Coin flipping

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. It is a form of sortition that by nature typically has only two possible outcomes.

History

A denarius by Maximinus.

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.

Process

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 mostly on nail of the thumb of the dominant hand with a small amount of the coin resting on the index finger,
• 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 winner's denial.
• 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 and ubiquity; in the UK a 2p, 10p or 50p piece is favoured. However, participants will use any coin that is handy. Americans may also use larger, though less common, coins such as the fifty-cent piece, Susan B. Anthony and golden dollars, and the largest of all coins still in general circulation, the increasingly rare Eisenhower Dollar.

Use in dispute resolution

The coin toss at the start of the Super Bowl XLIII American football game

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 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 team sports it is often the captain who makes 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 end of the field the teams will play to and/or which team gets first use of the ball, or similar questions in soccer matches, American football games, Australian rules football, volleyball, 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 at the Super Bowl, and other coins in that edition are sold as collectors items; the coin used, and the collector editions of the coins are minted by The Highland Mint. 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 and strength of schedule. The XFL, a short-lived American football league, attempted to avoid coin tosses by implementing a faceoff style "opening scramble," in which one player from each team tried to recover a loose football; the team whose player recovered the ball got first choice. Because of the high rate of injury in these events, it has not achieved mainstream popularity in any football league, and coin tossing remains the method of choice in virtually all of American football.

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.

Tossing a coin is common in many sports, such as cricket, where it used before the game to decide which team bats and which bowls

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 duels, a coin toss was sometimes used to determine which combatant had the sun at his back.. In some 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 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.

Major League Baseball once conducted a series of coin flips each September, the last month of its regular season, to determine home teams for any potential one-game playoff games that may need to be augmented to the regular season. This was done as a contingency only, and most of the one-game playoff scenarios for which coin flips were conducted did not occur. Starting with the 2009 season, the method to determine home-field advantage was changed to a performance-based criteria, starting with the head-to-head record between the tied clubs.

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.

Fédération Internationale d'Escrime rules use a coin toss to determine the winner of a fencing match that remains tied at the end of a sudden-death extra minute of competition.

One significant coin toss in United States history involved the naming of the city of Portland, Oregon. Asa Lovejoy and Francis W. Pettygrove, who owned the claim to the land that would later become Portland, each wanted to name their new town after their respective hometowns of Boston, Massachusetts and Portland, Maine. Pettygrove prevailed in the coin flip, and the town was named Portland.

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.

Physics

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. Further, in actual flipping, people exhibit slight bias – "coin tossing is fair to two decimals but not to three. That is, typical flips show biases such as .495 or .503."

In studying coin flipping, to observe the rotation speed of coin flips, Diaconis first used a strobe light and a coin with one side painted black, the other white, so that when the speed of the strobe flash equaled the rotation rate of the coin, it would appear to always show the same side. This proved difficult to use, and rotation rate was more accurately computed by attaching floss to a coin, such that it would wind around the coin – after a flip, one could count rotations by unwinding the floss, and then compute rotation rate as flips over air time.

Moreover, their theoretical analysis of the physics of coin tosses predicts 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, though a subsequent attempt to verify this experimentally gave ambiguous results. 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, 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.

Counterintuitive properties

Human intuition about conditional probability is often very poor and can give rise to some seemingly surprising observations. For example, if the successive tosses of a coin are recorded as a string of "H" and "T", then for any trial of tosses, it is twice as likely that the triplet TTH will occur before THT than after it. It is three times as likely that THH will precede HHT. (See Penney's game)

Mathematics

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 true 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. These are called p and q, and their product is "N." (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: Commitment scheme

In lotteries

The New Zealand lottery game Big Wednesday uses a coin toss. If a player matches all 6 of their numbers, the coin toss will decide whether they 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.

