An Euler diagram is a diagrammatic means of representing sets and their relationships. They involve overlapping shapes, may be scaled, such that the area of the shape is proportional to the number of elements it contains, they are useful for explaining complex hierarchies and overlapping definitions. They are confused with Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships; the first use of "Eulerian circles" is attributed to Swiss mathematician Leonhard Euler. In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since they have been adopted by other curriculum fields such as reading as well as organizations and businesses. Euler diagrams consist of simple closed shapes in a two dimensional plane that each depict a set or category. How or if these shapes overlap demonstrates the relationships between the sets.
There are only 3 possible relationships between any 2 sets. This is referred to as containment, overlap or neither or in mathematics, it may be referred to as subset and disjoint; each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, the exterior, which represents all elements that are not members of the set. Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets. A curve, contained within the interior zone of another represents a subset of it. Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color; when the number of sets grows beyond 3 a Venn diagram becomes visually complex compared to the corresponding Euler diagram.
The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets: A = B = C = The Euler and the Venn diagrams of those sets are: In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples below, the Euler diagram depicts that the sets Animal and Mineral are disjoint since the corresponding curves are disjoint, that the set Four Legs is a subset of the set of Animals; the Venn diagram, which uses the same categories of Animal and Four Legs, does not encapsulate these relationships. Traditionally the emptiness of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent emptiness either by the absence of a region. A set of well-formedness conditions are imposed. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations.
However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs; as shown in the illustration to the right, Sir William Hamilton in his posthumously published Lectures on Metaphysics and Logic erroneously asserts that the original use of circles to "sensualize... the abstractions of Logic" was not Leonhard Paul Euler but rather Christian Weise in his Nucleus Logicae Weisianae that appeared in 1712 posthumously, the latter book was written by Johann Christian Lange rather than Weise. He references Euler's Letters to a German Princess In Hamilton's illustration the four categorical propositions that can occur in a syllogism as symbolized by the drawings A, E, I and O are: A: The Universal Affirmative, Example: "All metals are elements". E: The Universal Negative, Example: "No metals are compound substances".
I: The Particular Affirmative, Example: "Some metals are brittle". O: The Particular Negative, Example: "Some metals are not brittle". In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn comments on the remarkable prevalence of the Euler diagram: "...of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose:-somewhat at random, as they happened to be most accessible:-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Scheme." But he contended, "the inapplicability of this scheme for the purposes of a general Logic" (pag
In probability theory and statistics, Bayes' theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age using Bayes' theorem, a person's age can be used to more assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person's age. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference; when applied, the probabilities involved in Bayes' theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics. Bayes' theorem is named after Reverend Thomas Bayes, who first used conditional probability to provide an algorithm that uses evidence to calculate limits on an unknown parameter, published as An Essay towards solving a Problem in the Doctrine of Chances.
In what he called a scholium, Bayes extended his algorithm to any unknown prior cause. Independently of Bayes, Pierre-Simon Laplace in 1774, in his 1812 "Théorie analytique des probabilités" used conditional probability to formulate the relation of an updated posterior probability from a prior probability, given evidence. Sir Harold Jeffreys put Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry". Bayes' theorem is stated mathematically as the following equation: where A and B are events and P ≠ 0. P is a conditional probability: the likelihood of event A occurring given. P is a conditional probability: the likelihood of event B occurring given that A is true. P and P are the probabilities of observing B independently of each other. Suppose that a test for using a particular drug is 99% sensitive and 99% specific; that is, the test will produce 99% true positive results for drug users and 99% true negative results for non-drug users.
Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a positive test is a drug user? P = P P P = P P P P + P P = 0.99 × 0.005 0.99 × 0.005 + 0.01 × 0.995 ≈ 33.2 % Even if an individual tests positive, it is more that they do not use the drug than that they do. This is; the number of false positives outweighs the number of true positives. For example, if 1000 individuals are tested, there are expected to be 5 users. From the 995 non-users, 0.01 × 995 ≃ 10 false positives are expected. From the 5 users, 0.99 × 5 ≈ 5 true positives are expected. Out of 15 positive results, only 5 are genuine; the importance of specificity in this example can be seen by calculating that if sensitivity is raised to 100% and specificity remains at 99% the probability of the person being a drug user only rises
Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, loosely speaking, 0 indicates impossibility and 1 indicates certainty; the higher the probability of an event, the more it is that the event will occur. A simple example is the tossing of a fair coin. Since the coin is fair, the two outcomes are both probable; these concepts have been given an axiomatic mathematical formalization in probability theory, used in such areas of study as mathematics, finance, science, artificial intelligence/machine learning, computer science, game theory, philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is used to describe the underlying mechanics and regularities of complex systems; when dealing with experiments that are random and well-defined in a purely theoretical setting, probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes.
For example, tossing a fair coin twice will yield "head-head", "head-tail", "tail-head", "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents possess different views about the fundamental nature of probability: Objectivists assign numbers to describe some objective or physical state of affairs; the most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome if it is performed only once.
