The Chicago Cubs are an American professional baseball team based in Chicago, Illinois. The Cubs compete in Major League Baseball as a member club of the National League Central division; the team plays its home games at Wrigley Field, located on the city's North Side. The Cubs are one of two major league teams in Chicago; the Cubs, first known as the White Stockings, were a founding member of the NL in 1876, becoming the Chicago Cubs in 1903. The Cubs have appeared in a total of eleven World Series; the 1906 Cubs won 116 games, finishing 116–36 and posting a modern-era record winning percentage of.763, before losing the World Series to the Chicago White Sox by four games to two. The Cubs won back-to-back World Series championships in 1907 and 1908, becoming the first major league team to play in three consecutive World Series, the first to win it twice. Most the Cubs won the 2016 National League Championship Series and 2016 World Series, which ended a 71-year National League pennant drought and a 108-year World Series championship drought, both of which are record droughts in Major League Baseball.
The 108-year drought was the longest such occurrence in all major North American sports. Since the start of divisional play in 1969, the Cubs have appeared in the postseason nine times through the 2017 season; the Cubs are known as "the North Siders", a reference to the location of Wrigley Field within the city of Chicago, in contrast to the White Sox, whose home field is located on the South Side. The Cubs have multiple rivalries. There is a divisional rivalry with the St. Louis Cardinals, a newer rivalry with the Milwaukee Brewers and an interleague rivalry with the Chicago White Sox; the Cubs began playing in 1870 as the Chicago White Stockings, joining the National League in 1876 as a charter member. Owner William Hulbert signed multiple star players, such as pitcher Albert Spalding and infielders Ross Barnes, Deacon White, Adrian "Cap" Anson, to join the team prior to the N. L.'s first season. The White Stockings played their home games at West Side Grounds and established themselves as one of the new league's top teams.
Spalding won forty-seven games and Barnes led the league in hitting at.429 as Chicago won the first National League pennant, which at the time was the game's top prize. After back-to-back pennants in 1880 and 1881, Hulbert died, Spalding, who had retired to start Spalding sporting goods, assumed ownership of the club; the White Stockings, with Anson acting as player-manager, captured their third consecutive pennant in 1882, Anson established himself as the game's first true superstar. In 1885 and'86, after winning N. L. pennants, the White Stockings met the champions of the short-lived American Association in that era's version of a World Series. Both seasons resulted in matchups with the St. Louis Brown Stockings, with the clubs tying in 1885 and with St. Louis winning in 1886; this was the genesis of what would become one of the greatest rivalries in sports. In all, the Anson-led Chicago Base Ball Club won six National League pennants between 1876 and 1886; as a result, Chicago's club nickname transitioned, by 1890 they had become known as the Chicago Colts, or sometimes "Anson's Colts", referring to Cap's influence within the club.
Anson was the first player in history credited with collecting 3,000 career hits. After a disappointing record of 59–73 and a ninth-place finish in 1897, Anson was released by the Cubs as both a player and manager. Due to Anson's absence from the club after 22 years, local newspaper reporters started to refer to the Colts as the "Orphans". After the 1900 season, the American Base-Ball League formed as a rival professional league, incidentally the club's old White Stockings nickname would be adopted by a new American League neighbor to the south. In 1902, who by this time had revamped the roster to boast what would soon be one of the best teams of the early century, sold the club to Jim Hart; the franchise was nicknamed the Cubs by the Chicago Daily News in 1902, although not becoming the Chicago Cubs until the 1907 season. During this period, which has become known as baseball's dead-ball era, Cub infielders Joe Tinker, Johnny Evers, Frank Chance were made famous as a double-play combination by Franklin P. Adams' poem Baseball's Sad Lexicon.
The poem first appeared in the July 1910 edition of the New York Evening Mail. Mordecai "Three-Finger" Brown, Jack Taylor, Ed Reulbach, Jack Pfiester, Orval Overall were several key pitchers for the Cubs during this time period. With Chance acting as player-manager from 1905 to 1912, the Cubs won four pennants and two World Series titles over a five-year span. Although they fell to the "Hitless Wonders" White Sox in the 1906 World Series, the Cubs recorded a record 116 victories and the best winning percentage in Major League history. With the same roster, Chicago won back-to-back World Series championships in 1907 and 1908, becoming the first Major League club to play three times in the Fall Classic and the first to win it twice. However, the Cubs would not win another World Series until 2016; the next season, veteran catcher Johnny Kling left the team to become a professional pocket billiards player. Some historians think Kling's absence was significant enough to prevent the Cubs from winning a third straight title in 1909, as they finished 6 games out of first place.
