Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to randomly choose between two alternatives. It is a form of sortition which inherently has two possible outcomes. The party who calls the side that is facing up when the coin lands wins.
Coin flipping 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 was referred to as cross and pile. [1] [2]
During a coin toss, the coin is thrown into the air such that it rotates edge-over-edge an unpredictable number of times. Either beforehand or when the coin is in the air, an interested party declares "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 correctly or was assigned the upper side is declared the winner.
It is possible for a coin to land on its side, usually by landing up against an object (such as a shoe) or by getting stuck in the ground. However, even on a flat surface it is possible for a coin to land on its edge. A computational model suggests that the chance of a coin landing on its edge and staying there is about 1 in 6000 for an American nickel. [3]
The coin may be any type as long as it has two distinct sides. Larger coins tend to be more popular than smaller ones. Some high-profile coin tosses, such as those in the Cricket World Cup and the Super Bowl, use custom-made ceremonial medallions. [4] [5]
Three-way coin flips are also possible, by a different process – this can be done either to choose one or two 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, the previous is either reversed (the odd coin out is the winner) or a regular two-way coin flip between the two remaining players can decide. The three-way flip is 75% likely to work each time it is tried (if all coins are heads or all are tails, each of which occur 1/8 of the time due to the chances being 0.5 by 0.5 by 0.5, the flip is repeated until the results differ), and does not require that "heads" or "tails" be called. A well-known 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), wherein three Texas high school football teams use a three-way coin flip. [6] [7] A legacy of that particular 1988 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.
"Heads and Tails" or "Heads or Tails" is an informal game of chance using repeated coin tosses, suitable for a roomful of seated people, typically a social or children's event. Initially all players stand. Before each coin toss, all still standing put their hands on either their head to indicate "heads" or their hips or buttocks to indicate "tails"; once the toss result is announced, those who guessed incorrectly sit down. The process repeats until the last player standing wins; often the last few players remaining are called to the announcer's table for the climax. A variant with faster elimination is played with two coins and players placing each hand separately. [8]
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 original 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. (The revived XFL, which launched in 2020, removed the coin toss altogether and allowed that decision to be made as part of a team's home field advantage.)
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 early-1970s introduction of the penalty shootout, coin tosses were occasionally needed to decide the outcome of drawn matches where a replay was not possible. The most famous instance of this was the semifinal game of the 1968 European Championship between Italy and the Soviet Union, which finished 0–0 after extra time. Italy won, and went on to become European champions. [9]
In cricket the toss is often significant, as the decision whether to bat or bowl first can influence the outcome of the game. The coin toss in cricket is more important than in other games because in many situations it can lead a team winning or losing the game. Factors such as pitch conditions, weather and the time of day are considered by the team captain who wins the toss. Now there are websites such as flip a coin online which domestic sports team use to toss the coin. [10]
Similarly, in tennis a coin toss is used in professional matches to determine which player serves first. The player who wins the toss decides whether to serve first or return, while the loser of the toss decides which end of the court each player plays on first.
In duels a coin toss was sometimes used to determine which combatant had the sun at his back. [11] 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. [12]
Fédération Internationale d'Escrime rules use a coin toss to determine the winner of some fencing matches that remain 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.[ citation needed ]
In the United States Asa Lovejoy and Francis W. Pettygrove, who each owned the claim to the land that would later become Portland, Oregon, wanted to name the new town after their respective hometowns of Boston, Massachusetts and Portland, Maine; Pettygrove won with the flip of a coin which has been preserved as the Portland Penny. [13]
Scientists sometimes use coin flipping to determine the order in which they appear on the list of authors of scholarly papers. [14]
In addition to its practical applications in sports, coin tossing is symbolic of the democratic principle of equal opportunity. When two parties face an impasse, the act of flipping a coin signifies a commitment to impartiality and a willingness to accept the outcome, no matter how arbitrary it may seem. This shared acceptance of chance as the ultimate arbiter can foster cooperation and conflict resolution in various aspects of life beyond sports, including business negotiations and interpersonal conflicts.
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. [15]
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).
