Disjoint Events Research Articles

Lessons from the famous 17th‐century paradox of the Chevalier de Méré

AbstractWe take advantage of a combinatorial misconception and the famous paradox of the Chevalier de Méré to present the multiplication rule for independent events; the principle of inclusion and exclusion in the presence of disjoint events; the median of a discrete‐type random variable, and a confidence interval for a large sample. Moreover, we pay tribute to our original bibliographic sources by providing two computational tools to facilitate the students' insights on these topics.

Analysis of the probability of connectivity of a telecommunications network based on the reduction of several non-connectivity events to a union of independent events

Introduction: For large and structurally complex telecommunication networks, calculating the connectivity probability turns out to be a very cumbersome and time-consuming process due to the huge number of elements in the resulting expression. The most expedient way out of this situation is a method based on the representation of a network connectivity event in the form of sums of products of incompatible events. However, this method also requires performing additional operations on sets in some cases. Purpose: To eliminate the main disadvantages of the method using multi-variable inversion. Results: It is shown that the connectivity event of a graph should be interpreted as a union of connectivity events of all its subgraphs, which leads to the validity of the expression for the connectivity event of the network in the form of a union of connectivity events of typical subgraphs (path, backbone, and in general, a multi-pole tree) of the original random graph. An iterative procedure is proposed for bringing a given number of connectivity events to the union of independent events by sequentially adding subgraph disjoint events. The possibility of eliminating repetitive routine procedures inherent in methods using multi-variable inversion is proved by considering not the union of connectivity events (incoherence) degenerating into the sum of incompatible products, but the intersection of opposite events, which also leads to a similar sum. However, to obtain this sum, there is no need to perform a multi-variable inversion for each of the terms over all those previously analyzed. Practical relevance: The obtained analytical relations can be applied in the analysis of reliability, survivability or stability of complex telecommunications networks.

Application of Z-Numbers Based Approach to Project Risks Assessment

At every phase of the project management process various risks originate owing to occurrence of uncertain events. In this research, we are analyzing potentialities of the Z-numbers in improving the quality of risk assessment. Risk assessment uses probability theory, theory of possibilities, fuzzy approach, Z-number based approach etc. Combined risk measure based on probability and consequence is calculated by applying the disjoint events probability formula or as a product of events. Reliability of relevant information unaccounted in this approach and this circumstance limits the descriptive power of the approach. Suggested by L. Zade a bi-component Z-number Z = (A, B) represents in a unified way a restriction on the values of the uncertain variable (A) and its certainty (B) and allows to take into account the reliability of information. Prediction identical to (High, Very Sure) can be formalized as a Z- evaluation “X is Z (A,B)”, where X is random variable of Risk Likelihood, A and B are fuzzy sets, describing soft constraints on a risk likelihood and a partial reliability, respectively. Usually, A and B are sense-based and in effect are imprecise.Z-number describes a probability of threat as: Likelihood =Z1(High,Very Sure),where A is expressed by linguistic terms High, Medium, Low, and B is expressed by terms Very Sure, Sure and so on. Similarly, Сonsequence measure is described as Сonsequence measure = Z2 (Low, Sure).Risk levels (Z12) is calculated as the product of the likelihood (Z1) and consequence measures (Z2).Effectiveness of the approach illustrated by examples. A general and computationally effective approach suggested to computation with Z-numbers allows using Z-information for the solving decision-making problem which can be utilized for risk factors estimation.. Application of the Z-number based approach for a project risks assessment increases adequacy of the risks representation due to better approximation of the combined effects

