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 of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
A game show is a type of radio, television, or stage show in which contestants, individually or as teams, play a game which involves answering questions or solving puzzles for money or prizes. Alternatively, a gameshow can be a demonstrative program about a game. In the former, contestants may be invited from a pool of public applicants. Game shows reward players with prizes such as cash and goods and services provided by the show's sponsor prize suppliers. Game shows began to appear on television in the late 1930s; the first television game show, Spelling Bee, as well as the first radio game show, Information Please, were both broadcast in 1938. Q. a radio quiz show that began in 1939. Truth or Consequences was the first game, its first episode aired in 1941 as an experimental broadcast. Over the course of the 1950s, as television began to pervade the popular culture, game shows became a fixture. Daytime game shows would be played for lower stakes to target stay-at-home housewives. Higher-stakes programs would air in primetime.
During the late 1950s, high-stakes games such as Twenty-One and The $64,000 Question began a rapid rise in popularity. However, the rise of quiz shows proved to be short-lived. In 1959, many of the higher stakes game shows were discovered to be rigged and ratings declines led to most of the primetime games being canceled. An early variant of the game show, the panel game, survived. On shows like What's My Line?, I've Got A Secret, To Tell the Truth, panels of celebrities would interview a guest in an effort to determine some fact about them. Panel games had success in primetime until the late 1960s, when they were collectively dropped from television because of their perceived low budget nature. Panel games made a comeback in American daytime television in the 1970s through comedy-driven shows such as Match Game and Hollywood Squares. In the UK, commercial demographic pressures were not as prominent, restrictions on game shows made in the wake of the scandals limited the style of games that could be played and the amount of money that could be awarded.
Panel have continued to thrive. The focus on quick-witted comedians has resulted in strong ratings, combined with low costs of production, have only spurred growth in the UK panel show phenomenon. Game shows remained a fixture of US daytime television through the 1960s after the quiz show scandals. Lower-stakes games made a slight comeback in daytime in the early 1960s. Let's Make a Deal began in 1963 and the 1960s marked the debut of Hollywood Squares, The Dating Game, The Newlywed Game. Though CBS gave up on daytime game shows in 1968, the other networks did not follow suit. Color television was introduced to the game show genre in the late 1960s on all three networks; the 1970s saw a renaissance of the game show as new games and massive upgrades to existing games made debuts on the major networks. The New Price Is Right, an update of the 1950s-era game show The Price Is Right, debuted in 1972 and marked CBS's return to the game show format in its effort to draw wealthier, suburban viewers; the Match Game became "Big Money" Match Game 73, which proved popular enough to prompt a spin-off, Family Feud, on ABC in 1976.
The $10,000 Pyramid and its numerous higher-stakes derivatives debuted in 1973, while the 1970s saw the return of disgraced producer and host Jack Barry, who debuted The Joker's Wild and a clean version of the rigged Tic-Tac-Dough in the 1970s. Wheel of Fortune debuted on NBC in 1975; the Prime Time Access Rule, which took effect in 1971, barred networks from broadcasting in the 7–8 p.m. time slot preceding prime time, opening up time slots for syndicated programming. Most of the syndicated programs were "nighttime" adaptations of network daytime game shows; these game shows aired once a week, but by the late 1970s and early 1980s most of the games had transitioned to five days a week. Game shows were the lowest priority of television networks and were rotated out every thirteen weeks if unsuccessful. Most tapes were destroyed until the early 1980s. Over the course of the late 1980s and early 1990s, as fewer new hits were produced, game shows lost their permanent place in the daytime lineup. ABC transitioned out of the daytime game show format in the mid-1980s.
