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Probability (other)
Probability is the branch of mathematics concerning Event (probability theory), events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an Probability axioms, axiomatic mathematical formalization in ''probability theory'', which is use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dice Distribution (bar)
Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance. A traditional die is a cube with each of its six faces marked with a different number of dots ( pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Dice may also have polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their numbers carved out from the material of the dice instead of marked on it. Loaded dice are designed to favor some results over others for cheating or entertainment. History Dice have been used since before recorded history, and it is uncertain where they originated. It is theorized that dice developed from the practice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Credence (statistics)
Credence is a statistical term that expresses how much a person believes that a proposition is true. As an example, a reasonable person will believe with 50% credence that a fair coin will land on heads the next time it is flipped. If the prize for correctly predicting the coin flip is $100, then a reasonable risk-neutral person will wager $49 on heads, but they will not wager $51 on heads. Credence is a measure of belief strength, expressed as a percentage. Credence values range from 0% to 100%. Credence is closely related to odds, and a person's level of credence is directly related to the odds at which they will place a bet. Credence is especially important in Bayesian statistics. If a bag contains 4 red marbles and 1 blue marble, and a person withdraws one marble at random, then they should believe with 80% credence that the random marble will be red. In this example, the probability of drawing a red marble is 80%. Credence values can be based entirely on subjective feelings. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subjective Probability
Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability. Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, relevant data (evidence). The Bayesian interpretation provides a standard set of procedures and formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propensity Probability
The propensity theory of probability is a probability interpretation in which the probability is thought of as a physical propensity, disposition, or tendency of a given type of situation to yield an outcome of a certain kind, or to yield a long-run relative frequency of such an outcome.'Interpretations of Probability', Stanford Encyclopedia of Philosophybr> Retrieved 23 December 2006. Propensities are not relative frequencies, but purported ''causes'' of the observed stable relative frequencies. Propensities are invoked to ''explain why'' repeating a certain kind of experiment will generate a given outcome type at a persistent rate. A central aspect of this explanation is the law of large numbers. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency (statistics)
In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form. Types The cumulative frequency is the total of the absolute frequencies of all events at or below a certain point in an ordered list of events. The (or empirical probability) of an event is the absolute frequency normalized by the total number of events: : f_i = \frac = \frac. The values of f_i for all events i can be plotted to produce a frequency distribution. In the case when n_i = 0 for certain i, pseudocounts can be added. Depicting frequency distributions A frequency distribution shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data notably to show results of an election, income of people for a certain region, sales of a product with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequentist Probability
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation. In the classical interpretation, probability was defined in terms of the principle of indifference, based on the natural symmetry of a problem, so, ''e.g.'' the probabilities of dice games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry for reasoning. Definition In the frequentist interpretation, probabilities a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Objectivity (philosophy)
In philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ..., objectivity is the concept of truth independent from individual subjectivity (bias caused by one's perception, emotions, or imagination). A proposition is considered to have objective truth when its truth conditions are met without bias caused by the mind of a sentience, sentient being. Objectivity (science), Scientific objectivity refers to the ability to judge without Impartiality, partiality or external influence. Objectivity in the moral framework calls for moral codes to be assessed based on the well-being of the people in the society that follow it. Moral objectivity also calls for moral codes to be compared to one another through a set of universal facts and not through subjectivity. Objectivity of knowledge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Probability
The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment. More generally, empirical probability estimates probabilities from experience and observation. Given an event ''A'' in a sample space, the relative frequency of ''A'' is the ratio ''m/n'', ''m'' being the number of outcomes in which the event ''A'' occurs, and ''n'' being the total number of outcomes of the experiment. In statistical terms, the empirical probability is an ''estimate'' or estimator of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modelling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likelihood estimate. It is the Bayesian estimate for the same case if certain assumptions are made fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-defined Expression
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if ''f'' takes real numbers as input, and if ''f''(0.5) does not equal ''f''(1/2) then ''f'' is not well defined (and thus not a function). The term ''well defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if ''f''(''x'') = 1/''x'', then the fact that ''f''(0) is undefined does not mean that the ''f'' is ''not'' well defined – but that 0 is simply not in the domain of ''f''. Example Let A_0,A_1 be sets, let A = A_0 \cup A_1 and "define" f: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy. The fields o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary greatly in goal and scale but always rely on repeatable procedure and logical analysis of the results. There also exist natural experimental studies. A child may carry out basic experiments to understand how things fall to the ground, while teams of scientists may take years of systematic investigation to advance their understanding of a phenomenon. Experiments and other types of hands-on activities are very important to student learning in the science classroom. Experiments can raise test scores and help a student become more engaged and interested in the material they are learning, especially when used over time. Experiments can vary from personal and informal natural comparisons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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