query: a list of all possible outcomes of an experiment is called

pos: Outcome (probability) upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. Outcome (probability) In probability theory, an outcome is a possible result of an experiment. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible "outcomes" that make up our "sample space" are (H, T), (T, H), (T, T) and (H,

pos: Outcome (probability) upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. Outcome (probability) In probability theory, an outcome is a possible result of an experiment. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible "outcomes" that make up our "sample space" are (H, T), (T, H), (T, T) and (H,

pos: Outcome (probability) Outcome (probability) In probability theory, an outcome is a possible result of an experiment. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible "outcomes" that make up our "sample space" are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Outcomes should not be confused with

pos: Sample space Sample space In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels "S", Ω, or "U" (for "universal set"). For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written

pos: Sample space Sample space In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels "S", Ω, or "U" (for "universal set"). For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written

pos: Empirical probability 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. In a more general sense, 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,

pos: Sample space the various outcomes of an experiment are always defined so as to be equally likely. However, there are experiments that are not easily described by a sample space of equally likely outcomes—for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since

neg: Probability theory discrete, continuous, a mix of the two, and 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 (or equivalently, the event 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 {1,3,5} is an element of the power set of the sample space of die rolls. These collections are called "events".

neg: Experiment (probability theory) Experiment (probability theory) In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be "random" if it has more than one possible outcome, and "deterministic" if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial. When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said

neg: Experiment (probability theory) Experiment (probability theory) In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be "random" if it has more than one possible outcome, and "deterministic" if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial. When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said

neg: Probability interpretations mathematically as follows: If a random experiment can result in N mutually exclusive and equally likely outcomes and if N of these outcomes result in the occurrence of the event A, the probability of A is defined by There are two clear limitations to the classical definition. Firstly, it is applicable only to situations in which there is only a 'finite' number of possible outcomes. But some important random experiments, such as tossing a coin until it rises heads, give rise to an infinite set of outcomes. And secondly, you need to determine in advance that all the possible outcomes

neg: Outcome (probability) containing exactly one outcome is called an elementary event. The event that contains all possible outcomes of an experiment is its sample space. A single outcome can be a part of many different events. Typically, when the sample space is finite, any subset of the sample space is an event ("i"."e". all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite (most notably when the outcome must be some real number). So, when defining a probability space it is possible,

neg: Experiment (probability theory) the relative frequencies of occurrence of each of the events would approach agreement with the values "P" assigns them. As a simple experiment, we may flip a coin twice. The sample space (where the order of the two flips is relevant) is "{(H, T), (T, H), (T, T), (H, H)}" where "H" means "heads" and "T" means "tails". Note that each of "(H, T), (T, H)", ... are possible "outcomes" of the experiment. We may define an "event" which occurs when a "heads" occurs in either of the two flips. This event contains all of the outcomes except "(T, T)".

neg: Bernoulli trial two possible outcomes, it can be framed as some "yes or no" question. For example: Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result meeting specified conditions, not in any moral judgement. More generally, given any probability space, for any event (set of outcomes), one can define a Bernoulli trial, corresponding to whether the event occurred or not (event or complementary event). Examples of Bernoulli trials include: Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials.

neg: Frequentist probability 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. In the frequentist interpretation, probabilities are discussed only when dealing with well-defined random experiments (or random samples). The set of all possible outcomes of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space to be considered. For any given

neg: Outcome (probability) "events", which are "sets" (or informally, "groups") of outcomes. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. This event would contain all outcomes in the sample space except the element (T, T). Since individual outcomes may be of little practical interest, or because there may be prohibitively (even infinitely) many of them, outcomes are grouped into sets of outcomes that satisfy some condition, which are called "events." The collection of all such events is a sigma-algebra. An event

neg: Random variable a real-numbered value. A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, due to imprecise measurements or quantum uncertainty). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself but is instead related to philosophical arguments over the interpretation

neg: Schrödinger's cat the course of developing the thought experiment. Schrödinger intended his thought experiment as a discussion of the EPR article—named after its authors Einstein, Podolsky, and Rosen—in 1935. The EPR article highlighted the counterintuitive nature of quantum superpositions, in which a quantum system such as an atom or photon can exist as a combination of multiple states corresponding to different possible outcomes. The prevailing theory, called the Copenhagen interpretation, said that a quantum system remains in superposition until it interacts with, or is observed by the external world. When this happens, the superposition collapses into one or another of the possible

neg: Quantum suicide and immortality The quantum suicide thought experiment attempts to isolate all possible outcomes for the duration of the thought experiment. That isolation delays decoherence in such a way that the subjective experience of the superposition is illustrated. It is only within the confines of such an abstract quantum scenario that an observer finds they defy all odds. Physicist David Deutsch, though a proponent of the MWI, states regarding quantum suicide that "that way of applying probabilities does not follow directly from quantum theory, as the usual one does. It requires an additional assumption, namely that when making decisions one should ignore the