When Kling returned the next year, the Cubs won the pennant again, but lost to the Philadelphia Athletics in the 1910 World Series. In 1914, adver
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be undefined. For a unimodal distribution, negative skew indicates that the tail is on the left side of the distribution, positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means. Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side; these tapering sides are called tails, they provide a visual means to determine which of the two kinds of skewness a distribution has: negative skew: The left tail is longer. The distribution is said to be left-skewed, left-tailed, or skewed to the left, despite the fact that the curve itself appears to be skewed or leaning to the right. A left-skewed distribution appears as a right-leaning curve.
Positive skew: The right tail is longer. The distribution is said to be right-skewed, right-tailed, or skewed to the right, despite the fact that the curve itself appears to be skewed or leaning to the left. A right-skewed distribution appears as a left-leaning curve. Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the numeric sequence, whose values are evenly distributed around a central value of 50. We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, e.g.. We can make the sequence positively skewed by adding a value far above the mean, e.g.. The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, for positive skew. In the older notion of nonparametric skew, defined as / σ, where μ is the mean, ν is the median, σ is the standard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than the median, while negative/left nonparametric skew means the mean is less than the median.
However, the modern definition of skewness and the traditional nonparametric definition do not in general have the same sign: while they agree for some families of distributions, they differ in general, conflating them is misleading. If the distribution is symmetric the mean is equal to the median, the distribution has zero skewness. If the distribution is both symmetric and unimodal the mean = median = mode; this is the case of a coin toss or the series 1,2,3,4... Note, that the converse is not true in general, i.e. zero skewness does not imply that the mean is equal to the median. A 2005 journal article points out: Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew, left of the median under left skew; this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal.
Such distributions not only contradict the textbook relationship between mean and skew, they contradict the textbook interpretation of the median. The skewness of a random variable X is the third standardized moment γ1, defined as: γ 1 = E = μ 3 σ 3 = E 3 / 2 = κ 3 κ 2 3 / 2 where μ is the mean, σ is the standard deviation, E is the expectation operator, μ3 is the third central moment, κt are the tth
Statistics is a branch of mathematics dealing with data collection, analysis and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics; when census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, inferential statistics, which draw conclusions from data that are subject to random variation. Descriptive statistics are most concerned with two sets of properties of a distribution: central tendency seeks to characterize the distribution's central or typical value, while dispersion characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets.
Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are subject to error. Many of these errors are classified as random or systematic, but other types of errors can be important; the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.
Some definitions are: Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis and presentation of masses of numerical data." Statistician Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."Statistics is a mathematical body of science that pertains to the collection, interpretation or explanation, presentation of data, or as a branch of mathematics. Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty. Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, measure-theoretic probability theory.
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population; this may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types, while frequency and percentage are more useful in terms of describing categorical data; when a census is not feasible, a chosen subset of the population called. Once a sample, representative of the population is determined, data is collected for the sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are due to uncertainty.
To still draw meaningful conclusions about the entire population, in
The National League of Professional Baseball Clubs, known as the National League, is the older of two leagues constituting Major League Baseball in the United States and Canada, the world's oldest current professional team sports league. Founded on February 2, 1876, to replace the National Association of Professional Base Ball Players of 1871–1875, the NL is sometimes called the Senior Circuit, in contrast to MLB's other league, the American League, founded 25 years later. Both leagues have 15 teams. After two years of conflict in a "baseball war" of 1901–1902, the two leagues of 8 team franchises each, agreed in a "peace pact" to recognize each other as "major leagues", draft rules regarding player contracts, prohibiting "raiding", regulating relationships with minor leagues and lower level clubs, with each establishing a team in the nation's largest metropolis of New York City, the league champions of 1903 arranged to compete against each other in the new professional baseball championship tournament with the inaugural "World Series" that Fall of 1903, succeeding earlier similar national series in previous decades since the 1880s.