In fiction

George Raft became famous as the coin-flipping gangster "Guino Rinaldo" in the 1932 Howard Hawks/Howard Hughes film Scarface (1932). Bugs Bunny parodies Raft in the classic 1946 animated short film Racketeer Rabbit. Raft himself later parodied his own gangster persona as the character "Spats Colombo" in Billy Wilder's 1959 comedy Some Like It Hot: Raft sees another mobster flipping a coin and responds, "Where did you pick up that cheap trick?" Raft's coin-tossing established a distinctive motif used in numerous later gangster movies.

In the 1939 film Mr. Smith Goes to Washington, 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 and 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 chooses Mr. Smith.

In the climax of Sholay, Veeru and Jaidev decide their next strategy over their encounter with the villains by tossing a coin (they are in habit of deciding over the affairs between themselves this way). It is revealed at the end that the coin used by him is actually a trick-coin (i.e. it would always show heads on tossing).

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 (most famously as a member of Batman's rogues gallery), 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. He will do evil if it lands on the defaced side, and good on the other side. The coin is also representative of alter-ego Harvey Dent's obsession with dualism and the number 2. In the movie The Dark Knight, the coin starts out clean, and Harvey Dent uses this trick coin to seemingly leave important decisions to chance ("Heads I go through with it"). The coin is later blackened in the explosion that maims his face.

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 he purposely knocks the coin off its edge, in order to get rid of his telepathic ability.

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. This coin is also seen if Edgar is present in the first encounter with the gambler Setzer who is highly amused by it when it is used to trick him into providing his airship.

In Futurama episode The Farnsworth Parabox, 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.

The final episode of the American television series JAG ends with an incomplete coin flip.

In an episode of Malcolm in the Middle, Malcolm decides to flip a coin in order to resolve a dispute about keeping a potentially offensive cardboard cut-out up in the store that he works in (citing that logic wasn't good enough). The coin is shown to land on its side, leaving Malcolm bemused as to what to do.

In both the book and the film of No Country for Old Men, Anton Chigurh, the story's primary antagonist, occasionally flips coins for potential victims. He allows people to place their life in the hands of divine providence, and those who refuse the chance to live are killed anyway, for their obstinancy and refusal to submit to Fate. The meaning of Chigurh's coin-flipping is left ambiguous (in both the book and the film), and has led to considerable discussion: commentators suggest, for example, that Chigurh views himself as simply following the will of the universe, or is "merely cruel," or that it is an inevitable outgrowth of his (perceived) atheism or that Chigurh is in fact a stand-in for fate, or alternatively that his adherence to chance is a way for him to deny responsibility for his actions and/or to displace that responsibility onto his victims.

In the manga/anime of Hunter x Hunter by Yoshihiro Togashi, a servant of the Zaolydeck family challenges Gon and his companions, Leorio and Kurapica, to a game involving a coin flip. The game is simple: The employee flips the coin in the air and then quickly grabs it before the coin falls, and then Gon or his companions have to figure out which hand did the employee catch the coin with. This proves to be incredibly difficult with the unrealistic speed of the coin flipper's hands. In any case Gon is very observant and is occasionally able to guess right. See Flipism.

In Scrubs episode "My Best Friend's Baby's Baby and My Baby's Baby", protagonist J.D. and Kim cannot decide whether to keep their baby or not, after gotten accidentally pregnant. When all else fails, they flip a coin, which ironically lands on its side.

See also

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

Footnotes

James Schoonover loses Super Bowl 44 coin toss and buys Dan Rogers Basil Thai

References

• Ford, Joseph (1983). "How random is a coin toss?". Physics Today. 36: 40–47. doi:10.1063/1.2915570.
• Keller, Joseph B. (1986). "The probability of heads". American Mathematical Monthly. 93: 191–197. doi:10.2307/2323340.
• Vulovic, Vladimir Z. (1986). "Randomness of a true coin toss". Physical Review A. 33: 576–582. doi:10.1103/PhysRevA.33.576. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

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 Theologica 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
• Leads in Coin Tossing by Fiona Maclachlan, The Wolfram Demonstrations Project.