neg: Experiment (probability theory) hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses. A random experiment is described or modelled by a mathematical construct known as a probability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind. A mathematical description of an experiment consists of three parts: An "outcome" is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complicated "events" are used to characterize groups of outcomes. The collection of all such

neg: Probability theory 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 usually denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events is obvious, namely 1 for 1, 2 for 2, etc. This is not always the case. For example, when flipping a coin the two possible outcomes are "heads" and

neg: Complementary event coin is tossed and one assumes that it cannot land on its edge, then it can either land showing "heads" or "tails." Because these two outcomes are mutually exclusive (i.e. the coin cannot simultaneously show both heads and tails) and collectively exhaustive (i.e. there are no other possible outcomes not represented between these two), they are therefore each other's complements. This means that [heads] is logically equivalent to [not tails], and [tails] is equivalent to [not heads]. In a random experiment, the probabilities of all possible events (the sample space) must total to 1— that is, some outcome must occur

neg: Binomial proportion confidence interval times, each trial of the experiment has two possible outcomes (success and failure), the probability of success is the same for each trial, and the trials are statistically independent. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approximations are used to calculate this confidence interval, all with their own tradeoffs in accuracy and computational intensity. A simple example of a binomial distribution is the set of various possible outcomes, and their probabilities, for the number of heads observed when a coin is flipped ten

neg: Probability space Probability space In probability theory, a probability space or a probability triple formula_1 is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same. A probability space consists of three parts: An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more

neg: Almost surely is formula_16 from which it follows that the complement event, flipping tails, has formula_17. Suppose we were to conduct an experiment where the coin is tossed repeatedly, with outcomes formula_18, and it is assumed each flip's outcome is independent of all the others. That is, they are "i.i.d.". Define the sequence of random variables on the coin toss space, formula_19 where formula_20. "i.e." each formula_21 records the outcome of the formula_22'th flip. Any infinite sequence of heads and tails is a possible outcome of the experiment. However, any particular infinite sequence of heads and tails has probability zero of being

neg: Counterfactual definiteness would be inconsistent with the predictions of quantum mechanical theory. Experiments have shown that quantum mechanical results predictably exceed those classical limits. Calculating expectations based on Bell's work implies that for quantum physics the assumption of "local realism" must be abandoned. In Bell's derivation it is explicitly assumed that every possible measurement, even if not performed, can be included in statistical calculations. The calculation involves averaging over sets of outcomes that cannot all be simultaneously factual—if some are assumed to be factual outcomes of an experiment others have to be assumed counterfactual. (Which ones are designated as factual is determined

neg: Probability distribution probability distribution is defined in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a higher-dimensional vector space, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be . Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a

neg: Factorial experiment Factorial experiment In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. For the vast majority of factorial experiments, each factor has only two levels. For example, with

neg: Natural experiment was not under the control of any scientist. Therefore, this exposure has been recognized as being a natural experiment. An aim of a study Angrist and Evans (1998) was to estimate the effect of family size on the labor market outcomes of the mother. For at least two reasons, the correlations between family size and various outcomes (e.g., earnings) do not inform us about how family size causally affects labor market outcomes. First, both labor market outcomes and family size may be affected by unobserved "third" variables (e.g., personal preferences). Second, labor market outcomes themselves may affect family size (called

neg: Negative multinomial distribution Negative multinomial distribution In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB("r", "p")) to more than two outcomes. Suppose we have an experiment that generates "m"+1≥2 possible outcomes, {"X",…,"X"}, each occurring with non-negative probabilities {"p",…,"p"} respectively. If sampling proceeded until "n" observations were made, then {"X",…,"X"} would have been multinomially distributed. However, if the experiment is stopped once "X" reaches the predetermined value "k", then the distribution of the "m"-tuple {"X",…,"X"} is "negative multinomial". These variables are not multinomially distributed because their sum "X"+…+"X" is not fixed, being a draw from

neg: Negative multinomial distribution with the negative multi- nomial distribution. Biometrics 53: 971-82. Negative multinomial distribution In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB("r", "p")) to more than two outcomes. Suppose we have an experiment that generates "m"+1≥2 possible outcomes, {"X",…,"X"}, each occurring with non-negative probabilities {"p",…,"p"} respectively. If sampling proceeded until "n" observations were made, then {"X",…,"X"} would have been multinomially distributed. However, if the experiment is stopped once "X" reaches the predetermined value "k", then the distribution of the "m"-tuple {"X",…,"X"} is "negative multinomial". These variables are not multinomially distributed because their

neg: Probability distribution Probability distribution In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 for , and 0.5 for (assuming the coin is fair). Examples of random phenomena can include the results of an experiment or survey. A

neg: Expectation value (quantum mechanics) Expectation value (quantum mechanics) In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the "most" probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics. Consider an operator formula_1. The expectation value is then formula_2

neg: Divergence-from-randomness model with possible configurations of the outcomes. The one-to-one correspondence for sample space can be defined as: Where ld is the number of trials of the experiment or in this example, the length of a document. We can assume that each outcome may or may not depend on the outcomes of the previous experiments. If the experiments are designed so that an outcome is influencing the next outcomes, then the probability distribution on V is different at each trial. But, more commonly, in order to establish the simpler case when the probability space is invariant in IR, the term independence assumption