After the 1904 champions failed to reach a similar agreement, the two leagues formalized the new World Series tournament beginning in 1905 as an arrangement between the leagues themselves. National League teams have won 48 of the 114 World Series championships contested from 1903 to 2018. Due to its length, the National League's full name is used. Up until about the 1970's, the term National League was considered an informal term to be used for any North American major sports league that included those two words in its name the National Football League and National Hockey League. By the 21st century, that practice had fallen out of favor in North America, with the terms National League and NL reserved for the baseball league and similarly-named leagues in other sports being referred to by their full names or initials. By 1875, the National Association of Professional Base Ball Players, founded four years earlier, was suffering from a lack of strong authority over clubs, unsupervised scheduling, unstable membership of cities, dominance by one team, an low entry fee that gave clubs no incentive to abide by league rules when it was inconvenient to them.
William A. Hulbert, a Chicago businessman and an officer of the Chicago White Stockings of 1870–1889, approached several NA clubs with the plans for a professional league for the sport of base ball with a stronger central authority and exclusive territories in larger cities only. Additionally, Hulbert had a problem: five of his star players were threatened with expulsion from the NAPBBP because Hulbert had signed them to his club using what were considered questionable means. Hulbert had a great vested interest in creating his own league, after recruiting St. Louis four western clubs met in Louisville, Kentucky, in January 1876. With Hulbert speaking for the four in New York City on February 2, 1876, the National League of Professional Base Ball Clubs was established with eight charter members, as follows: Chicago White Stockings from the NA Philadelphia Athletics from the NA Boston Red Stockings, the dominant team in the NA Hartford Dark Blues from the NA Mutual of New York from the NA St. Louis Brown Stockings from the NA Cincinnati Red Stockings, a new franchise Louisville Grays, a new franchise The National League's formation meant the end of the old National Association after only five seasons, as its remaining clubs shut down or reverted to amateur or minor league status.
The only strong club from 1875 excluded in 1876 was a second one in Philadelphia called the White Stockings or Phillies. The first game in National League history was played on April 22, 1876, at Philadelphia's Jefferson Street Grounds, at 25th & Jefferson Streets, between the Philadelphia Athletics and the Boston baseball club. Boston won the game 6–5; the new league's authority was soon tested after the first season. The Athletic and Mutual clubs fell behind in the standings and refused to make western road trips late in the season, preferring to play games against local non-league competition to recoup some of their financial losses rather than travel extensively incurring more costs. Hulbert reacted to the clubs' defiance by expelling them, an act which not only shocked baseball followers and the sports world, but made it clear to clubs that league schedule commitments, a cornerstone of competition integrity, were not to be ignored; the National League operated with only six clubs during 1877 and 1878.
Over the next several years, various teams left the struggling league. By 1880, six of the eight charter members had folded; the two remaining original NL franchises and Chicago, remain still in operation today as the Atlanta Braves and the Chicago Cubs. When all eight participants for 1881 returned for 1882—the first off-season without turnover in membership—the "circuit" consist
Atomic bombings of Hiroshima and Nagasaki
During the final stage of World War II, the United States detonated two nuclear weapons over the Japanese cities of Hiroshima and Nagasaki on August 6 and 9, 1945, respectively. The United States dropped the bombs after obtaining the consent of the United Kingdom, as required by the Quebec Agreement; the two bombings killed 129,000 -- 226,000 people. They remain the only use of nuclear weapons in the history of armed conflict. In the final year of the war, the Allies prepared for what was anticipated to be a costly invasion of the Japanese mainland; this undertaking was preceded by a conventional and firebombing campaign that devastated 67 Japanese cities. The war in Europe had concluded when Germany signed its instrument of surrender on May 8, 1945; as the Allies turned their full attention to the Pacific War, the Japanese faced the same fate. The Allies called for the unconditional surrender of the Imperial Japanese armed forces in the Potsdam Declaration on July 26, 1945—the alternative being "prompt and utter destruction".
The Japanese ignored the war continued. By August 1945, the Allies' Manhattan Project had produced two types of atomic bombs, the 509th Composite Group of the United States Army Air Forces was equipped with the specialized Silverplate version of the Boeing B-29 Superfortress that could deliver them from Tinian in the Mariana Islands. Orders for atomic bombs to be used on four Japanese cities were issued on July 25. On August 6, one of the modified B-29s dropped a uranium gun-type bomb on Hiroshima. Three days on August 9, a plutonium implosion bomb was dropped by another B-29 on Nagasaki; the bombs devastated their targets. Over the next two to four months, the acute effects of the atomic bombings killed 90,000–146,000 people in Hiroshima and 39,000–80,000 people in Nagasaki. Large numbers of people continued to die from the effects of burns, radiation sickness, other injuries, compounded by illness and malnutrition, for many months afterward. In both cities, most of the dead were civilians. Japan announced its surrender to the Allies on August 15, six days after the bombing of Nagasaki and the Soviet Union's declaration of war.
