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The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event has occurred less frequently than expected, it is more likely to happen again in the future. The fallacy is commonly associated with gambling, where it may be believed, for example, that the next dice roll is more than usually likely to be six because there have recently been fewer than the expected number of sixes.
The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability. Depending on context or application it can be considered a valid common-sense observation or a misunderstanding of probability. This notion can lead to the gambler's fallacy when one becomes convinced that a particular outcome must come soon simply because it has not occurred recently.
The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory.
In probability theory and statistics, Bayes' theorem, named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the individual is typical of the population as a whole.
Fixed-odds betting is a form of gambling where individuals place bets on the outcome of an event, such as sports matches or horse races, at predetermined odds. In fixed-odds betting, the odds are fixed and determined at the time of placing the bet. These odds reflect the likelihood of a particular outcome occurring. If the bettor's prediction is correct, they receive a payout based on the fixed odds. This means that the potential winnings are known at the time of placing the bet, regardless of any changes in the odds leading up to the event.
In probability theory, odds provide a measure of the likelihood of a particular outcome. When specific events are equally likely, odds are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.
The theory of belief functions, also referred to as evidence theory or Dempster–Shafer theory (DST), is a general framework for reasoning with uncertainty, with understood connections to other frameworks such as probability, possibility and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed by Glenn Shafer into a general framework for modeling epistemic uncertainty—a mathematical theory of evidence. The theory allows one to combine evidence from different sources and arrive at a degree of belief that takes into account all the available evidence.
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.
Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a degree of belief in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation, which views probability as the limit of the relative frequency of an event after many trials. More concretely, analysis in Bayesian methods codifies prior knowledge in the form of a prior distribution.
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 randomly 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.
Subjectivism is the doctrine that "our own mental activity is the only unquestionable fact of our experience", instead of shared or communal, and that there is no external or objective truth.
In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and probability theory can provide guidance on two types of question; specifically those of how many trials to undertake and of the accuracy of an estimate of the probability of turning up heads, derived from a given sample of trials.
Cromwell's rule, named by statistician Dennis Lindley, states that the use of prior probabilities of 1 or 0 should be avoided, except when applied to statements that are logically true or false, such as 2 + 2 equaling 4.
Credibility theory is a branch of actuarial mathematics concerned with determining risk premiums. To achieve this, it uses mathematical models in an effort to forecast the (expected) number of insurance claims based on past observations. Technically speaking, the problem is to find the best linear approximation to the mean of the Bayesian predictive density, which is why credibility theory has many results in common with linear filtering as well as Bayesian statistics more broadly.
Projectivism or projectionism in philosophy involves attributing (projecting) qualities to an object as if those qualities actually belong to it. It is a theory for how people interact with the world and has been applied in both ethics and general philosophy. It is derived from the Humean idea that all judgements about the world derive from internal experience, and that people therefore project their emotional state onto the world and interpret it through the lens of their own experience. Projectivism can conflict with moral realism, which asserts that moral judgements can be determined from empirical facts, i.e., some things are objectively right or wrong.
The Sleeping Beauty problem, also known as the Sleeping Beauty paradox, is a puzzle in decision theory in which an ideally rational epistemic agent is told she will be awoken from sleep either once or twice according to the toss of a coin. Each time she will have no memory of whether she has been awoken before, and is asked what her degree of belief that the outcome of the coin toss is Heads ought to be when she is first awakened.
The mathematics of gambling is a collection of probability applications encountered in games of chance and can get included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, and it is possible to calculate by using the properties of probability on a finite space of possibilities.
The ludic fallacy, proposed by Nassim Nicholas Taleb in his book The Black Swan (2007), is "the misuse of games to model real-life situations". Taleb explains the fallacy as "basing studies of chance on the narrow world of games and dice". The adjective ludic originates from the Latin noun ludus, meaning "play, game, sport, pastime".
In philosophy, Pascal's mugging is a thought experiment demonstrating a problem in expected utility maximization. A rational agent should choose actions whose outcomes, when weighted by their probability, have higher utility. But some very unlikely outcomes may have very great utilities, and these utilities can grow faster than the probability diminishes. Hence the agent should focus more on vastly improbable cases with implausibly high rewards; this leads first to counter-intuitive choices, and then to incoherence as the utility of every choice becomes unbounded.
Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. It is based on the idea that beliefs can be interpreted as subjective probabilities. As such, they are subject to the laws of probability theory, which act as the norms of rationality. These norms can be divided into static constraints, governing the rationality of beliefs at any moment, and dynamic constraints, governing how rational agents should change their beliefs upon receiving new evidence. The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. Bayesians have applied these fundamental principles to various epistemological topics but Bayesianism does not cover all topics of traditional epistemology. The problem of confirmation in the philosophy of science, for example, can be approached through the Bayesian principle of conditionalization by holding that a piece of evidence confirms a theory if it raises the likelihood that this theory is true. Various proposals have been made to define the concept of coherence in terms of probability, usually in the sense that two propositions cohere if the probability of their conjunction is higher than if they were neutrally related to each other. The Bayesian approach has also been fruitful in the field of social epistemology, for example, concerning the problem of testimony or the problem of group belief. Bayesianism still faces various theoretical objections that have not been fully solved.
References
- 1 2 Critch, Andrew. "Credence – a measure of belief strength" . Retrieved 18 December 2014.
- ↑ Strevens, Michael. "Notes on Bayesian Confirmation Theory" (PDF). New York University. Archived from the original on 29 August 2014. Retrieved 18 December 2014.
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