neg: Bayesian inference in marketing determine the possible consequences of each action into some measure of success (or loss) with which a certain objective is achieved. This 3 component example explains how the payoffs are conditional upon which outcomes occur. The advertising manager can characterize the outcomes based on past experience and knowledge and devise some possible events that are more likely to occur than others. He can then assign to these events prior probabilities, which would be in the form of numerical weights. He can test out his predictions (prior probabilities) through an experiment. For example, he can run a test campaign to decide

neg: Multiverse the multiverse is not a legitimate topic of scientific inquiry. Concerns have been raised about whether attempts to exempt the multiverse from experimental verification could erode public confidence in science and ultimately damage the study of fundamental physics. Some have argued that the multiverse is a philosophical rather than a scientific hypothesis because it cannot be empirically falsified. The ability to disprove a theory by means of scientific experiment has always been part of the accepted scientific method. Paul Steinhardt has famously argued that no experiment can rule out a theory if the theory provides for all possible outcomes. In

neg: Von Neumann–Morgenstern utility theorem that of Bentham's Greatest Happiness Principle. In the theorem, an individual agent is faced with options called "lotteries". Given some mutually exclusive outcomes, a lottery is a scenario where each outcome will happen with a given probability, all probabilities summing to one. For example, for two outcomes "A" and "B", denotes a scenario where "P"("A") = 25% is the probability of "A" occurring and "P"("B") = 75% (and exactly one of them will occur). More generally, for a lottery with many possible outcomes "A", we write with the sum of the formula_3s equalling 1. The outcomes in a lottery can

neg: Spillover (experiment) outcomes that limit the type of interference. As an example, consider an experiment that sends out political information to undergraduate students to increase their political participation. If the study population consists of all students living with a roommate in a college dormitory, one can imagine four sets of potential outcomes, depending on whether the student or their partner received the information (assume no spillover outside of each two-person room): Now an individual’s outcomes are influenced by both whether they received the treatment and whether their roommate received the treatment. We can estimate one type of spillover effect by looking at

neg: Shot noise outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot. It can be proven that the relative fluctuations reduce as the reciprocal square root of the number of throws, a result valid for all statistical fluctuations, including shot noise. Shot noise exists because phenomena such as light and electric current consist of the movement of discrete (also called "quantized") 'packets'. Consider light—a stream of discrete photons—coming out of a laser pointer and hitting a wall to create a

neg: Probability all possible results 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 a dice can produce six possible results. One collection of possible results gives an odd number on the dice. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the dice falls on some odd number. If the results that actually occur fall in a given event, the

neg: Experiment From God we derive support in all things."" According to his explanation, a strictly controlled test execution with a sensibility for the subjectivity and susceptibility of outcomes due to the nature of man is necessary. Furthermore, a critical view on the results and outcomes of earlier scholars is necessary: Thus, a comparison of earlier results with the experimental results is necessary for an objective experiment - the visible results being more important. In the end, this may mean that an experimental researcher must find enough courage to discard traditional opinions or results, especially if these results are not experimental but

neg: Gibbard–Satterthwaite theorem "regardless of the preferences of other voters". If the dictator has several ex-aequo most-liked alternatives among the possible outcomes, then the winning alternative is simply one of them. The "serial dictatorship" is defined as follows. If voter 1 has a unique most-liked candidate, then this candidate is elected. Otherwise, possible outcomes are restricted to her ex-aequo most-liked candidates, whereas the other candidates are eliminated. Then voter 2's ballot is examined: if she has a unique best-liked candidate among the non-eliminated ones, then this candidate is elected. Otherwise, the list of possible outcomes is reduced again, etc. If there is still

neg: Bernoulli trial Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his "Ars Conjectandi" (1713). The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment. Since a Bernoulli trial has only

neg: Simulation-based optimization solved iteratively where in each iteration the solution moves closer to the optimum solution. Such methods are known as ‘numerical optimization’ or ‘simulation-based optimization’. In simulation experiment, the goal is to evaluate the effect of different values of input variables on a system. However, the interest is sometimes in finding the optimal value for input variables in terms of the system outcomes. One way could be running simulation experiments for all possible input variables. However, this approach is not always practical due to several possible situations and it just makes it intractable to run experiments for each scenario. For example,

neg: Null hypothesis or "simple" proposed change in the null hypothesis ((new vs old) rather than (new vs placebo)) can have a dramatic effect on the utility of a test for complex non-statistical reasons. The choice of null hypothesis ("H") and consideration of directionality (see "one-tailed test") is critical. Consider the question of whether a tossed coin is fair (i.e. that on average it lands heads up 50% of the time) and an experiment where you toss the coin 5 times. A possible result of the experiment that we consider here is 5 heads. Let outcomes be considered unlikely with respect to an

