Coin flipping

Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air to choose between two alternatives, sometimes to resolve a dispute between two parties. It is a form of sortition which inherently has only two possible and equally likely outcomes.

History

The historical origin of coin flipping is the interpretation of a chance outcome as the expression of divine will.

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.^[1] In England, this game was referred to as cross and pile.^[1][2] The expression Heads or Tails results from heads and tails being considered opposite body parts.

Process

During a coin toss, the coin is thrown into the air such that it rotates edge-over-edge several times. Either beforehand or when the coin is in the air, an interested party calls "heads" or "tails", indicating which side of the coin that party is choosing. The other party is assigned the opposite side. Depending on custom, the coin may be caught, caught and inverted, or allowed to land on the ground. When the coin comes to rest, the toss is complete and the party who called or was assigned the face-up side is declared the winner.

It is theoretically possible for a coin to land on its edge, either by landing up against an object (such as a shoe) or by getting stuck in the ground (as famously happened during the December 8, 2013 NFL match up between the Philadelphia Eagles and Detroit Lions, which took place during a heavy snowstorm). Angular momentum typically prevents most coins from landing on their edges unsupported if flipped. Such cases in which a coin does land on its edge are exceptionally rare and in most cases the coin is simply re-flipped.^[3]

The coin may be any type as long as it has two distinct sides; it need not be a circulating coin as such. Larger coins tend to be more popular than smaller ones. Most high-profile coin tosses use custom-made ceremonial medallions.

Three-way

Three-way coin flips are also possible, by a different process – this can be done either to choose two out of three, or to choose one out of three. To choose two out of three, three coins are flipped, and if two coins come up the same and one different, the different one loses (is out), leaving two players. To choose one out of three, either reverse this (the odd coin out is the winner), or add a regular two-way coin flip between the remaining players as a second step. Note that the three-way flip is 75% likely to work each time it is tried (if all coins are heads or all are tails, which occurs 1/4 of the time, the flip is repeated until the results differ), and does not require that "heads" or "tails" be called. A famous example of such a three-way coin flip (choose two out of three) is dramatized in Friday Night Lights (originally a book, subsequently film and TV series), where three high school football teams with identical records use a three-way coin flip – at a truck stop – to determine which two will advance to the playoffs.^[4][5] A legacy of this coin flip was to reduce the use of coin flips to break ties in Texas sports, instead using point-systems to reduce the frequency of ties.

Use 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 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; these decisions may tend to favor one side, or may be neutral. Factors such as wind direction, the position of the sun, and other conditions may affect the decision. 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 face-off 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 football matches, American football games, Australian rules football, volleyball, and other sports requiring such decisions. In the U.S. a specially minted coin is flipped in National Football League games; the coin is then sent to the Pro Football Hall of Fame, and other coins of the special series minted at the same time are sold to collectors. The XFL, a short-lived American football league, attempted to avoid coin tosses by implementing a face-off 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 (a modified version was adopted by X-League Indoor Football, in which each player pursued his own ball), and coin tossing remains the method of choice in American football.

In an association football 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. Coin tosses are also used to decide which team has the pick of going first or second in a penalty shoot-out. Before the introduction of the penalty shoot-out, coin tosses were occasionally needed to decide the outcome of tied matches. The most famous instance of this was the semifinal game of the 1968 European Championship in Italy between Italy and the Soviet Union, which finished 0-0 after extra time. Italy won, and went on to become European champions.^[6]

In cricket the toss is often significant, as the decision whether to bat or bowl first can influence the outcome of the game.

In duels a coin toss was sometimes used to determine which combatant had the sun at his back.^[7] 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 for tie-breaking among teams for playoff berths and seeding, but the rules make the need for coin toss, which is random rather than competitive, very unlikely. A similar procedure breaks ties for the purposes of seeding in the NFL Draft; these coin tosses are more common, since the tie-breaking procedure for the draft is much less elaborate than the one used for playoff seeding.

Major League Baseball once conducted a series of coin flips as a contingency on the last month of its regular season to determine home teams for any potential one-game playoff games that might need to be added to the regular season. Most of these cases did not occur. From the 2009 season, the method to determine home-field advantage was changed.^[8]

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. Although in most international matches this is now done electronically by the scoring apparatus.

In the United States Asa Lovejoy and Francis W. Pettygrove, who owned the claim to the land that would later become Portland, Oregon, each wanted to name the new town after their respective hometowns of Boston, Massachusetts and Portland, Maine; Pettygrove won the coin flip.^[9]

Scientists sometimes use coin flipping to determine the order in which they appear on the list of authors of scholarly papers.^[10]

Politics

Australia

In December 2006 Australian television networks Seven and Ten, which shared the broadcasting of the 2007 AFL Season, decided who would broadcast the Grand Final with the toss of a coin. Network Ten won. in Australia it is commonly used in cricket.

