What is Coin Toss?

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

Coin flipping is based on pure luck, under the assumption that 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, and or touching the coin) is considered cheating. It is generally assumed that the outcome is unpredictable, with equal probabilities for the two outcomes.

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 sometimes 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 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;
• if the coin falls to the ground, despite an attempt by the person flipping the coin to catch it, the process is usually not repeated, and if the winner wishes he may pass the win on the person receiving the loss, but the loser can not make any choices otherwise, meaning he must accept the winners deny.
• 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 tossing is a simple and unbiased way of settling a dispute or deciding between two or more arbitrary options. In a game theoretic analysis it provides even odds to all both sides involved, requiring little effort and preventing the dispute from escalating into a struggle. It is used widely in sports and other games to decide arbitrary factors such as which side of the field a team will play from, or which side will attack or defend initially. In teams sports it is often the captain who make the call, while the umpire or referee usually oversees such proceedings. A competitive method may be used instead of a toss in some situations, for example in basketball the jump ball is employed, while the faceoff plays a similar role in ice hockey.

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. Before the start of the game, and before overtime (if needed), the visiting team (or one so designated in a neutral site) calls said coin toss. A coin toss is also used to determine which team gets the higher draft pick if there are two teams with identical win-loss records.

In a soccer match, the team winning the coin toss chooses which goal to attack in the first half; the opposing team kicks off for the first half. For the second half, the teams switch ends, and the team that won the coin toss kicks off.

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. Factors such as wind and other conditions may affect the decision, for example in outdoor sports a player or team may choose to have the wind at their backs initially, hoping it will change direction later in 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.

The National Football League also has a coin toss as the very last resort in tie-breaking among teams for playoff berths and seeding. Because of the complicated rules for such tie-breaking, it is quite unlikely a coin toss would be needed. The coin toss is the very last tie-breaker because of its being non-competitive. There was a close call in 1970, with a relatively-simple tie-breaking system in effect, where the reversal of just one game's outcome would have led to a coin toss to decide the NFC wildcard team.

In the 1968 European Football Championship the semi-final between Italy and the Soviet Union finished 0-0 after extra-time. Penalty shoot-outs had not been invented and it was decided to toss a coin to see who reached the final, rather than play a replay. Italy won, and went on to become European champions.

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 December 2006 Australian television networks Seven and Ten resolved the issue of who would be broadcasting the 2007 AFL Grand Final with a toss of a coin. This decision was necessary because both networks would be sharing the broadcasting of the 2007 AFL Season. Network Ten subsequently won the toss.

In more casual settings, coin flipping is used simply to resolve arguments between friends or family members. Unlike Rock, Paper, Scissors, coin tossing is not usually invoked purely for amusement.

Fair results from a biased coin

If a cheater has altered a coin to prefer one side over another (a biased coin), surprisingly the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure:

1. Toss the coin twice.
2. If the results match, start over, forgetting both results.
3. If the results differ, use the first result, forgetting the second.

The reason this process produces a fair result is that the probability of getting heads and then tails must be the same as the probability of getting tails and then heads, as the coin is not changing its bias between flips. By excluding the events of two heads and two tails by repeating the procedure, the coin flipper is left with the only two remaining outcomes having equivalent probability. This procedure only works if the tosses are paired properly; if part of a pair is reused in another pair, the fairness may be ruined.

Some coins have been alleged to be unfair when spun on a table, but the results have not been substantiated or are not significant. Ref.

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 — the phase space is fairly regular.

Moreover, 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. Stage magicians and gamblers, with practice, are 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, it is possible that a coin will come to rest on its edge (estimated at roughly 1/6000 for a U.S. nickel). [1]

Coin flipping in fiction

At the beginning of an award winning 1939 movie, a state governor has to select an interim Senator, and he is being pressured by two opposing factions to choose between their respective candidates, 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. Consequently, he makes neither choice, and thus Mr. Smith Goes to Washington.

The 1972 movie adaptation 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, leaving their fate unresolved.