Application of Z-Numbers Based Approach to Project Risks Assessment

At every phase of the project management process various risks originate owing to occurrence of uncertain events. In this research, we are analyzing potentialities of the Z-numbers in improving the quality of risk assessment. Risk assessment uses probability theory, theory of possibilities, fuzzy approach, Z-number based approach etc. Combined risk measure based on probability and consequence is calculated by applying the disjoint events probability formula or as a product of events. Reliability of relevant information unaccounted in this approach and this circumstance limits the descriptive power of the approach. Suggested by L. Zade a bi-component Z-number Z = (A, B) represents in a unified way a restriction on the values of the uncertain variable (A) and its certainty (B) and allows to take into account the reliability of information. Prediction identical to (High, Very Sure) can be formalized as a Z- evaluation “X is Z (A,B)”, where X is random variable of Risk Likelihood, A and B are fuzzy sets, describing soft constraints on a risk likelihood and a partial reliability, respectively. Usually, A and B are sense-based and in effect are imprecise.Z-number describes a probability of threat as: Likelihood =Z1(High,Very Sure),where A is expressed by linguistic terms High, Medium, Low, and B is expressed by terms Very Sure, Sure and so on. Similarly, Сonsequence measure is described as Сonsequence measure = Z2 (Low, Sure).Risk levels (Z12) is calculated as the product of the likelihood (Z1) and consequence measures (Z2).Effectiveness of the approach illustrated by examples. A general and computationally effective approach suggested to computation with Z-numbers allows using Z-information for the solving decision-making problem which can be utilized for risk factors estimation.. Application of the Z-number based approach for a project risks assessment increases adequacy of the risks representation due to better approximation of the combined effects

Two Proofs and One Algorithm Related to the Analytic Hierarchy Process

36 years ago, Thomas Saaty introduced a new mathematical methodology, called Analytic Hierarchy Process (AHP), regarding the decision-making processes. The methodology was widely applied by Saaty and by other authors in the different human activity areas, like planning, business, education, healthcare, etc. but, in general, in the area of management. In this paper, we provide two new proofs for well-known statement that the maximal eigenvalue λmax is equal to n for the eigenvector problem Aw=λw, where A is, so-called, the consistent matrix of pairwise comparisons of type n×n (n≥ 2) with the solution vector w that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution Aw=nw as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix A can be simply programmed and used.

The conjunction fallacy in profiles of victims of homicide

AbstractUsing basic probability theory, estimates of the characteristics of the average homicide victim are calculated using the notion of disjoint probability. The assumption of disjoint events (e.g., the victim's race bears no effect on the offender's weapon choice) is then tested empirically using the Uniform Crime Report‐Supplementary Homicide Report. Exploratory results suggest that many demographic and situational characteristics taken together are only slightly more related than chance. Put simply, the average profile of homicide victims portrayed by the media becomes less likely as demographic variables are added. A survey was conducted to test whether individuals conjoined these characteristics, thinking they were more likely to occur together. Consistent with the conjunction fallacy, many participants overestimated the likelihood that certain demographic or situational characteristics will occur together, and some overestimated it to a mathematically impossible degree. These two experiments showcase the difficulty in displaying statistical profiling and how it affects the public's perception of offenders.

Fishnet statistics for probabilistic strength and scaling of nacreous imbricated lamellar materials

Similar to nacre (or brick masonry), imbricated (or staggered) lamellar structures are widely found in nature and man-made materials, and are of interest for biomimetics. They can achieve high defect insensitivity and fracture toughness, as demonstrated in previous studies. But the probability distribution with a realistic far-left tail is apparently unknown. Here, strictly for statistical purposes, the microstructure of nacre is approximated by a diagonally pulled fishnet with quasibrittle links representing the shear bonds between parallel lamellae (or platelets). The probability distribution of fishnet strength is calculated as a sum of a rapidly convergent series of the failure probabilities after the rupture of one, two, three, etc., links. Each of them represents a combination of joint probabilities and of additive probabilities of disjoint events, modified near the zone of failed links by the stress redistributions caused by previously failed links. Based on previous nano- and multi-scale studies at Northwestern, the strength distribution of each link, characterizing the interlamellar shear bond, is assumed to be a Gauss-Weibull graft, but with a deeper Weibull tail than in Type 1 failure of non-imbricated quasibrittle materials. The autocorrelation length is considered equal to the link length. The size of the zone of failed links at maximum load increases with the coefficient of variation (CoV) of link strength, and also with fishnet size. With an increasing width-to-length aspect ratio, a rectangular fishnet gradually transits from the weakest-link chain to the fiber bundle, as the limit cases. The fishnet strength at failure probability 10−6 grows with the width-to-length ratio. For a square fishnet boundary, the strength at 10−6 failure probability is about 11% higher, while at fixed load the failure probability is about 25-times higher than it is for the non-imbricated case. This is a major safety advantage of the fishnet architecture over particulate or fiber reinforced materials. There is also a strong size effect, partly similar to that of Type 1 while the curves of log-strength versus log-size for different sizes could cross each other. The predicted behavior is verified by about a million Monte Carlo simulations for each of many fishnet geometries, sizes and CoVs of link strength. In addition to the weakest-link or fiber bundle, the fishnet becomes the third analytically tractable statistical model of structural strength, and has the former two as limit cases.