Canada

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 ($100.3800623 per km), $146,584.65 ($91.33 per km, exactly), and two equal bids of $111,242.55 ($69.31 per km, exactly).

Philippines

"Drawing of lots" is one of the methods to break ties to determine a winner in an election; the coin flip is considered an acceptable variant. Each candidate will be given five chances to flip a coin; the candidate with the most number of "heads" wins. The 2013 mayoral election in San Teodoro, Oriental Mindoro was decided on a coin flip, with a winner being proclaimed after the second round when both candidates remained tied in the first round.^[11]

United Kingdom

In the United Kingdom, if a local or national election has resulted in a tie where candidates receive exactly the same number of votes, then the winner can be decided either by drawing straws/lots, coin flip, or drawing a high card in pack of cards.^[12][13]

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."^[14]

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.^[14]

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,^[15] though a subsequent attempt to verify this experimentally gave ambiguous results.^[16] Stage magicians and gamblers, with practice, are able to greatly increase this bias, whilst still making throws which are visually indistinguishable from normal throws.^[14]

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.^[17] (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.

Telecommunications

There is no reliable way to use a true coin flip to settle a dispute between two parties if they cannot both see the coin—for example, over 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. Alice and Bob each choose a random string, "ljngjkrjgnfdudiudd" and "gfdgdfjkherfsfsd" respectively.
2. Alice chooses an outcome for an imaginary coin flip, such as "tail"
3. Bob sends Alice his random string "gfdgdfjkherfsfsd"
4. Alice immediately computes a cryptographic hash of the string "tail ljngjkrjgnfdudiudd gfdgdfjkherfsfsd", which is 59dea408d43183a3937957e71a4bcacc616d9cbc and sends it to Bob
5. Alice asks Bob: "heads or tails"?
6. Bob says, for instance, "heads".
7. Alice tells him she's just won, and proves it by showing the string "tail ljngjkrjgnfdudiudd gfdgdfjkherfsfsd".
8. Bob can check that Alice didn't lie by computing the SHA-1 of the string himself
9. Furthermore Bob by providing his own randomly generated string guarantees that Alice wasn't able to precompute an image pair of "tail/random string" or "head/random string".

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.

Clarifying feelings

A technique attributed to Sigmund Freud to help in making difficult decisions is to toss a coin not actually to determine the decision, but to clarify the decision-maker's feelings. He explained: "I did not say you should follow blindly what the coin tells you. What I want you to do is to note what the coin indicates. Then look into your own reactions. Ask yourself: Am I pleased? Am I disappointed? That will help you to recognize how you really feel about the matter, deep down inside. With that as a basis, you'll then be ready to make up your mind and come to the right decision."^[18]

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.^[19]

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 that always come up heads.

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.

The DC Comics supervillain Two-Face, (most famously as a member of Batman's rogues gallery), has a double-headed coin with one side defaced—a parallel to his actual character, because one side of his face is deformed from acid throwing—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 film The Dark Knight, the coin starts out clean, and Harvey Dent (played by Aaron Eckhart) uses this trick coin to seemingly leave important decisions to chance ("Heads I go through with it"). The coin is later blackened on one side in the explosion that kills his fiancée Rachel Dawes and burns half of his face. In Batman Forever, Two Face, (portrayed by Tommy Lee Jones), flips his coin multiple times in one scene to see how far Bruce Wayne can come within range of his gun. After a number of tries, the scarred side finally comes up and he fires anyway.

Tom Stoppard's Rosencrantz & Guildenstern Are Dead begins with a series of coin tosses that all come up heads, causing the characters to question the nature of chance and randomness.

In the video game Final Fantasy VI 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 the video game "Shenmue 2" gang leader Wuying Ren carries a double-sided coin in each pocket, asking for a head or tail call before pulling the coin out and flipping the coin. This process guarantees him victory in the outcome of the coin toss, usually forcing protagonist Ryo into a dangerous situation. The trickery behind this method is revealed as the characters part ways at the end of the game.

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.

