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Martingale (betting System)
A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy is an instantiation of the St. Petersburg paradox. Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler has infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who chose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betting Strategy
A betting strategy (also known as betting system) is a structured approach to gambling, in the attempt to produce a profit. To be successful, the system must change the house edge into a player advantage — which is impossible for pure games of probability with fixed odds, akin to a perpetual motion machine. Betting systems are often predicated on statistical analysis. Mathematically, no betting system can alter long-term expected results in a game with random, independent trials, although they can make for higher odds of short-term winning at the cost of increased risk, and are an enjoyable gambling experience for some people. Strategies which take into account the changing odds that exist in some games (e.g. card counting and handicapping), can alter long-term results. This is formally stated by game theorist Richard Arnold Epstein in ''The Theory of Gambling and Statistical Logic'' as: Examples Common betting systems include: * Card games – Card counting * Roulette – M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pacific and Indian Oceans. Its Metropolitan France, metropolitan area extends from the Rhine to the Atlantic Ocean and from the Mediterranean Sea to the English Channel and the North Sea; overseas territories include French Guiana in South America, Saint Pierre and Miquelon in the North Atlantic, the French West Indies, and many islands in Oceania and the Indian Ocean. Due to its several coastal territories, France has the largest exclusive economic zone in the world. France borders Belgium, Luxembourg, Germany, Switzerland, Monaco, Italy, Andorra, and Spain in continental Europe, as well as the Kingdom of the Netherlands, Netherlands, Suriname, and Brazil in the Americas via its overseas territories in French Guiana and Saint Martin (island), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coin Flipping
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking obverse and reverse, which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute between two parties. It is a form of sortition which inherently has two possible outcomes. The party who calls the side that is facing up when the coin lands wins. History Coin flipping 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 Roman Emperor, emperor on the other. In England, this was referred to as ''cross and pile''. Process During a coin toss, the coin is thrown into the air such that it rotates edge-over-edge several times. Either beforehand or when the coin is in the air, an interested party declares "heads" or "tails", indicating which side of the coin that party is choosing. The other party is assigned the opposite side. Depending on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression. The formula for exponential growth of a variable at the growth rate , as time goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is x_t = x_0(1+r)^t where is the value of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roulette
Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the color red or black, whether the number is odd or even, or if the numbers are high (19–36) or low (1–18). To determine the winning number, a croupier spins a wheel in one direction, then spins a ball in the opposite direction around a tilted circular track running around the outer edge of the wheel. The ball eventually loses momentum, passes through an area of deflectors, and falls onto the wheel and into one of thirty-seven (single-zero, French or European style roulette) or thirty-eight (double-zero, American style roulette) or thirty-nine (triple-zero, "Sands Roulette") colored and numbered pockets on the wheel. The winnings are then paid to anyone who has placed a successful bet. History The first form of roulette was devised in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed Random Variables
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winning Strategy
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness". The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws. The field of game theory studies more general kinds of games, including games with draws such as tic-tac-toe, chess, or infinite chess, or games with imperfect information such as poker. Basic notions Games The first sort of game we shall consider is the two-player game of perfect information of length ω, in which the players play natural numbers. These games are often called Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optional Stopping Theorem
In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ... of a martingale at a stopping time is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies. The optional stopping theorem is an important tool of mathematical finance i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representativeness Heuristic
The representativeness heuristic is used when making judgments about the probability of an event under uncertainty. It is one of a group of heuristics (simple rules governing judgment or decision-making) proposed by psychologists Amos Tversky and Daniel Kahneman in the early 1970s as "the degree to which n event(i) is similar in essential characteristics to its parent population, and (ii) reflects the salient features of the process by which it is generated". Heuristics are described as "judgmental shortcuts that generally get us where we need to go – and quickly – but at the cost of occasionally sending us off course." Heuristics are useful because they use effort-reduction and simplification in decision-making. When people rely on representativeness to make judgments, they are likely to judge wrongly because the fact that something is more representative does not actually make it more likely. The representativeness heuristic is simply described as assessing similarity of ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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