NBC's game block lasted until 1991, but the network attempted to bring them back in 1993 before cancelling its game show block again in 1994. CBS phased out most of its game shows, except for The Price Is Right, by 1993. To the benefit of the genre, the moves of Wheel of Fortune and a modernized revival of Jeopardy! to syndication in 1983 and 1984 was and remains successful. Cable television allowed for the debut of game shows such as Supermarket Sweep, Trivial Pursuit and Family Challenge, Double Dare, it opened up a underdeveloped ma
The bean machine known as the Galton Board or quincunx, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution. Among its applications, it afforded insight into regression to the mean or "regression to mediocrity"; the Galton Board consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top and, when the device is level, bounce either left or right as they hit the pegs, they are collected into bins at the bottom, where the height of bead columns accumulated in the bins will approximate a bell curve. Overlaying Pascal's triangle onto the pins shows the number of different paths that can be taken to get to each bin. Large-scale working models of this device created by Charles and Ray Eames can be seen in the Mathematica: A World of Numbers... and Beyond exhibits permanently on view at the Boston Museum of Science, the New York Hall of Science, or the Henry Ford Museum.
Another large-scale version is displayed in the lobby of Index Fund Advisors in California. If a bead bounces to the right k times on its way down it ends up in the kth bin counting from the left. Denoting the number of rows of pegs in a Galton Board by n, the number of paths to the kth bin on the bottom is given by the binomial coefficient. If the probability of bouncing right on a peg is p the probability that the ball ends up in the kth bin equals p k n − k; this is the probability mass function of a binomial distribution. According to the central limit theorem, the binomial distribution approximates the normal distribution provided that both the number of rows and balls is large. Varying the rows will result in different standard deviations or widths of the bell-shaped curve or the normal distribution in the bins. Sir Francis Galton was fascinated with the order of the bell curve that emerges from the apparent chaos of beads bouncing off of pegs in the Galton Board, he eloquently described this relationship in his book Natural Inheritance: Order in Apparent Chaos: I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the Law of Frequency of Error.
The law would have been deified, if they had known of it. It reigns in complete self-effacement amidst the wildest confusion; the huger the mob, the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along. Several games have been developed utilizing the idea of pins changing the route of balls or other objects: Pachinko Payazzo Peggle Pinball Plinko The Wall Galton Board informational website with resource links An 8-foot-tall Probability Machine comparing stock market returns to the randomness of the beans dropping through the quincunx pattern. From Index Fund Advisors IFA.com Quincunx and its relationship to normal distribution from Math Is Fun A multi-stage bean machine simulation Pascal's Marble Run: a deterministic Galton board
Sir Francis Galton, FRS was an English Victorian era statistician, polymath, psychologist, eugenicist, tropical explorer, inventor, proto-geneticist, psychometrician. He was knighted in 1909. Galton produced over books, he created the statistical concept of correlation and promoted regression toward the mean. He was the first to apply statistical methods to the study of human differences and inheritance of intelligence, introduced the use of questionnaires and surveys for collecting data on human communities, which he needed for genealogical and biographical works and for his anthropometric studies, he was a pioneer in eugenics, coining the term itself and the phrase "nature versus nurture". His book Hereditary Genius was the first social scientific attempt to study greatness; as an investigator of the human mind, he founded psychometrics and differential psychology and the lexical hypothesis of personality. He devised a method for classifying fingerprints, he conducted research on the power of prayer, concluding it had none by its null effects on the longevity of those prayed for.
His quest for the scientific principles of diverse phenomena extended to the optimal method for making tea. As the initiator of scientific meteorology, he devised the first weather map, proposed a theory of anticyclones, was the first to establish a complete record of short-term climatic phenomena on a European scale, he invented the Galton Whistle for testing differential hearing ability. He was Charles Darwin's half-cousin. Galton was born at "The Larches", a large house in the Sparkbrook area of Birmingham, built on the site of "Fair Hill", the former home of Joseph Priestley, which the botanist William Withering had renamed, he was Charles Darwin's half-cousin. His father was son of Samuel "John" Galton; the Galtons were Quaker gun-manufacturers and bankers, while the Darwins were involved in medicine and science. He was half-cousin of Charles Darwin. Both families had members who loved to invent in their spare time. Both Erasmus Darwin and Samuel Galton were founding members of the Lunar Society of Birmingham, which included Boulton, Wedgwood, Edgeworth.
