http://www.npr.org/display_pages/features/feature_1697475.html) (Brief blurb about Persi Diaconis' work, with a photograph of the coin-tossing machine)
• Dynamical Bias in the Coin Toss (http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf) (by Persi Diaconis, Susan Holmes and Richard Montgomery; very detailed)
• The Casting of Lots (http://www.jameslindlibrary.org/essays/casting_of_lots/casting.html) (Discussion of making decisions by chance outcomes throughout history)

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