neg: CLs method (particle physics) any frequentist method if the conditionality principle is regarded as necessary). Birnbaum himself suggested in his 1962 paper that the CLs ratio formula_14 should be used as a measure of the strength of "statistical evidence" provided by significance tests, rather than formula_9 alone. This followed from a simple application of the likelihood principle: if the outcome of an experiment is to be only reported in a form of a "accept"/"reject" decision, then the overall procedure is equivalent to an experiment that has only two possible outcomes, with probabilities formula_9,formula_63 and formula_34,formula_65 under formula_66. The likelihood ratio associated with the outcome

neg: Gauge theory theory of gravity known as general relativity. Gauge theories used to model the results of physical experiments engage in: We cannot express the mathematical descriptions of the "setup information" and the "possible measurement outcomes", or the "boundary conditions" of the experiment, without reference to a particular coordinate system, including a choice of gauge. One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. Mishandling gauge dependence calculations in boundary conditions is a frequent source of anomalies, and approaches to anomaly avoidance classifies gauge theories. The two gauge theories mentioned above, continuum electrodynamics and general relativity,

neg: The Coming of the Quantum Cats can - given different social conditions - develop completely different character traits (timid or assertive, benevolent or predatory, etc.) The title refers to the famous thought-experiment known as "Schrödinger's Cat", where the outcome of a real-world event depends on a quantum event. By some interpretations, the real-world object, in this case a cat, must be in two states simultaneously until it is observed. However, the story is more closely related to the many-worlds interpretation of quantum mechanics where all possible outcomes of an event occur, each in its own parallel universe. The same concept was treated, somewhat more pessimistically, in

neg: Random variable a crack in the floor, but such a possibility is excluded from consideration. The domain of a random variable is the set of possible outcomes. In the case of the coin, there are only two possible outcomes, namely heads or tails. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. A random variable has a probability distribution, which specifies the probability of its values. Random variables can be discrete, that is, taking any of a specified finite or countable list of values,

neg: Expectation value (quantum mechanics) or the operator formula_1 is taken to be time-dependent, depending on whether the Schrödinger picture or Heisenberg picture is used. The evolution of the expectation value does not depend on this choice, however. If formula_1 has a complete set of eigenvectors formula_18, with eigenvalues formula_19, then (1) can be expressed as (2) formula_20. This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues formula_19 are the possible outcomes of the experiment, and their corresponding coefficient formula_22 is the probability that this outcome will occur; it is often called the "transition probability".

neg: Scenario analysis Scenario analysis Scenario analysis is a process of analyzing possible future events by considering alternative possible outcomes (sometimes called "alternative worlds"). Thus, scenario analysis, which is one of the main forms of projection, does not try to show one exact picture of the future. Instead, it presents several alternative future developments. Consequently, a scope of possible future outcomes is observable. Not only are the outcomes observable, also the development paths leading to the outcomes. In contrast to prognoses, the scenario analysis is not based on extrapolation of the past or the extension of past trends. It does not rely on

neg: Free recall of each list, subjects were asked to recall as many words as possible in any order. Results from the experiment showed that all groups expressed both primacy effects and recency effects. Recency effects were exhibited regardless of the length of the list, and it was strongest for the words in the last eight serial positions. The primacy effect extended over the first four serial positions. Another evidence of the recency effect is found in the way that participants initiate recall of a list: they most often start with terminal (recent) list items (an early description of the recency effect in

neg: CPO-STV show only the comparison of those outcomes which include both of these candidates as winners. The list of possible outcomes may therefore be reduced to three: Each of these outcomes will be compared, in turn, with each other outcome, in order to find a winner. Three comparisons are therefore necessary. A fourth comparison will also be demonstrated, that between Outcome A and Outcome D. We know that Outcome D cannot win but this will be done in order to completely clarify CPO-STV's rule for when and when not to transfer surpluses. The first preferences cast for all of the candidates

neg: Beta distribution condition 1 < "s" < "n" − 1, necessary for a mode to exist between both ends, is fulfilled). In Chapter 12 (p. 385) of his book, Jaynes asserts that the "Haldane prior" Beta(0,0) describes a "prior state of knowledge of complete ignorance", where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure, while the "Bayes (uniform) prior Beta(1,1) applies if" one knows that "both binary outcomes are possible". Jaynes states: ""interpret the Bayes-Laplace (Beta(1,1)) prior as describing not a state of complete ignorance", but the state of

neg: Experiment (probability theory) to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis. Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. A fixed number of repetitions of the same experiment can be thought of as a composed experiment, in which case the individual repetitions are called trials. For example, if one were to toss the same coin one

neg: Empirical research are derived (e.g., "People who study a word list while listening to vocal music will remember fewer words on a later memory test than people who study a word list in silence."). These predictions can then be tested with a suitable experiment. Depending on the outcomes of the experiment, the theory on which the hypotheses and predictions were based will be supported or not, or may need to be modified and then subjected to further testing. The term empirical was originally used to refer to certain ancient Greek practitioners of medicine who rejected adherence to the dogmatic doctrines of the