Two-Face, the comic book supervillain, 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 upon for all of his decisions.

In The Twilight Zone episode "A Penny for Your Thoughts," the main character buys a newspaper, and flips a coin into the collection pan, where it lands on its edge. As a consequence, he can hear people's thoughts, but at the end of the day the coin gets knocked off its edge, and he is no longer telepathic.

Tom Stoppard's Rosencrantz & Guildenstern Are Dead begins with a series of coin tosses that all come up heads, implying that the characters are suspended in one unchanging moment of time before becoming part of the play.

In the video game Final Fantasy VI, the brothers Edgar and Sabin flip a coin in order to determine who succeeds the throne of Figaro. It is later revealed that Edgar used a double-headed coin in order to win, allowing Sabin to live without the burden of the kingdom.

In the animated series Futurama, Professor Farnsworth creates a parallel universe. The only difference between our universe and the other is that every time someone flipped a coin, it landed on the opposite side. This leads to extremely different worlds and humorous confusion.

In the American comedy film Mouse Hunt, out of work brothers Lars and Ernie toss a coin to decide who gets to sleep in the only bed in the inherited house. The coin ends up spinning on the floor and coming to rest on edge—an extremely rare and unlikely occurrence—so the brothers share the bed.

The Hong Kong-made film Shaolin Soccer contains a scene in which one of Sing's brothers is being asked to join Sing's soccer team, and he refuses because he mathematically predicts the team will fail; he uses a coin toss to demonstrate his point, saying it has zero chance of landing on its edge. When the coin is carelessly dropped later in the scene, the brother is amazed to discover that it has, indeed, landed on its edge and gotten stuck inside a small crack in the asphalt.

The DVD of Final Destination 3, has a special feature allowing the viewer to flip a coin to determine the outcome of the movie; however, the outcome is fixed to maintain the plot, and the coin flip is meaningless.

Isaac Asimov's short story The Machine that Won the War ends with a character revealing that he made his decisions based on coin tosses.

Mathematics of coin flipping

The mathematical abstraction of the statistics of coin flipping is described by means of the Bernoulli process; a single flip of a coin is a Bernoulli trial. In the study of statistics, coin-flipping plays the role of being an introductory example of the complexities of statistics. A commonly treated textbook topic is that of checking if a coin is fair.

Coin flipping in telecommunications

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. In telecommunications and cryptography, 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 = pq; then N is communicated to party B, but p and q are not. It follows N will be congruent to 1 mod 4. The primes should be chosen large enough that factoring of N is not computationally feasible. The exact size will depend on how much time party B is to be given to make the choice in the next step, and on party B's expected resources.
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 B called "3", B 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.

Further information: bit commitment

In lotteries

The New Zealand lottery game Big Wednesday is probably the only lottery in the world to use a coin toss. It works like this: If you match all 6 of your numbers, the coin toss will decide whether you win a cash jackpot (minimum of NZ$25,000) or a bigger jackpot with luxury prizes (minimum of NZ$2 million cash, plus value of luxury prizes.) The coin toss is also used in determining the Second Chance winner's prize.

See also

• Two-up
• Heads or Tails
• Checking if a coin is fair
• Gambler's fallacy
• Bernoulli process
• Rock paper scissors

References

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

External links

• Heads or Tails? (A discussion of the predictability of a coin toss; with references)
• The Not So Random Coin Toss (Brief blurb about Persi Diaconis' work, with a photograph of the coin-tossing machine)
• Dynamical Bias in the Coin Toss (by Persi Diaconis, Susan Holmes, and Richard Montgomery; very detailed)
• Whether divination by drawing lots is unlawful? (From the Summa Theologiae of Thomas Aquinas)
• The Casting of Lots (Discussion of making decisions by chance outcomes throughout history)
• Coin Tossing — mathworld.com (Contains information about counterintuitive properties of coin tossing)
• Persi Diaconis' website — including the paper Dynamical Bias in the Coin Toss PDF
• Random.org: flip a virtual coin
• Flip-Coin.com: Simple web app to flip a virtual coin