Subjectivists assign numbers per subjective probability. The degree of belief has been interpreted as, "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E." The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some prior probability distribution; these data are incorporated in a likelihood function. The product of the prior and the likelihood, results in a posterior probability distribution that incorporates all the information known to date. By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions regardless of how much information the agents share; the word probability derives from the Latin probabilitas, which can mean "probity", a measure of the authority of a witness in a legal case in Europe, correlated with the witness's nobility.
In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, is arrived at from inductive reasoning and statistical inference. The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers. According to Richard Jeffrey, "Before the middle of the seventeenth century, the term'probable' meant approvable, was applied in that sense, unequivocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances." However, in legal contexts especially,'probable' could apply to propositions for which there was good evidence.
The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal. Christiaan Huygens gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi and Abraham de Moivre's Doctrine of Chances treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the concept of mathematical probability; the theory of errors may be traced back to Roger Cotes's Opera Miscellanea, but a memoir prepared by Thomas Simpson in 1755 first applied the theory to the discussion of errors of observation. The reprint of this memoir lays down the axioms that positive and negative errors are probable, that certain assignable limits define the range of all errors.
Simpson discusses c
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves circles, each representing a set; the points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends to read visualizations. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets, they are thus a special case of Euler diagrams, which do not show all relations. Venn diagrams were conceived around 1880 by John Venn, they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, statistics and computer science. A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram.
This example involves A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures; the blue circle, set B, represents the living creatures. Each separate type of creature can be imagined as a point somewhere in the diagram. Living creatures that both can fly and have two legs—for example, parrots—are in both sets, so they correspond to points in the region where the blue and orange circles overlap, it is important to note that this overlapping region would only contain those elements that are members of both set A and are members of set B Humans and penguins are bipedal, so are in the orange circle, but since they cannot fly they appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have six legs, fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles.
The combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that can fly; the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. Venn diagrams were introduced in 1880 by John Venn in a paper entitled On the Diagrammatic and Mechanical Representation of Propositions and Reasonings in the "Philosophical Magazine and Journal of Science", about the different ways to represent propositions by diagrams; the use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, is "not an easy history to trace, but it is certain that the diagrams that are popularly associated with Venn, in fact, originated much earlier. They are rightly associated with Venn, because he comprehensively surveyed and formalized their usage, was the first to generalize them".
Venn himself did not use the term "Venn diagram" and referred to his invention as "Eulerian Circles". For example, in the opening sentence of his 1880 article Venn writes, "Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that called'Eulerian circles,' has met with any general acceptance..." Lewis Carroll includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book "Symbolic Logic". The term "Venn diagram" was used by Clarence Irving Lewis in 1918, in his book "A Survey of Symbolic Logic". Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. M. E. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler, but much of it was unpublished.
She observes earlier Euler-like diagrams by Ramon Llull in the 13th Century. In the 20th century, Venn diagrams were further developed. D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number, he showed that such symmetric Venn diagrams exist when n is five or seven. In 2002 Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs and Savage showed that symmetric Venn diagrams exist for all other primes, thus rotationally symmetric Venn diagrams exist. Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since they have been adopted in the curriculum of other fields such as reading. A Venn diagram is constructed with a collection of simple closed curves drawn in a plane. According to Lewis, the "principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.
That is, the diagram leaves room for any possible relation
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g. the value assigned to "1 or 2" in a throw of a die should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to biology; the requirements for a function μ to be a probability measure on a probability space are that: μ must return results in the unit interval, returning 0 for the empty set and 1 for the entire space.μ must satisfy the countable additivity property that for all countable collections of pairwise disjoint sets: μ = ∑ i ∈ I μ.
For example, given three elements 1, 2 and 3 with probabilities 1/4, 1/4 and 1/2, the value assigned to is 1/4 + 1/2 = 3/4, as in the diagram on the right. The conditional probability based on the intersection of events defined as: P = P P. satisfies the probability measure requirements so long as P is not zero. Probability measures are distinct from the more general notion of fuzzy measures in which there is no requirement that the fuzzy values sum up to 1, the additive property is replaced by an order relation based on set inclusion. Market measures which assign probabilities to financial market spaces based on actual market movements are examples of probability measures which are of interest in mathematical finance, e.g. in the pricing of financial derivatives. For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the expected value of the future payoff taken with respect to that same risk neutral measure, discounted at the risk-free rate.
If there is a unique probability measure that must be used to price assets in a market the market is called a complete market. Not all measures that intuitively represent chance or likelihood are probability measures. For instance, although the fundamental concept of a system in statistical mechanics is a measure space, such measures are not always probability measures. In general, in statistical physics, if we consider sentences of the form "the probability of a system S assuming state A is p" the geometry of the system does not always lead to the definition of a probability measure under congruence, although it may do so in the case of systems with just one degree of freedom. Probability measures are used in mathematical biology. For instance, in comparative sequence analysis a probability measure may be defined for the likelihood that a variant may be permissible for an amino acid in a sequence. Borel measure Fuzzy measure Haar measure Martingale measure Lebesgue measure Probability and Measure by Patrick Billingsley, 1995 John Wiley ISBN 978-0-471-00710-4 Probability & Measure Theory by Robert B.
Ash, Catherine A. Doléans-Dade 1999 Academic Press ISBN 0-12-065202-1. Media related to Probability measure at Wikimedia Commons