neg: Statistical ensemble (mathematical physics) to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics. The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes. The notional size of ensembles in thermodynamics, statistical mechanics and quantum statistical mechanics can be very large, including every possible microscopic state the system could be in, consistent with its observed macroscopic properties. For many important physical cases, it is possible to calculate averages directly over the whole of the

neg: Bernoulli trial the coin is assumed to be fair, the probability of success is formula_20. Thus the probability of failure, formula_2, is given by Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by: Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in

neg: Push of the past and then counting how many branches are present at each time interval. As we know retrospectively that all such lineages survived until the present, it follows that no extinction is possible along them. It can be shown that the rate of production of new lineages through time is given by formula_6. Rather than considering the distribution of all possible stochastic outcomes for given values of formula_7 and formula_2 it is also possible to consider what happens when certain conditions of survivorship are imposed on the possible outcomes. If a BDM is forward-modelled, i.e. if the fate of an original single

neg: Info-gap decision theory "local" decision theory, beginning with an estimate and considering "deviations" from it; this contrasts with "global" methods such as minimax, which considers worst-case analysis over the entire space of outcomes, and probabilistic decision theory, which considers all possible outcomes, and assigns some probability to them. In info-gap, the universe of possible outcomes under consideration is the union of all of the nested subsets: formula_4 Info-gap analysis gives answers to such questions as: It can be used for satisficing, as an alternative to optimizing in the presence of uncertainty or bounded rationality; see robust optimization for an alternative approach. In contrast

neg: Single-parameter utility Single-parameter utility In mechanism design, an agent is said to have single-parameter utility if his valuation of the possible outcomes can be represented by a single number. For example, in an auction for a single item, the utilities of all agents are single-parametric, since they can be represented by their monetary evaluation of the item. In contrast, in a combinatorial auction for two or more related items, the utilities are usually not single-parametric, since they are usually represented by their evaluations to all possible bundles of items. There is a set formula_1 of possible outcomes. There are formula_2 agents which

neg: Thought experiment pattern that thought experiments display is their orientation in time. They are either: The second characteristic pattern is their movement in time in relation to “the present moment standpoint” of the individual performing the experiment; namely, in terms of: Generally speaking, there are seven types of thought experiments in which one reasons from causes to effects, or effects to causes: "Prefactual (before the fact) thought experiments" — the term prefactual was coined by Lawrence J. Sanna in 1998 — speculate on possible future outcomes, given the present, and ask "What will be the outcome if event E occurs?" "Counterfactual (contrary

neg: Event (probability theory) < "X" ≤ "v" is an example of an inverse image under the mapping "X" because formula_6 if and only if formula_7. Event (probability theory) In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event "not" occurring), and together these define a

neg: Stern–Gerlach experiment deflected either up or down by a specific amount. This was a measurement of the quantum observable now known as spin angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or point spectrum. Although some discrete quantum phenomena, such as atomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to observe separation between discrete quantum states for the first time in the history of science. By now, it is known that, theoretically, quantum angular momentum "of any kind" has a discrete spectrum, which is sometimes briefly expressed as "angular

neg: Event (probability theory) Event (probability theory) In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event "not" occurring), and together these define a Bernoulli trial: did the event occur or not? Typically, when the sample space is finite, any subset of the sample space is

neg: Two-balloon experiment one on the right. Equilibria in which both balloons are on the "right" of the pressure peak also exist but are unstable. This is easy to verify by squeezing the air back and forth between two interconnected balloons. At large extensions, the pressure inside a natural rubber balloon once again goes up. This is due to a number of physical effects that were ignored in the James/Guth theory: crystallization, imperfect flexibility of the molecular chains, steric hindrances and the like. As a result, if the two balloons are initially very extended, other outcomes of the two-balloon experiment are possible, and

neg: Simplex an "n"-simplex is equivalent to an "n"-ball. Every "n"-simplex is an "n"-dimensional manifold with corners. In probability theory, the points of the standard "n"-simplex in ("n" + 1)-space are the space of possible parameters (probabilities) of the categorical distribution on "n" + 1 possible outcomes. Since all simplices are self-dual, they can form a series of compounds; In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology

neg: Bayesian statistics of an experiment, then, formula_25 When there are an infinite number of outcomes, it is necessary to integrate over all outcomes to calculate formula_23 using the law of total probability. Often, formula_23 is difficult to calculate as the calculation would involve sums or integrals that would be time-consuming to evaluate, so often only the product of the prior and likelihood is considered, since the evidence does not change in the same analysis. The posterior is proportional to this product: formula_28 The maximum a posteriori, which is the mode of the posterior and is often computed in Bayesian statistics using mathematical

neg: Statistical fluctuations all, if a fair coin is tossed many times and the number of heads and tails counted, the ratio of heads to tails will be very close to 1 (about as many heads as tails); but after only a few throws, outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot. An electric current so small that not many electrons are involved flowing through a p-n junction is susceptible to statistical fluctuations as the actual number of electrons