Topological and Orthomodular Modeling of Context in Behavioral Science

Two non-boolean methods are discussed for modeling context in behavioral data and theory. The first is based on intuitionistic logic, which is similar to classical logic except that not every event has a complement. Its probability theory is also similar to classical probability theory except that the definition of probability function needs to be generalized to unions of events instead of applying only to unions of disjoint events. The generalization is needed, because intuitionistic event spaces may not contain enough disjoint events for the classical definition to be effective. The second method develops a version of quantum logic for its underlying probability theory. It differs from Hilbert space logic used in quantum mechanics as a foundation for quantum probability theory in variety of ways. John von Neumann and others have commented about the lack of a relative frequency approach and a rational foundation for this probability theory. This article argues that its version of quantum probability theory does not have such issues. The method based on intuitionistic logic is useful for modeling cognitive interpretations that vary with context, for example, the mood of the decision maker, the context produced by the influence of other items in a choice experiment, etc. The method based on this article's quantum logic is useful for modeling probabilities across contexts, for example, how probabilities of events from different experiments are related.

Interpretation of Confidence Interval Facing the Conflict

As literature has reported, it is usual that university students in statistics courses, and even statistics teachers, interpret the confidence level associated with a confidence interval as the probability that the parameter value will be between the lower and upper interval limits. To confront this misconception, class activities have been designed with the aim of realizing that this application of confidence level violates the basic laws of probability, when considering two non-overlapping confidence intervals, that could plausibly correspond to two random samples from the same population, where the probability of events within this interpretation contradicts the probability rule for disjoint events and the rule of monotonicity (P[E] ≤ P[F] if E ⊂ F). Afterwards, we use simulation to help students shift to a frequentist interpretation of confidence intervals. Although the expected questioning was generated in students, it does not look like it is enough to establish a solid re-conceptualization of confidence level. We believe that this is due in part, to the slight language used in the teaching process and to incipient conceptions about the probability notion.

Reliability assessment for three-state element systems using ARBITR software

The article deals with reliability assessment methods for systems with three-state elements. It is shown that further development of conventional logic-and-probabilistic methods (LPM) eliminates the LPM deficiencies such as analysis of only two element states and assumption of their independency. Two models of element failure impact on the system reliability are shown which demonstrate the principles of reliability assessment using the ARBITR software which incorporates capabilities of algebra of disjoint event groups and is based on the general logic-and-probabilistic method (GLPM).

Automated Modeling of Reliability of Structurally Complex Systems Consisting of Elements with Three Disjoint states

This paper presents a techniques assessing the reliability of the complex structures which can not be reduced to a series-parallel connection of elements. In the case where the system elements can have three mutually exclusive (disjoint) states – up state, “fail-closed” mode, “fail-open” mode there is a need and an opportunity to separate the system failure probability calculations for the break and closure. It is shown that the developed method of estimating the reliability can be used for structures of any complexity with very weak restrictions. Orthogonalization are examples of logic functions based on the incompatibility of individual variables. Correctness of the proposed methods was confirmed by the decision of problems by full exhaustive search. Examples of automated modeling of bridge structure and the structure of the two stars connected to a delta solved using software ARBITR in which software is implemented algebra of mutually exclusive (disjoint) events.

Оценка надежности системы из элементов с тремя состояниями с использованием ПК АРБИТР

The article deals with reliability assessment methods for systems with three-state elements. It is shown that further development of conventional logic-and-probabilistic methods (LPM) eliminates the LPM deficiencies such as analysis of only two element states and assumption of their independency. Two models of element failure impact on the system reliability are shown which demonstrate the principles of reliability assessment using the ARBITER software which incorporates capabilities of algebra of disjoint event groups and is based on the general logic-and-probabilistic method (GLPM). Description of logic and probabilistic function transformations for disjoint events is shown in the annexes.