On September 2, the Japanese government signed the instrument of surrender ending World War II. The effects of the bombings on the social and political character of subsequent world history and popular culture has been studied extensively, the ethical and legal justification for the bombings is still debated to this day. In 1945, the Pacific War between the Empire of Japan and the Allies entered its fourth year. Most Japanese military units fought fiercely, ensuring that the Allied victory would come at an enormous cost; the 1.25 million battle casualties incurred in total by the United States in World War II included both military personnel killed in action and wounded in action. Nearly one million of the casualties occurred during the last year of the war, from June 1944 to June 1945. In December 1944, American battle casualties hit an all-time monthly high of 88,000 as a result of the German Ardennes Offensive. America's reserves of manpower were running out. Deferments for groups such as agricultural workers were tightened, there was consideration of drafting women.
At the same time, the public was becoming war-weary, demanding that long-serving servicemen be sent home. In the Pacific, the Allies returned to the Philippines, recaptured Burma, invaded Borneo. Offensives were undertaken to reduce the Japanese forces remaining in Bougainville, New Guinea and the Philippines. In April 1945, American forces landed on Okinawa. Along the way, the ratio of Japanese to American casualties dropped from 5:1 in the Philippines to 2:1 on Okinawa. Although some Japanese soldiers were taken prisoner, most fought until they were killed or committed suicide. Nearly 99% of the 21,000 defenders of Iwo Jima were killed. Of the 117,000 Okinawan and Japanese troops defending Okinawa in April–June 1945, 94% were killed; as the Allies advanced towards Japan, conditions became worse for the Japanese people. Japan's merchant fleet declined from 5,250,000 gross tons in 1941 to 1,560,000 tons in March 1945, 557,000 tons in August 1945. Lack of raw materials forced the Japanese war economy into a steep decline after the middle of 1944.
The civilian economy, which had deteriorated throughout the war, reached disastrous levels by the middle of 1945. The loss of shipping affected the fishing fleet, the 1945 catch was only 22% of that in 1941; the 1945 rice harvest was the worst since 1909, hunger and malnutrition became widespread. U. S. industrial production was overwhelmingly superior to Japan's. By 1943, the U. S. produced 100,000 aircraft a year, compared to Japan's production of 70,000 for the entire war. By the middle of 1944, the U. S. had a hundred aircraft carriers in the Pacific, far more than Japan's twenty-five for the entire war. In February 1945, Prince Fumimaro Konoe advised Emperor Hirohito that defeat was inevitable, urged him to abdicate. Before the surrender of Nazi Germany on May 8, 1945, plans were underway for the largest operation of the Pacific War, Operation Downfall, the Allied invasion of Japan; the operation had two parts: Operation Coronet. Set to begin in October 1945, Olympic involved a series of landings by the U.
S. Sixth Army intended to capture the southern third of the southernmost main Japanese island, Kyūshū. Operation Olympic was to be followed in March 1946 by Operation Coronet, the capture of t
A slot machine, known variously as a fruit machine, the slots, poker machine/pokies, or slot, is a casino gambling machine which creates a game of chance for its customers. Its standard layout is a display with three or more reels which rotate when a lever is pulled or button pushed. Slot machines are known as one-armed bandits because they were operated by pulling upon a large mechanical lever on the side of the machine, because of their ability to empty a player's pockets and wallet as a thief would. Many modern machines are still equipped with a legacy lever in addition to the button. Slot machines include one or more currency detectors that validate the form of payment, whether coin, cash, or token; the machine pays off according to patterns of symbols appearing on its display. Slot machines are the most popular gambling method in casinos and constitute about 70 percent of the average US casino's income. Digital technology has resulted in variations on the original slot machine concept. Since the player is playing a video game, manufacturers are able to offer more interactive elements, such as advanced bonus rounds and more varied video graphics.
The "slot machine" term derives from the slots on the machine for retrieving coins. "Fruit machine" comes from the traditional fruit images on the spinning reels, such as lemons and cherries. Sittman and Pitt of Brooklyn, New York, U. S. developed a gambling machine in 1891, a precursor to the modern slot machine. It contained five drums holding a total of 50 card was based on poker; this machine proved popular and soon many bars in the city had one or more of the machines. Players would insert a nickel and pull a lever, which would spin the drums and the cards they held, the player hoping for a good poker hand. There was no direct payout mechanism, so a pair of kings might get the player a free beer, whereas a royal flush could pay out cigars or drinks, the prizes wholly dependent on what was on offer at the local establishment. To make the odds better for the house, two cards were removed from the deck: the ten of spades and the jack of hearts, which doubles the odds against winning a royal flush.