Both families were known for their literary talent. Erasmus Darwin composed lengthy technical treatises in verse. Galton's aunt Mary Anne Galton wrote on aesthetics and religion, her autobiography detailed the environment of her childhood populated by Lunar Society members. Galton was a child prodigy – he was reading by the age of two. In life, Galton proposed a connection between genius and insanity based on his own experience:Men who leave their mark on the world are often those who, being gifted and full of nervous power, are at the same time haunted and driven by a dominant idea, are therefore within a measurable distance of insanity. Galton attended King Edward's School, but chafed at the narrow classical curriculum and left at 16, his parents pressed him to enter the medical profession, he studied for two years at Birmingham General Hospital and King's College London Medical School. He followed this up with mathematical studies at Trinity College, University of Cambridge, from 1840 to early 1844.
According to the records of the United Grand Lodge of England, it was in February 1844 that Galton became a freemason at the Scientific lodge, held at the Red Lion Inn in Cambridge, progressing through the three masonic degrees: Apprentice, 5 February 1844. A note in the record states: "Francis Galton Trinity College student, gained his certificate 13 March 1845". One of Galton's masonic certificates from Scientific lodge can be found among his papers at University College, London. A nervous breakdown prevented Galton's intent to try for honours, he elected instead to take a "poll" B. A. degree, like his half-cousin Charles Darwin. He resumed his medical studies but the death of his father in 1844 left him destitute, though financially independent, he terminated his medical studies turning to foreign travel and technical invention. In his early years Galton was an enthusiastic traveller, made a notable solo trip through Eastern Europe to Constantinople, before going up to Cambridge. In 1845 and 1846, he went to Egypt and travelled up the Nile to Khartoum in the Sudan, from there to Beirut and down the Jordan.
In 1850 he joined the Royal Geographical Society, over the next two years mounted a long and difficult expedition into little-known South West Africa. He wrote a book on his experience, "Narrative of an Explorer in Tropical South Africa", he was awarded the Royal Geographical Society's Founder's Gold Medal in 1853 and the Silver Medal of the French Geographical Society for his pioneering cartographic survey of the region. This established his reputation as a explorer, he proceeded to write the best-selling The Art of Travel, a handbook of practical advice for the Victorian on the move, which went through many editions and is still in print. In January 1853, Galton met Louisa Jane Butler at his neighbour's home and they were married on 1 August 1853; the union of 43
Central limit theorem
In probability theory, the central limit theorem establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution if the original variables themselves are not distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. For example, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the distribution of the average will be approximated by a normal distribution. A simple example of this is that if one flips a coin many times the probability of getting a given number of heads in a series of flips will approach a normal curve, with mean equal to half the total number of flips in each series.
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions; the earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is now known as the de Moivre–Laplace theorem. In more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory, they all express the fact that a sum of many independent and identically distributed random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. Variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d.
Random variables with power law tail distributions decreasing as |x|−α − 1 where 0 < α < 2 will tend to an alpha-stable distribution with stability parameter of α as the number of variables grows. Let be a random sample of size n—that is, a sequence of independent and identically distributed random variables drawn from a distribution of expected value given by µ and finite variance given by σ2. Suppose we are interested in the sample average S n:= X 1 + ⋯ + X n n of these random variables. By the law of large numbers, the sample averages converge in probability and surely to the expected value µ as n → ∞; the classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence. More it states that as n gets larger, the distribution of the difference between the sample average Sn and its limit µ, when multiplied by the factor √n, approximates the normal distribution with mean 0 and variance σ2.