neg: Equipossibility Equipossibility Equipossibility is a philosophical concept in possibility theory that is a precursor to the notion of equiprobability in probability theory. It is used to distinguish what "can" occur in a probability experiment. For example, when considering rolling a six-sided die, why do we typically view the possible outcomes as {1,2,3,4,5,6} rather than, say, {6, not 6}? The former set contains equally possible alternatives, while the latter does not because there are five times as many alternatives inherent in 'not 6' as in 6. This is true even if the die is biased so that 6 and 'not 6' are

neg: Self-experimentation in medicine of what are now called hormones, was a nineteenth century proponent of the practice: Self-experimentation has value in rapidly obtaining the first results. In some cases, such as with Forssmann's experiments done in defiance of official permission, results may be obtained that would never otherwise have come to light. However, self-experiment lacks the statistical validity of a larger experiment. It is not possible to generalise from an experiment on a single person. For instance, a single successful blood transfusion does not indicate, as we now know from the work of Karl Landsteiner, that all such transfusions between any two random

neg: Overtaking criterion to do this comparison. formula_1 is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual gross domestic product. It is normalized formula_2 is the set of infinite sequences of possible outcomes. Each element in formula_2 is of the form: formula_4. formula_5 is a partial order. Given two infinite sequences formula_6, it is possible that formula_7 is weakly better (formula_8) or that formula_9 is weakly better (formula_10) or that they are incomparable. formula_11 is the strict variant of formula_5, i.e., formula_13 if formula_14 and not formula_15. formula_11 is called the "overtaking

neg: Bell test experiments local hidden variable theories, and recently an experiment has been performed that is not subject to either the locality loophole or the detection loophole (Hensen et al.). An experiment free of the locality loophole is one where for each separate measurement and in each wing of the experiment, a new setting is chosen and the measurement completed before signals could communicate the settings from one wing of the experiment to the other. An experiment free of the detection loophole is one where close to 100% of the successful measurement outcomes in one wing of the experiment are paired with a

neg: Gibbard's theorem other candidates are eliminated. Then voter 2's ballot is examined: if he has a unique best-liked candidate among the non-eliminated ones, then this candidate is elected. Otherwise, the list of possible outcomes is reduced again, etc. If there is still several non-eliminated candidates after all ballots have been examined, then an arbitrary tie-breaking rule is used. This game form is straightforward: whatever the preferences of a voter, he has a dominant strategy that consists in declaring his sincere preference order. It is also dictatorial, and its dictator is voter 1: if his wishes to see candidate formula_4 elected, then he

neg: Binomial distribution Binomial distribution </math> In probability theory and statistics, the binomial distribution with parameters "n" and "p" is the discrete probability distribution of the number of successes in a sequence of "n" independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability "p") or failure/no/false/zero (with probability "q" = 1 − "p"). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., "n" = 1, the binomial

neg: Liquidity risk time period. The BIS mentions "... a number of institutions are exploring the use of liquidity adjusted-VAR, in which the holding periods in the risk assessment are adjusted by the length of time required to unwind positions." Alan Greenspan (1999) discusses management of foreign exchange reserves and suggested a measure called Liquidity at risk. A country's liquidity position under a range of possible outcomes for relevant financial variables (exchange rates, commodity prices, credit spreads, etc.) is considered. It might be possible to express a standard in terms of the probabilities of different outcomes. For example, an acceptable debt structure could

neg: Experiment results, often through a comparison between control measurements and the other measurements. Scientific controls are a part of the scientific method. Ideally, all variables in an experiment are controlled (accounted for by the control measurements) and none are uncontrolled. In such an experiment, if all controls work as expected, it is possible to conclude that the experiment works as intended, and that results are due to the effect of the tested variable. In the scientific method, an experiment is an empirical procedure that arbitrates competing models or hypotheses. Researchers also use experimentation to test existing theories or new hypotheses to

neg: E. H. Carr an essential part of the historians' work. Trevor-Roper argued that only by considering all possible outcomes of a given historical situation could a historian properly understand the period under study. In Trevor-Roper's opinion, only by looking at all possible outcomes and all sides could a historian properly understand history, and those historians who adopted Carr's perspective of only seeking to understand the "winners" of history, and treating the outcome of a particular set of events as the only possible outcome were "bad historians". In a review in 1963 in "Historische Zeitschrift", Andreas Hillgruber wrote favourably of Carr's "geistvoll-ironischer" (ironically spirited)

neg: Morton's List "Quests." The players then discuss how to interpret and execute the Quest. During this discussion the players are encouraged to use creativity to find not only a viable interpretation, but one which is both fun for all players, and in accordance with the moral code of all players. Morton's List claims to include any possible activity that people might do for fun. As such, it includes directives that might be interpreted as encouraging illegal or dangerous activities such as "vigilantism, real life spell casting, and to experiment with potentially illegal substances." For this reason the game was banned from Gen

neg: Many-minds interpretation decisions. In general, the Many-minds theory holds that a conscious being who observes the outcome of a random zero-sum experiment will evolve into two successors in different observer states, each of whom observes one of the possible outcomes. Moreover, the theory advises you to favour choices in such situations in proportion to the probability that they will bring good results to your various successors. But in a life-or-death case like getting into the box with Schrödinger's cat, you will only have one successor, since one of the outcomes will ensure your death. So it seems that the Many-minds Interpretation advises