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

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 the book No Country for Old Men (and the film made of it), Anton Chigurh, the story's villain, occasionally flips coins for potential victims to decide whether or not to kill them. He allows people to place their life in the hands of divine providence, and those who refuse to choose are killed anyway, for their obstinacy 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,"^[20] 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 or to displace that responsibility onto his victims.^[21]

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: Yoshihiro flips the coin in the air and quickly snatches it before the coin falls, then Gon or his companions have to say which hand the employee caught the coin with. This proves to be incredibly difficult due to the unrealistic speed of the coin flipper's hands. Gon is very observant and is occasionally able to guess right. See Flipism.

In The Mentalist episode "Blood In, Blood Out" during season 2, CBI consultant Patrick Jane wins a wager by flipping a coin and it landing on heads 20 times in a row. It is later shown that he rigged the coin in his favor.

In the video game Bioshock Infinite there is a coin-flipping sequence early in the game. Booker DeWitt is asked by the Lutece Twins to choose a side. He sometimes chooses "heads" but other times "tails" - but the coin always lands on heads. This is tightly connected to a central subject of the narrative (constants and variables in different realities) and to a central thematic of the game (the illusion of one's choices having an impact).

Coin landing on its edge in fiction

A coin toss has a theoretical third outcome, in which the coin comes to rest upright on its edge, rather than falling to either heads or tails. Such an outcome is fairly unlikely, having been estimated at approximately 1 in 6000 tosses on a hard and flat surface,^[22] but is seen in fiction, often for comedic effect. Such an outcome usually results in either a tied coin toss, or victory to a person who successfully called "edge". One can also do the math to find the measurements of the coin such that the probability of it landing on its side is 1/3 when the coin only rotates around 1 horizontal axis. That is, P(Heads) = P(Tails) = P(Landing on Side) = 1/3. We define r to be the radius of the coin and ξ to be the ratio of the height of the coin to the diameter of the coin, so 2ξr = h. From geometry, we also have that tan(θ) = r/rξ = 1/ξ. Let the angle between the normal vector to the heads surface = θ. Then for the coin to land on heads it must land in a region of size 2θ. The same applies to tails. Thus, the probability of the coin landing on its side is equal to 2π - 4θ. This gives us that θ = π - 2θ, and thus θ = π/3. Thus, tan(θ) = tan(π/3) = √3 = 1/ξ, and therefore ξ = 1/√3.

Examples

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 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 knocks the coin off its edge when dropping another coin into the pan, which causes him to lose his telepathic ability.

In the Smooth Criminal music video, Michael Jackson flips a coin through a jukebox which plays the tune of Smooth Criminal and he starts dancing.

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, 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 become stuck inside a small crack in the asphalt.

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 was not good enough). The coin is shown to land on its edge, leaving Malcolm uncertain what to do.

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

In How I Met Your Mother episode Blitzgiving, Steve Henry flipped a coin just as Barney left the room. The coin landed on its side and remained there until Barney returned.

In The Simpsons episode "Waverly Hills 9-0-2-1-D'oh", A city inspector asks Homer to call a coin toss, to which Homer calls as the last moment as side, proving to be correct.

The Peter Serafinowicz Show features a sketch parodying Who Wants to Be a Millionaire?, where contestants must correctly guess the outcome of a series coin tosses to win. Presented with only the options 'heads' or 'tails', one contestant is forced to lose when the coin lands on its edge.

In the video game Soul Reaver 2, Kain proposes the possibility of a coin landing on its edge when discussing a fateful outcome critical to plot development.

In the Terry Pratchett book The Colour of Magic, Rincewind the Wizzard (sic) uses this event to show the strength of the ambient magic around them.

See also

• Bernoulli process
• Checking whether a coin is fair
• Flipism
• Penney's game
• Gambler's fallacy
• Rock-paper-scissors
• Two-up
• Two-Face
• Obverse and reverse

Footnotes

References

• Ford, Joseph (1983). "How random is a coin toss?". Physics Today 36 (4): 40–47. doi:10.1063/1.2915570. 
• Keller, Joseph B. (1986). "The probability of heads". American Mathematical Monthly (Mathematical Association of America) 93 (3): 191–197. doi:10.2307/2323340. JSTOR 2323340. 
• Vulovic, Vladimir Z.; Prange, Richard E. (1986). "Randomness of a true coin toss". Physical Review A 33 (1): 576–582. doi:10.1103/PhysRevA.33.576. PMID 9896645. 

External links

Wikimedia Commons has media related to Coin flipping.

• 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)
• CoinToss website (Online coin toss)
• Flip a coin website (Flip a coin virtually)
• 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)
• Leads in Coin Tossing by Fiona Maclachlan, The Wolfram Demonstrations Project.
• Simple "Flip coin" php implementation example.(Php code sample based on random function)