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

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

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

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, usually by resting it on a bent index finger (usually of the dominant hand) and hitting the edge of it with the thumb, or a similar motion. During important events this may be done by a third person who is uninvolved or impartial to the dispute, although more commonly one of the disputants will do the flipping.

While the coin is in midair one of the parties calls "heads" or "tails". The coin may be left to fall freely, or caught by the tosser, either in the open palm or by bringing one hand down over it as it lands on the other hand or arm. In any case the participants then see whether it has landed with the upper side showing "heads"--the side with the portrait or profile on it, or "tails"--the opposite side. If the caller was correctly able to guess the side, then they win that round, otherwise they lose.

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 the United States is the UK a 2p, 10p or 50p piece is favoured. However, participants will use any coin that is handy.

Coin flipping in dispute resolution

Coin flipping is used to decide which team gets the kickoff, which way the teams will play, or similar questions in soccer matches, American football games, and almost any other sport requiring such decisions. The most famous case of this in the U.S. is the use of coin flipping in major league football games, especially the Super Bowl. A special mint coin, which later goes to the Football Hall of Fame, is used for this purpose, and other coins in that edition are sold as collectors items. The actual NFL rule is that the team winning the coin toss elects whether to choose which team kicks off, or whether to choose which team defends which end, in the first quarter; the other team makes the other one of the two choices, and then makes the same election at the start of the third quarter.

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

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

Physics of coin flipping

Experimental and theoretical analysis of coin tossing has shown that the outcome is predictable, to some degree at least, if the initial conditions of the toss 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. A mechanical coin flipper which imparts the same momentum to every toss has a predictable outcome.

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

Although it is extremely rare, there is an extremely slight possibility that a coin will come to rest on (and remain on) its edge. In such an instance, while it may cause temporary distraction, the only fair course of action would be to toss the coin again.

Coin flipping in fiction

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

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

[Someone familiar with Greene's novel or the stage version should edit this to say whether they end similarly.]

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

Checking if a coin is biased

Sometimes when given a coin, we wish to find out if the coin is fair (ie. the probability of obtaining head (or tail) in a toss is 50%.

One way of verifing this is to calculate the probability density function using the bayesian probability theory.

First perform a test by tossing the coin N times and carefully note down the number of heads H. So we have :

n = N Total number of toss is N

h = H Total number of head is H

T = N _ H Total number of tails

Next, let r be the actual probability of obtaining head in a single toss of the coin. This is the value which we wishes to find. Using the bayesian probability theory, we have :

The prior summarizes what we know about the distribution of r in the absence of any observation. We will assume in this case that the prior distribution of r is uniform over the interval [0, 1]. That is, f(r) = 1. That assumption should be considered provisional __ if some additional background information is found, we should modify the prior accordingly.

The probability of obtaining H heads in N toss of a coin with any value of r is given by

Putting it together we have :

Now using the identity

We obtained have the final formula for the probability density function:

Example

For example: N=10 H=7 ie. We toss the coin 10 times and get 7 heads

Graph of Example

Plot a graph of y= 1320 * x^7 * (1_x)^3 with x ranging from 0 to 1

This graph is the graph of the probability density function of r given that we had obtained 7 heads in 10 tosses (Note: r is the actual probability of obtaining head when tossing that coin).

So is the coin bias? I would be pretty confident that the coin is indeed bias because the actual value of r is more likely to be in the vicinity of 0.7 than 0.5

Shape of curve

The astute person would notice that the shape of the plotted curve is solely determined by the numerator while the denominator determines only the scaling of the plotted curve.

This means that you can plot the shape of the curve using just the equation and by observing the plotted curve, you can ascertain whether the coin is bias and roughly how much bias it is.

The value of r where f(r) have the maximum value is r_max = H / N as you would have expected.

How many times should the coin be toss

To determine the actual probability of obtaining head with an accuracy of within 10% , you should toss the coin 100 times. For a much better accuracy of within 1% , you need to toss the coin 10,000 times.

Number_theoretic version of "flipping"

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

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

See Also

• Two-up
• Heads_or_Tails

References

• Joseph Ford. "How random is a coin toss?" Physics Today, 36:40-47, 1983.

• Joseph B. Keller. "The probability of heads". American Mathematical Monthly, 93:191-197, 1986.

• Vladimir Z. Vulovic and Richard E. Prange. "Randomness of a true coin toss". Physical Review A, 33:576-582, 1986.

External links

• Heads or Tails? (http://www.sciencenews.org/articles/20040228/mathtrek.asp) (A discussion of the predictability of a coin toss; with references)

• The Not So Random Coin Toss (http://www.npr.org/display_pages/features/feature_1697475.html) (Brief blurb about Persi Diaconis' work, with a photograph of the coin-tossing machine)

• Dynamical Bias in the Coin Toss (http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf) (by Persi Diaconis, Susan Holmes and Richard Montgomery; very detailed)

• Whether divination by drawing lots is unlawful? (http://www.newadvent.org/summa/309508.htm) (From the Summa Theologica of St. Thomas Aquinas)

• The Casting of Lots (http://www.jameslindlibrary.org/essays/casting_of_lots/casting.html) (Discussion of making decisions by chance outcomes throughout history)

• Persi Diaconis' website (http://stat.stanford.edu/~cgates/PERSI/index.html) - including the paper Dynamical Bias in the Coin Toss PDF (http://stat.stanford.edu/~cgates/PERSI/papers/headswithJ.pdf)