neg: Thought experiment room is nomologically possible. Some thought experiments present scenarios that are not nomologically possible. In his Twin Earth thought experiment, Hilary Putnam asks us to imagine a scenario in which there is a substance with all of the observable properties of water (e.g., taste, color, boiling point), but is chemically different from water. It has been argued that this thought experiment is not nomologically possible, although it may be possible in some other sense, such as metaphysical possibility. It is debatable whether the nomological impossibility of a thought experiment renders intuitions about it moot. In some cases, the hypothetical scenario

neg: Info-gap decision theory considering all possible outcomes, as in minimax. Further, it considers a fixed universe formula_6 so it is not robust to unexpected (not modeled) events. The connection to minimax analysis has occasioned some controversy: (Ben-Haim 1999, pp. 271–2) argues that info-gap's robustness analysis, while similar in some ways, is not minimax worst-case analysis, as it does not evaluate decisions over all possible outcomes, while (Sniedovich, 2007) argues that the robustness analysis can be seen as an example of maximin (not minimax), applied to maximizing the horizon of uncertainty. This is discussed in criticism, below, and elaborated in the classical decision theory

neg: Grey goo goo scenario. Grey goo is a useful construct for considering low-probability, high-impact outcomes from emerging technologies. Thus, it is a useful tool in the ethics of technology. Daniel A. Vallero applied it as a worst-case scenario thought experiment for technologists contemplating possible risks from advancing a technology. This requires that a decision tree or event tree include even extremely low probability events if such events may have an extremely negative and irreversible consequence, i.e. application of the precautionary principle. Dianne Irving admonishes that "any error in science will have a rippling effect..." Vallero adapted this reference to chaos theory to

neg: Education in South Africa set about reforming the educational system by first removing all racially offensive and outdated content and then introducing continuous assessment into schools. The South African Schools Act, 1996 was promulgated to "provide for a uniform system for the organisation, governance and funding of schools". In 1997, the government launched its new education system called Curriculum 2005, which would be based on "outcomes based education" (OBE). By 2006 it was clear that OBE as a social experiment had failed, and it was quietly shelved. South Africa has 11 official languages. and the first year of schooling is provided in all these

neg: Visual novel game, leading to many different possible outcomes. An acclaimed example is "", where nearly every action and dialogue choice can lead to entirely new branching paths and endings. Each path only reveals certain aspects of the overall storyline and it is only after uncovering all the possible different paths and outcomes, through multiple playthroughs, that every component comes together to form a coherent well-written story. The branching path stories found in visual novels represent an evolution of the "Choose Your Own Adventure" concept. The digital medium allows for significant improvements, such as being able to fully explore multiple aspects and

neg: Combinatorial principles used to manipulate sequences, and can describe if not resolve many combinatorial situations. The rule of sum is an intuitive principle stating that if there are "a" possible outcomes for an event (or ways to do something) and "b" possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are "a + b" total possible outcomes for the events (or total possible ways to do one of the things). More formally, the sum of the sizes of two disjoint sets is equal

neg: Bayes' theorem that degree of belief may rise, fall or remain the same depending on the results. For proposition "A" and evidence "B", For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see Bayesian inference. In the frequentist interpretation, probability measures a "proportion of outcomes." For example, suppose an experiment is performed many times. "P"("A") is the proportion of outcomes with property "A", and "P"("B") that with property "B". "P"("B" | "A" ) is the proportion of outcomes with property "B" "out of" outcomes with property "A", and "P"("A" | "B" ) the proportion of those with

neg: Hindsight bias In an elaboration of these heuristics, Beyth and Fischhoff devised the first experiment directly testing the hindsight bias. They asked participants to judge the likelihood of several outcomes of US president Richard Nixon's upcoming visit to Beijing (then romanized as Peking) and Moscow. Some time after president Nixon's return, participants were asked to recall (or reconstruct) the probabilities they had assigned to each possible outcome, and their perceptions of likelihood of each outcome was greater or overestimated for events that actually had occurred. This study is frequently referred to in definitions of the hindsight bias, and the title of the

neg: Loopholes in Bell test experiments in all physical experiments. Whether or not any of those presented here have been found important enough to be called loopholes, in general or because of possible mistakes by the performers of some known experiment found in literature, is discussed in the subsequent sections. There are also non-optical Bell test experiments, which are not discussed here. As a basis for our description of experimental errors let us consider a typical experiment of CHSH type (see picture to the right). In the experiment the source is assumed to emit light in the form of pairs of particle-like photons with each photon

neg: Psychology random allocation of subjects to conditions allows researchers to make strong inferences about causal relationships. In an experiment, the researcher alters parameters of influence, called independent variables, and measures resulting changes of interest, called dependent variables. Prototypical experimental research is conducted in a laboratory with a carefully controlled environment. Repeated-measures experiments are those which take place through intervention on multiple occasions. In research on the effectiveness of psychotherapy, experimenters often compare a given treatment with placebo treatments, or compare different treatments against each other. Treatment type is the independent variable. The dependent variables are outcomes, ideally assessed in several ways

