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Probability and making decisions
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
At the beginning of each shift the safety of the walls and roof in a mine gallery are assessed by a laser system. The laser system provides either a warning (W) of instability or no warning (W¯). If a Warning is issued an engineer will perform a detailed check and declare the gallery as safe (S) or unsafe (S¯). The probability that the laser system issues a warning if the gallery is safe is 0.02, and the probability it fails to issue a warning if the gallery is unsafe is 0.01. Assume that at the beginning of a shift, before the laser system is used, the probability that the mine is unsafe is 0.004. Represent the sample space by drawing a tree diagram. Show the probabilities given in the question on your diagram.At the beginning of the shift, what is the probability the system will issue a warning?Given that the laser system issues a warning at the beginning of the shift, what is the probability that the engineer finds the gallery to be unsafe?
Probability and Its Distribution in Statistics
Published in Seong-woo Woo, Design of Mechanical Systems Based on Statistics, 2021
All statistical experiments have commonly three properties: (1) the experiment might have more than one outcome, (2) each outcome can be stated in advance, and (3) the outcome of the experiment relies on chance. For example, tossing a coin is a kind of statistical experiment. There is more than one possible outcome that might be tail or head. And there is an element of chance since the outcome is unknown. A statistical experiment can provide a random outcome. The set of all possible outcomes is defined as the sample space. An event is a subset of outcomes from the sample space. Examples of events are getting tail when tossing a coin or getting ‘5’ when rolling a dice.
Project Control System
Published in Adedeji B. Badiru, Project Management, 2019
A sample space of an experiment is the set of all possible distinct outcomes of the experiment. An experiment is some process that generates distinct sets of observations. The simplest and most common example is the experiment of tossing a coin to observe whether heads or tails will show up. An outcome is a distinct observation resulting from a single trial of an experiment. In the experiment of tossing a coin, heads and tails are the two possible outcomes. Thus, the sample space consists of only two items.
Independent events and their complements. Part II
Published in International Journal of Mathematical Education in Science and Technology, 2022
Gabriel Perez Mizuno, Carolina Martins Crispim, Adrian Pizzinga
Consider a random experiment. That is, we cannot correctly guess any outcome whatsoever. A probability space, the axiomatic structure designed to handle situations like this, reads as a triple . The set Ω is the sample space, whose points represent possible outcomes of the random experiment and whose subsets are termed events. The non-empty class is an appropriate σ-field of events. Finally, the probability measure P is defined on and, allegedly, serves well as the mathematical understanding of chance. In practical problems, P is frequently achieved, or selected, after a suitable statistical data analysis. The events that belong to and, as such, the ones that have probabilities are called random events. The motivation for , which by definition has the complement of each of its events and is closed under every sort of countable union, is to make the most comprising and hence rich mathematical/probabilistic model possible.1
Method for Restoring Consistency in Probabilistic Knowledge Bases
Published in Cybernetics and Systems, 2018
Van Tham Nguyen, Ngoc Thanh Nguyen, Trong Hieu Tran, Do Kieu Loan Nguyen
Let be a sample space that includes all possible outcomes of a statistical experiment (Walpole et al. 2012). Let be a finite set of events, where each event is a subset of the sample space . For , the intersection of two events F and G, signified by FG, is the event containing all elements that are common to F and G; negation of F, signified by F, is abbreviated by . A complete conjunction Θ of is an expression of the form with . Let be a set which includes all complete conjunctions of , therefore . A complete conjunction satisfies an event F, signified by iff F positively appears in Θ. Let with H denote an event or a set of events. Let be the numbers of complete conjunctions of . Let be a set which includes all non-negative real values from 0 to +∞. Let be a set which includes all real values from 0 to 1. Let be a probability function of a complete conjunction. Let be a set which includes all probability functions defined by . Let be a column vector, where an auxiliary variable corresponding to a probabilistic . With , .