For large enough n, the distribution of Sn is close to the normal distribution with mean µ and variance σ2/n. The usefulness of the theorem is that the distribution of √n approaches normality regardless of the shape of the distribution of the individual Xi. Formally, the theorem can be stated as follows: Lindeberg–Lévy CLT. Suppose is a sequence of i.i.d. Random variables with E = µ and Var = σ2 < ∞. As n approaches infinity, the random variables √n converge in distribution to a normal N: n → d N. In the case σ > 0, convergence in distribution means that the cumulative distribution functions of √n converge pointwise to the cdf of the N distribution: for every real number z, lim n → ∞ Pr = Φ, where Φ is the standard normal cdf evaluated at x. Note that the convergence is uniform in z in the sense that lim n → ∞ sup z ∈ R | Pr − Φ | = 0, where sup denotes the least upper bound of the se
The Wall (game show)
The Wall is an American television game show broadcast by NBC, which premiered on December 19, 2016. The show is hosted by Chris Hardwick, who serves as executive producer on the show along with LeBron James, Maverick Carter, Andrew Glassman. On January 18, 2017, NBC ordered 20 additional episodes bringing the episode count to 30. On March 12, 2018, The Wall was renewed for the third season with 20 episodes. Two special episodes aired on November 22, 2017 and December 27, 2017. A special Christmas themed episode aired on December 4, 2017; the second half of the second season premiered on January 1, 2018. In 2018, Telemundo announced plans for a Spanish-language version of The Wall to premiere in 2019; the game show will be hosted by Marco Antonio Regil. On April 1, 2019, it was announced that Season 3 will premiere on June 20, 2019; the Wall is a four-story-tall pegboard, similar to a pachinko bean machine. The bottom of the board is divided into 15 slots marked with various U. S. dollar amounts. Seven numbered "drop zones" are centered at the top of the board, from which balls can be dropped into play.
A team of two contestants plays each game, with a potential top prize of $12,374,994. Green balls dropped on the board will add to the team's bank, while red balls dropped on the board will subtract from it. Throughout the game, the bank has a floor of $0. In Free Fall, the team is asked a series of each with two answer choices; as each question is asked, three balls are released from drop zones 1, 4, 7. The team must select one answer and lock it in before any portion of a ball crosses the threshold of a money slot. If the team's answer is correct, the balls turn green and their values are added to the team's bank. If the team answers incorrectly or fails to lock in an answer, the balls turn red and their values are subtracted from the team's bank. If the team's bank balance is zero at the end of this round, the game is over and they leave without any winnings. If not, their earnings become part of a guaranteed payout to be offered to them at the end of the game; the highest amount that a team can bank in this round is $375,000.
The values on the board range from $1 to $25,000, are arranged as follows: At the start of the second round, the contestants are separated from each other for the remainder of the game. One enters an isolation chamber behind The Wall. Two green balls are played dropped from zones chosen by the onstage player. Three multiple-choice questions are played, each with three answer choices; the onstage player is shown only the answers to each question, must decide which zone to use, based on how confident he/she is that the isolated player can answer correctly. The question and answers are presented to the isolated player. A correct answer turns the ball green and adds the value of the slot it lands in to the team bank, while a miss turns the ball red and deducts the value; the isolated player is not told which of his/her answers are correct or given any information on the team bank. The onstage player is offered an opportunity to "Double Up" on the second question and "Triple Up" on the third. After the third question, if the banked total is at least $3, two red balls are dropped from the same zones that were chosen for the initial two green balls.
The maximum amount that a team can bank in this round is $1,999,998. The values on the board range from $1 to $250,000, are arranged as follows: Gameplay proceeds as in Round 2, but each of the three questions now has four answer choices. In addition, four green and four red balls are played at the start and end of the round and are dropped one at a time, rather than simultaneously; the "Double Up" and "Triple Up" options are available as before. The maximum amount that a team can bank in this round is $9,999,996; the values on the board range from $1 to $1,000,000, are arranged as follows: After the third question in Round 3, the isolated player is sent a contract by the host, he/she must sign it or tear it up. Signing the contract gives up the team bank in favor of a guaranteed payout, equal to the Free Fall winnings plus an additional $20,000 for every question answered in Rounds 2 and 3. If the isolated player tears up the contract, the team receives their final bank total instead. After the four red balls have dropped in Round 3 and the final bank is calculated, the isolated player returns to the stage to reveal his/her decision.
Only at this point does he/she learn the number of correct answers given, the payout total, the team's final bank. The maximum possible guaranteed payout is $495,000, obtained by scoring $375,000 in Free Fall and answering all questions in Rounds 2 and 3; the maximum possible bank total is $12,374,994, obtained by answering every question using every Double Up and Triple Up option, having every green ball drop into the highest-valued slot, having the six mandatory red balls each drop into a $1 slot. Color key The contestants left with at least $1,000,000; the contestants left with the larger possible amount. The contestants left with the smaller possible amount; the contestants left with nothing. The Wa