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probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, a sequence of
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial 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 c ...
s 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. John Edmund Kerrich performed experiments in
coin flipping 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 choose between two alternatives, heads or tails, sometimes used to resolve a dispute betwe ...
and found that a coin made from a wooden disk about the size of a
crown A crown is a traditional form of head adornment, or hat, worn by monarchs as a symbol of their power and dignity. A crown is often, by extension, a symbol of the monarch's government or items endorsed by it. The word itself is used, partic ...
and coated on one side with
lead Lead is a chemical element with the symbol Pb (from the Latin ) and atomic number 82. It is a heavy metal that is denser than most common materials. Lead is soft and malleable, and also has a relatively low melting point. When freshly cut, ...
landed heads (wooden side up) 679 times out of 1000. In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table.
Edwin Thompson Jaynes Edwin Thompson Jaynes (July 5, 1922 – April 30, 1998) was the Wayman Crow Distinguished Professor of Physics at Washington University in St. Louis. He wrote extensively on statistical mechanics and on foundations of probability and statist ...
claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time. Exploring the problem of checking whether a coin is fair is a well-established pedagogical tool in teaching
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
.


Role in statistical teaching and theory

The probabilistic and statistical properties of coin-tossing games are often used as examples in both introductory and advanced text books and these are mainly based in assuming that a coin is fair or "ideal". For example, Feller uses this basis to introduce both the idea of
random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
s and to develop tests for
homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
within a sequence of observations by looking at the properties of the runs of identical values within a sequence. The latter leads on to a runs test. A
time-series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
consisting of the result from tossing a fair coin is called a
Bernoulli process In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) 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. T ...
.


Fair results from a biased coin

If a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly.
John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
gave the following procedure: # Toss the coin twice. # If the results match, start over, forgetting both results. # 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 and the two flips are independent. This works only if getting one result on a trial doesn't change the bias on subsequent trials, which is the case for most non- malleable coins (but ''not'' for processes such as the Pólya urn). 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. Also, the coin must not be so biased that one side has a probability of zero. This method may be extended by also considering sequences of four tosses. That is, if the coin is flipped twice but the results match, and the coin is flipped twice again but the results match now for the opposite side, then the first result can be used. This is because HHTT and TTHH are equally likely. This can be extended to any power of 2. The expected value of flips at the n game E(F_n) is not hard to calculate, first notice that in step 3 whatever the event HT or TH we have flipped the coin twice so E(F_n, HT,TH)=2 but in step 2 (TT or HH) we also have to redo things so we will have 2 flips plus the expected value of flips of the next game that is E(F_n, TT,HH)=2+E(F_) but as we start over the expected value of the next game is the same as the value of the previous game or any other game so it doesn't really depend on n thus E(F)=E(F_n)=E(F_) (this can be understood the process being a martingale E(F_, F_n,...,F_1)=F_n where taking the expectation again get us that E(E(F_, F_n,...,X_1))=E(F_n) but because of the
law of total expectation The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...
we get that E(F_)=E(E(F_, F_n,...,F_1))=E(F_n)) hence we have: \begin E(F) &=E(F_n)\\ &=E(F_n, TT,HH)P(TT,HH)+E(F_n, HT,TH)P(HT,TH)\\ &=(2+E(F_))P(TT,HH)+2P(HT,TH)\\ &=(2+E(F))P(TT,HH))+2P(HT,TH)\\ &=(2+E(F))(P(TT)+P(HH))+2(P(HT)+P(TH))\\ &=(2+E(F))(P(T)^2+P(H)^2)+4P(H)P(T)\\ &=(2+E(F))(1-2P(H)P(T))+4P(H)P(T)\\ &=2+E(F)-2P(H)P(T)E(F)\\ \end \therefore E(F)=2+E(F)-2P(H)P(T)E(F)\Rightarrow E(F)=\frac=\frac The more biased our coin is, the more likely it is that we will have to perform a greater number of trials before a fair result.


See also

* Checking whether a coin is fair *
Coin flipping 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 choose between two alternatives, heads or tails, sometimes used to resolve a dispute betwe ...
*
Feller's coin-tossing constants Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in ''n'' independent tosses of a fair coin, no run of ''k'' consecutive heads (or, equally, tails) appears. William Feller showed that if ...


References


Further reading


Available
from
Andrew Gelman Andrew Eric Gelman (born February 11, 1965) is an American statistician and professor of statistics and political science at Columbia University. Gelman received bachelor of science degrees in mathematics and in physics from MIT, where he w ...
's website *{{cite news, title=Lifelong debunker takes on arbiter of neutral choices: Magician-turned-mathematician uncovers bias in a flip of a coin , url=http://news-service.stanford.edu/news/2004/june9/diaconis-69.html , work=Stanford Report, date= 2004-06-07 , access-date=2008-03-05 *John von Neumann, "Various techniques used in connection with random digits," in A.S. Householder, G.E. Forsythe, and H.H. Germond, eds., ''Monte Carlo Method'', National Bureau of Standards Applied Mathematics Series, 12 (Washington, D.C.: U.S. Government Printing Office, 1951): 36-38. Experiment (probability theory) Gambling mathematics Coin flipping