The drums could be rearranged to further reduce a player's chance of winning. Due to the vast number of possible wins with the original poker card-based game, it proved impossible to come up with a way to make a machine capable of making an automatic payout for all possible winning combinations. Somewhere between 1887 and 1895, Charles Fey of San Francisco, California, U. S. devised a much simpler automatic mechanism with three spinning reels containing a total of five symbols – horseshoes, spades, a Liberty Bell. The bell gave the machine its name. By replacing ten cards with five symbols and using three reels instead of five drums, the complexity of reading a win was reduced, allowing Fey to devise an effective automatic payout mechanism. Three bells in a row produced ten nickels. Liberty Bell spawned a thriving mechanical gaming device industry; when the use of these gambling devices was banned in his home state after a few years, Fey still could not keep up with demand for the game elsewhere.
The Liberty Bell machine was so popular. Thus in 1907, manufacturer Herbert Mills from Chicago produced a slot machine called the Operator Bell. By 1908 lots of "bell" machines were installed in most cigar stores, bowling alleys and barber shops. Early machines, including an 1899 "Liberty Bell", are now part of the Nevada State Museum's Fey Collection. Other early machines, such as the trade stimulator, gave out winnings in the form of fruit-flavoured chewing gums with pictures of the flavours as symbols on the reels; the popular cherry and melon symbols derive from this machine. The BAR symbol now common in slot machines was derived from an early logo of the Bell-Fruit Gum Company; the payment of food prizes was a used technique to avoid laws against gambling in a number of states, for this reason a number of gumball and other vending machines were regarded with mistrust by the courts. The two Iowa cases of State v. Ellis and State v. Striggles are both used in classes on criminal law to illustrate the concept of reliance upon authority as it relates to the axiomatic ignorantia juris non excusat.
In these cases, a mint vending machine was declared to be a gambling device because by chance the machine would give the next user a number of tokens exchangeable for more candy. Despite the fact that the result of the next use would be displayed on the machine, the courts ruled that "he machine appealed to the player's propensity to gamble, and, vice."In 1963, Bally developed the first electromechanical slot machine, called Money Honey. The electromechanical approach of the 1960s allowed Money Honey to be the first slot machine with a bottomless hopper and automatic payout of up to 500 coins without the help of an attendant; the popularity of this machine led to the increasing predominance of electronic games, the side lever soon became vestigial. The first true video slot machine was developed in 1976 in Kearny Mesa, California by the Las-Vegas based Fortune Coin Co; this slot machine used
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 unfair coin. In theoretical studies, the assumption that a coin is fair is made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads 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 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.
The probabilistic and statistical properties of coin-tossing games are used as examples in both introductory and advanced text books and these are based in assuming that a coin is fair or "ideal". For example, Feller uses this basis to introduce both the idea of random walks and to develop tests for homogeneity 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 consisting of the result from tossing a fair coin is called a Bernoulli process. If a cheat has altered a coin to prefer one side over another, the coin can still be used for fair results by changing the game slightly. John von Neumann 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 tails must be the same as the probability of getting tails and 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, the case for most non-malleable coins. 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. The coin must not be so biased that one side has a probability of zero; this method may be extended by considering sequences of four tosses. That is, if the coin is flipped twice but the results match, the coin is flipped twice again but the results match now for the opposite side the first result can be used; this is because HHTT and TTHH are likely. This can be extended to any power of 2. Checking whether a coin is fair Coin flipping Feller's coin-tossing constants Gelman, Andrew. "Teacher's Corner: You Can Load a Die, But You Can't Bias a Coin". American Statistician. 56: 308–311. Doi:10.1198/000313002605. Available from Andrew Gelman's website "Lifelong debunker takes on arbiter of neutral choices: Magician-turned-mathematician uncovers bias in a flip of a coin".
Stanford Report. 2004-06-07. Retrieved 2008-03-05. John von Neumann, "Various techniques used in connection with random digits," in A. S. Householder, G. E. Forsythe, H. H. Germond, eds. Monte Carlo Method, National Bureau of Standards Applied Mathematics Series, 12: 36-38