The Calibration Simplex: A Generalization of the Reliability Diagram for Three-Category Probability Forecasts

Abstract Full exposition of the performance of a set of forecasts requires examination of the joint frequency distribution of those forecasts and their corresponding observations. In settings involving probability forecasts, this joint distribution has a high dimensionality, and communication of its information content is often best achieved graphically. This paper describes an extension of the well-known reliability diagram, which displays the joint distribution for probability forecasts of dichotomous events, to the case of probability forecasts for three disjoint events, such as “below,” “near,” and “above normal.” The resulting diagram, called the calibration simplex, involves a discretization of the 2-simplex, which is an equilateral triangle. Characteristics and interpretation of the calibration simplex are illustrated using both idealized verification datasets, and the 6–10- and 8–14-day temperature and precipitation forecasts produced by the U.S. Climate Prediction Center.

A Lower Bound of Direction-of-Arrival Estimation for an Acoustic Vector Sensor Subject to Sensor Breakdown

The work presented here derives an approximate lower bound (ALB) for the error variance of direction-finding using a single acoustic vector sensor subject to random breakdown in its sensors. This ALB equals a weighted sum of Cramer-Rao bounds (CRBs), conditioned on disjoint events of sensor breakdown, that together define the overall phenomenon of random sensor breakdown. Its tightness is verified by Monte Carlo simulation of a maximum likelihood estimator.

Document ranking with quantum probabilities

In this thesis we investigate the use of quantum probability theory for ranking documents. Quantum probability theory is used to estimate the probability of relevance of a document given a user's query. We posit that quantum probability theory can lead to a better estimation of the probability of a document being relevant to a user's query than the common IR approach, i. e. the Probability Ranking Principle (PRP), which is based upon Kolmogorovian probability theory. Following our hypothesis, we formulate an analogy between the document retrieval scenario and a physical scenario, that of the double slit experiment. Through the analogy, we propose a novel ranking approach, the quantum probability ranking principle (qPRP). Key to our proposal is the presence of quantum interference. Mathematically, this is the statistical deviation between empirical observations and expected values predicted by the Kolmogorovian rule of additivity of probabilities of disjoint events in configurations such that of the double slit experiment. While PRP explicitly assumes that the relevancy of a document is independent of that of other documents, we suggest that qPRP implicitly models interdependent document relevance through quantum interference and thus is suited to those document ranking tasks where the independence assumption fails. Throughout the thesis, we also suggest how quantum interference can be estimated for effective document ranking. To validate our proposal and to gain more insights about approaches for document ranking, we (1) analyse PRP, qPRP and other ranking approaches, exposing the assumptions underlying their ranking criteria and formulating the conditions for the optimality of the two ranking principles, (2) empirically compare three ranking principles (i. e. PRP, interactive PRP, and qPRP) and two state-of-the-art ranking strategies in two retrieval scenarios, those of ad-hoc retrieval and diversity retrieval, (3) analytically contrast the ranking criteria of the examined approaches, exposing similarities and differences, (4) study the ranking behaviours of approaches alternative to PRP in terms of the kinematics they impose on relevant documents, i. e. by considering the extent and direction of the movements of relevant documents across the ranking recorded when comparing PRP against its alternatives. Our findings show that the effectiveness of the examined ranking approaches strongly depends upon the evaluation context. In the traditional evaluation context of ad-hoc retrieval, PRP is empirically shown to be better than or comparable to alternative ranking approaches. However, when evaluation contexts that account for interdependent document relevance are examined (i. e. when the relevance of a document is assessed also with respect to other retrieved documents, as it is the case in the diversity retrieval scenario), the use of quantum probability theory and thus of qPRP is shown to improve retrieval and ranking effectiveness over the traditional PRP and alternative ranking strategies, such as Maximal Marginal Relevance, Portfolio theory, and Interactive PRP. This work represents a significant step forward regarding the use of quantum theory in information retrieval. It demonstrates that the application of quantum theory to problems within information retrieval can lead to improvements both in modelling power and retrieval effectiveness, allowing the constructions of models that capture the complexity of information retrieval situations. Furthermore, the thesis opens up a number of lines of future research. These include investigating estimations and approximations of quantum interference in qPRP, exploiting complex numbers for the representation of documents and queries, and applying the concepts underlying qPRP to tasks other than document ranking. This dissertation was completed at School of Computing Science, University of Glasgow under the advise of Dr. Leif Azzopardi and Prof. Keith van Rijsbergen. Prof. Norbert Fuhr, Dr. Iadh Ounis, and Dr. John O'Donnell served as dissertation committee members. For the full dissertation, visit: http://theses.gla.ac.uk/3463.

