China
Africa
Explore chapters and articles related to this topic
Elements of Complexity Theory
Published in F. P. Tarasenko, Applied Systems Analysis, 2020
The probability of outcome is defined as a measure of its possibility to occur under a given set of conditions. This set of conditions includes the laws of the stochastic nature of the phenomena (which determine the objective component of probability); however, the level of uncertainty (and hence the value of the probability) is also affected by the degree of knowledge by the subject about objective conditions, which gives rise to the subjective component of probability. (A good example of controlling a random object due to the subjective part of the probability is given by a sharper in card or dice games; improving the quality of the weather forecast is carried out by taking into account additional factors affecting the weather; etc.).
Basic Concepts in Probability
Published in X. Rong Li, Probability, Random Signals, and Statistics, 2017
Since A ∩ S = A and P{S} = 1, we have P{A}=P{A∩S}P{S}=P{A|S} A comparison of this with (2.14) indicates that conditional probability of A given B is simply the probability of A assuming the entire sample space is B, which is reasonable since event B has occurred (and the occurrence of any outcome outside B is impossible), as illustrated below.
Advanced Project Planning and Risk Managemen
Published in Timothy J. Havranek, Modern Project Management Techniques for the Environmental Remediation Industry, 2017
The second important point is that the sum of the probabilities of rolling the numbers 2 through 12 is one. This is a fundamental requirement for probabilities of a set of events which are mutually exclusive and collectively exhaustive. Mutually exclusive means that one and only one possible outcome can occur in a given trial. This is obviously true for a single roll of the dice where the sum of the two die can result in only one of the numbers 2 through 12. Collectively exhaustive means that there are no possible outcomes other than those in our set. This is also true for a pair of dice (i.e., it is not possible to roll the number 13). The second fundamental concept of probability theory for mutually exclusive and collectively exhaustive events is mathematically stated in Equation 10.2: () ∑i=1nP(ei)=1
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