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When Expected values people play Lotteries lotteries or purchase Insurance insurance, they are investing money for a chance of some future financial return that may or may not occur. From the lottery or insurance company’s perspective, money comes in from multiple purchasers and is paid out to the winners or claimants. Both sides may have questions regarding whether the investments are worthwhile or the payments are fair. These questions appear to date back to antiquity. Evidence of Gambling gambling games has been found in archaeological excavations of caves and in many ancient civilizations, including Egypt, Greece, and the Roman Empire. Babylonians used a form of maritime insurance and the Romans paid some investments in annuities.

The Nielsen ratings Film and television Television and film Nielsen Ratings are a measure of how many people are watching certain television programs. When Arthur Nielsen began measuring television viewing in 1950, there were three networks and about 9% of U.S. households had a television. In the twenty-first century, homes often have multiple televisions receiving scores of channels. Even when the number of television sets was small, it was not possible to gather complete viewing data for every single person who owns a television. Instead, Nielsen Media Research uses statistical sampling methods to take a representative subset of viewers and then extrapolates from the sample’s viewing activities to the whole population of viewers. The statistical methods Nielsen uses to collect its data have been refined several times in response to changes in viewer behavior. People who develop and analyze ratings typically have expertise in analytics, metrics, and statistical modeling.

The Military draft Lotteries U.S. military is made up of volunteers. However, if more people are needed than the number of people who volunteer, there needs to be a method for procuring enlistment. The method used is called a “military draft.” It is the law that all male citizens ages 18–25 are to register with the Selective Service. If a need arises, the U.S. Congress would have to pass legislation instituting a draft. The U.S. president would have to sign the bill into law.

In a very broad sense, combinatorics is about counting. The mathematical discipline of combinatorics addresses the enumeration, permutation, and combination of sets of objects, as well as their relations and properties. Combinatorial problems can be found in many areas of pure and applied mathematics, including algebra, topology, geometry, probability, graph theory, optimization, computer science, and statistical physics. One of the earliest problems in combinatorics is found in the work of Greek biographer Plutarch, who described mathematician Xenocrates of Chalcedon’s work on calculating how many syllables could be produced by taking combinations of the letters of the alphabet. This occurred between 400 and 300 b.c.e. Millennia later, mathematician William Gowers won the 1998 Fields Medal, widely regarded as the most prestigious prize in mathematics, for his “contributions to functional analysis, making extensive use of methods from combination theory.” In twenty-first-century school curricula, primary school children study number combinations to facilitate learning basic operations like addition, subtraction, multiplication, and division. High school students often study permutations and combinations as counting techniques. Permutations and combinations were fundamental for cracking the World War II Enigma code and continue to remain vital in cryptography, among other fields.