"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 "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. [16]
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. [17] [18]
In the United States, when a new state is added to the Union, a coin toss determines the class of the senators (i.e., the election cycle in which the term each of the new state's senators will expire) in the US Senate. [19] Also, a number of states provide for "drawing lots" in the event an election ends in a tie, and this is usually resolved by a coin toss or picking names from a hat.[ citation needed ] A 2017 election to the 94th District of the Virginia House of Delegates resulted in a tie between Republican incumbent David Yancey and Democratic challenger Shelly Simmonds, with exactly 11,608 votes each. Under state law, the election was to be decided by drawing a name from a bowl, although a coin toss would also have been an acceptable option. The chair of the Board of Elections drew the film canister with Yancey's name, and he was declared the winner. [20] Additionally, the outcome of the draw determined control of the entire House, as Republicans won 50 of the other 99 seats and Democrats 49. A Yancey win extended the Republican advantage to 51–49, whereas a Simmonds win would have resulted in a 50–50 tie. As there is no provision for breaking ties in the House as a whole, this would have forced a power sharing agreement between the two parties. [21]
The outcome of coin flipping has been studied by the mathematician and former magician 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 0.495 or 0.503." [22]
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. [22]
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, [23] though a 2009 attempt to verify this experimentally at Berkeley with 40,000 tosses gave ambiguous results. [24] A much larger 2023 University of Amsterdam study (which won a 2024 Ig Nobel Prize [25] ) performed 350,757 tosses, finding an average same-side bias of 50.8%, but which varied from person to person. [26] [27]
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.[ citation needed ]
Stage magicians and gamblers, with practice, are able to greatly increase the same-side-up bias, whilst still making throws which are visually indistinguishable from normal throws. [22] Conjurers trim the edges of coins so that when spun on a surface, they usually land on a particular face.[ citation needed ]
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" than that "THH will follow HHT"; [29] see also Penney's game.
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.
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:
bubblejet and knockback respectively.knockback.bubblejet knockback tails), and sends that hash (e.g. 3fe97d8e6456a1dce4508d00251345e3) to Bob.bubblejet knockback tails, including both of their words and a call of tails3fe97d8e6456a1dce4508d00251345e3 is the hash of bubblejet knockback tailsBob, by providing his own random word, guarantees that Alice is not able to precompute an image pair of "tail/random string" or "head/random string", for two different random words. Bob is also unable to reverse Alice's hash to see what her chosen outcome was before flipping the coin, and to lie effectively about its outcome, because he does not know Alice's random word at that point in the process.
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.
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." [30]
Danish poet Piet Hein's 1966 book Grooks includes a poem, "A Psychological Tip", on a similar theme:
Whenever you're called on to make up your mind,
And you're hampered by not having any,
The best way to solve the dilemma, you'll find,
Is simply by spinning a penny.
No—not so that chance shall decide the affair
While you're passively standing there moping;
But the moment the penny is up in the air,
You suddenly know what you're hoping.
The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event has occurred less frequently than expected, it is more likely to happen again in the future. The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the expected number of sixes.
In probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die.
In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variablesXi are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin. Every variable Xi in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes ; this generalization is known as the Bernoulli scheme.
In the theory of probability and statistics, a Bernoulli trial is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his Ars Conjectandi (1713).
In logic and probability theory, two events are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.
In probability theory, an event is said to happen almost surely if it happens with probability 1. In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely ; however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0.
In cricket, the toss is the flipping of a coin to determine which captain will have the right to choose whether their team will bat or field at the start of the match.
I Ching divination is a form of cleromancy applied to the I Ching. The text of the I Ching consists of sixty-four hexagrams: six-line figures of yin (broken) or yang (solid) lines, and commentaries on them. There are two main methods of building up the lines of the hexagram, using either 50 yarrow stalks or three coins. Some of the lines may be designated "old" lines, in which case the lines are subsequently changed to create a second hexagram. The text relating to the hexagram(s) and old lines is studied, and the meanings derived from such study can be interpreted as an oracle.