neg: Experiment be identified. Researchers attempt to reduce the biases of observational studies with complicated statistical methods such as propensity score matching methods, which require large populations of subjects and extensive information on covariates. Outcomes are also quantified when possible (bone density, the amount of some cell or substance in the blood, physical strength or endurance, etc.) and not based on a subject's or a professional observer's opinion. In this way, the design of an observational study can render the results more objective and therefore, more convincing. By placing the distribution of the independent variable(s) under the control of the researcher, an

neg: Minimax loss function. In this framework, formula_37 is called minimax if it satisfies An alternative criterion in the decision theoretic framework is the Bayes estimator in the presence of a prior distribution formula_39. An estimator is Bayes if it minimizes the "average" risk A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using expected value or expected utility, it makes no assumptions about the probabilities of various outcomes, just scenario analysis of what the possible outcomes are. It is thus robust to changes in the assumptions, as these other decision techniques are not. Various extensions of

neg: Von Neumann–Wigner interpretation possible situations. However, during observation, the wavefunction describing the system collapses to one of several options. If there is no observation, this collapse does not occur, and none of the options ever becomes less likely. It can be predicted using quantum mechanics, absent a collapse postulate, that an observer observing a quantum superposition will turn into a superposition of different observers seeing different things. The observer will have a wavefunction which describes all the possible outcomes. Still, in actual experience, an observer never senses a superposition, but always senses that one of the outcomes has occurred with certainty. This apparent

neg: Differential outcomes effect of the differential outcomes effect for populations with intellectual disabilities and pervasive developmental disabilities. Scientist have theorised that this procedure may be useful in overcoming barriers to learning. Differential outcomes effect The differential outcomes effect is a theory in behaviorism, a branch of psychology, that shows that a positive effect on accuracy occurs in discrimination learning between different stimuli when unique rewards are paired with each individual stimulus. The DOE was first demonstrated in 1970 by Milton Trapold on an experiment with rats. Rats were trained to discriminate between a clicker and a tone by pressing the left and right

neg: Differential outcomes effect Differential outcomes effect The differential outcomes effect is a theory in behaviorism, a branch of psychology, that shows that a positive effect on accuracy occurs in discrimination learning between different stimuli when unique rewards are paired with each individual stimulus. The DOE was first demonstrated in 1970 by Milton Trapold on an experiment with rats. Rats were trained to discriminate between a clicker and a tone by pressing the left and right levers. Half of the rats were trained using the differential outcomes procedure, where the clicker was paired with sucrose and tone with food pellets. The remaining rats were

neg: Mathematics of bookmaking an equal chance of winning. The situation is the same in Match 2 between players C and D. In a "fair" book in each of their matches, i.e. each has a book of 100%, all players would be offered at odds of Evens. However, a bookmaker would probably offer odds of 5-6 (for example) on each of the two possible outcomes in each event (each tennis match). This results in a book for each of the tennis matches of 109.09...%, calculated by 100 × ( + ) i.e. 9.09% overround. There are four possible outcomes from combining the results from

neg: Inductive probability are defined as the ratio of the number of outcomes within an event to the total number of outcomes. In the possible world model each possible world is an outcome, and statements about possible worlds define events. The probability of a statement being true is the number of possible worlds where the statement is true divided by the total number of possible worlds. The probability of a statement formula_11 being true about possible worlds is then, For a conditional probability. then Using symmetry this equation may be written out as Bayes' law. This law describes the relationship between prior and

neg: Conditional variance recall, is a random variable itself (a function of "X", determined up to probability one). As a result, formula_4 itself is a random variable (and is a function of "X"). Recall that variance is the expected squared deviation between a random variable (say, "Y") and its expected value. The expected value can be thought of as a reasonable prediction of the outcomes of the random experiment (in particular, the expected value is the best constant prediction when predictions are assessed by expected squared prediction error). Thus, one interpretation of variance is that it gives the smallest possible expected squared prediction

neg: Probability distribution discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

neg: Almost surely lands exactly on a diagonal is zero. So, the dart will almost never land on a diagonal (i.e. it will almost surely "not" land on a diagonal). Nonetheless the set of points on the diagonals is not empty and a point on a diagonal is no less possible than any other point: the diagonal does contain valid outcomes of the experiment. Consider the case where a (possibly biased) coin is tossed, corresponding to the probability space formula_13, where the event formula_14 occurs if heads is flipped, and formula_15 if tails. For this particular coin, assume the probability of flipping heads

neg: Confusion matrix combined. Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 "confusion matrix", as follows: Confusion matrix In the field of machine learning and specifically the problem of statistical classification, a confusion matrix, also known as an error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one (in unsupervised learning it is usually called a matching matrix). Each row of the matrix represents the instances in a predicted class while each column represents the