On smoothness of black saturns

We prove smoothness of the domain of outer communications (d.o.c.) of the Black Saturn solutions of Elvang and Figueras. We show that the metric on the d.o.c. extends smoothly across two disjoint event horizons with topology \( \mathbb{R} \times {S^3} \) and \( \mathbb{R} \times {S^1} \times {S^2} \). We establish stable causality of the d.o.c. when the Komar angular momentum of the spherical component of the horizon vanishes, and present numerical evidence for stable causality in general.

Utility of gambling under p(olynomial)-additive joint receipt and segregation or duplex decomposition

This article continues our exploration of the utility of gambling, but here under the assumption of a non-additive but p(olynomial)-additive representation of joint receipt. That assumption causes the utility of gambling term to have a multiplicative impact, rather than just an additive one, on what amounts to ordinary weighted utility. Assuming the two rational recursions known as branching and upper gamble decomposition, we investigate separately the rational property of segregation and the non-rational one of duplex decomposition. Under segregation, we show that each pair of disjoint events either exhibits weight complementarity or both support no utility of gambling, a property called UofG-singular. We develop representations for both cases. The former representation is a simple weighted utility with each term multiplied by a function of the event underlying that branch. The latter representation is an ordinary rank-dependent utility with Choquet weights but with no utility of gambling. Under duplex decomposition, we show that weights have the intuitively unacceptable property that there is essentially no dependence upon the events, making duplex decomposition, in this context, of little behavioral interest.

Quantification of fault trees containing mutually exclusive events

This paper presents a method for analysing fault trees that contain independent sets of mutually exclusive (disjoint) events of different cardinality. Disjoint events can be used to model several issues, e.g. multistate systems with multistate components, different attack alternatives in security-related studies, and components in phased mission systems. Basic events of coherent and non-coherent binary trees can be considered as belonging to sets of cardinality 2. Each event is associated with a binary variable, and a labelling technique is used to distinguish the variables belonging to different sets. The proposed analysis method is based on the approach of binary decision diagrams (BDDs). The application of the rules for the construction of the BDD is driven by the labels associated with disjoint variables. The BDD is then transformed into a ternary decision diagram (TDD). The TDD represents a straightforward data structure for performing the probabilistic analysis. Equations are provided to determine the system unavailability and the probability of critical states. Numerical examples clarify the application of the proposed equations.

Quantitative analysis of a fault tree with priority AND gates

A method for calculating the exact top event probability of a fault tree with priority AND gates and repeated basic events is proposed when the minimal cut sets are given. A priority AND gate is an AND gate where the input events must occur in a prescribed order for the occurrence of the output event. It is known that the top event probability of such a dynamic fault tree is obtained by converting the tree into an equivalent Markov model. However, this method is not realistic for a complex system model because the number of states which should be considered in the Markov analysis increases explosively as the number of basic events increases. To overcome the shortcomings of the Markov model, we propose an alternative method to obtain the top event probability in this paper. We assume that the basic events occur independently, exponentially distributed, and the component whose failure corresponds to the occurrence of the basic event is non-repairable. First, we obtain the probability of occurrence of the output event of a single priority AND gate by Markov analysis. Then, the top event probability is given by a cut set approach and the inclusion–exclusion formula. An efficient procedure to obtain the probabilities corresponding to logical products in the inclusion–exclusion formula is proposed. The logical product which is composed of two or more priority AND gates having at least one common basic event as their inputs is transformed into the sum of disjoint events which are equivalent to a priority AND gate in the procedure. Numerical examples show that our method works well for complex systems.

Illustrating basic probability calculations using ‘Craps’

Instructors may use the gambling game of craps to illustrate the use of a number of fundamental probability identities. For the ‘pass-line’ bet we focus on the chance of winning and the expected game length. To compute these, probabilities of unions of disjoint events, probabilities of intersections of independent events, conditional probabilities and conditional expectations are encountered.