In statistical significance testing, a one-tailed test and a two-tailed test are alternative ways of computing the statistical significance of a parameter inferred from a data set, in terms of a test statistic. A two-tailed test is appropriate if the estimated value is greater or less than a certain range of values, for example, whether a test taker may score above or below a specific range of scores. This method is used for null hypothesis testing and if the estimated value exists in the critical areas, the alternative hypothesis is accepted over the null hypothesis. A one-tailed test is appropriate if the estimated value may depart from the reference value in only one direction, left or right, but not both. An example can be whether a machine produces more than one-percent defective products. In this situation, if the estimated value exists in one of the one-sided critical areas, depending on the direction of interest, the alternative hypothesis is accepted over the null hypothesis. Alternative names are one-sided and two-sided tests; the terminology "tail" is used because the extreme portions of distributions, where observations lead to rejection of the null hypothesis, are small and often "tail off" toward zero as in the normal distribution, colored in yellow, or "bell curve", pictured on the right and colored in green.
Gameplay in American football consists of a series of downs, individual plays of short duration, outside of which the ball is or is not in play. These can be plays from scrimmage – passes, runs, punts or field goal attempts – or free kicks such as kickoffs and fair catch kicks. Substitutions can be made between downs, which allows for a great deal of specialization as coaches choose the players best suited for each particular situation. During a play, each team should have no more than 11 players on the field, and each of them has specific tasks assigned for that specific play.
In game theory, the battle of the sexes is a two-player coordination game that also involves elements of conflict. The game was introduced in 1957 by R. Duncan Luce and Howard Raiffa in their classic book, Games and Decisions. Some authors prefer to avoid assigning sexes to the players and instead use Players 1 and 2, and some refer to the game as Bach or Stravinsky, using two concerts as the two events. The game description here follows Luce and Raiffa's original story.
In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and probability theory can provide guidance on two types of question; specifically those of how many trials to undertake and of the accuracy of an estimate of the probability of turning up heads, derived from a given sample of trials.
The sign test is a statistical test for consistent differences between pairs of observations, such as the weight of subjects before and after treatment. Given pairs of observations for each subject, the sign test determines if one member of the pair tends to be greater than the other member of the pair.
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin.
Penney's game, named after its inventor Walter Penney, is a binary (head/tail) sequence generating game between two players. Player A selects a sequence of heads and tails, and shows this sequence to player B. Player B then selects another sequence of heads and tails of the same length. Subsequently, a fair coin is tossed until either player A's or player B's sequence appears as a consecutive subsequence of the coin toss outcomes. The player whose sequence appears first wins.
Byzantine fault tolerant protocols are algorithms that are robust to arbitrary types of failures in distributed algorithms. The Byzantine agreement protocol is an essential part of this task. The constant-time quantum version of the Byzantine protocol, is described below.
Flipism, sometimes spelled "flippism", is a personal philosophy under which decisions are made by flipping a coin. It originally appeared in the Donald Duck Disney comic "Flip Decision" by Carl Barks, published in 1953. Barks called a practitioner of "flipism" a "flippist".
In probability theory, an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two possible outcomes is known as a Bernoulli trial.
The "hot hand" is a phenomenon, previously considered a cognitive social bias, that a person who experiences a successful outcome has a greater chance of success in further attempts. The concept is often applied to sports and skill-based tasks in general and originates from basketball, where a shooter is more likely to score if their previous attempts were successful; i.e., while having the "hot hand.” While previous success at a task can indeed change the psychological attitude and subsequent success rate of a player, researchers for many years did not find evidence for a "hot hand" in practice, dismissing it as fallacious. However, later research questioned whether the belief is indeed a fallacy. Some recent studies using modern statistical analysis have observed evidence for the "hot hand" in some sporting activities; however, other recent studies have not observed evidence of the "hot hand". Moreover, evidence suggests that only a small subset of players may show a "hot hand" and, among those who do, the magnitude of the "hot hand" tends to be small.
Consider two remote players, connected by a channel, that don't trust each other. The problem of them agreeing on a random bit by exchanging messages over this channel, without relying on any trusted third party, is called the coin flipping problem in cryptography. Quantum coin flipping uses the principles of quantum mechanics to encrypt messages for secure communication. It is a cryptographic primitive which can be used to construct more complex and useful cryptographic protocols, e.g. Quantum Byzantine agreement.
There are two methods to decide the outcome in the event of a draw - either a coin is flipped or the parties draw straws.
He or she [the returning officer] can use any random method such as tossing a coin, but the recommended way is to ask each candidate to write their name on a blank slip of paper and place it in a container.
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