Chance events, probability theory

If the experiment is highly repeatable, the repeatability of the resulting state of the system can be measured by the use of probability theory. We have seen that it is possible to calculate a probability of a fuzzy event as well as the probability of a precisely defined event. In this sense, probability is a measure of chance or frequency of occurrence in a sequence of trials. Probability itself can be used for the parameters of a system, as we have seen in the voting example given in Section 2.11. It is therefore possible to have an imprecise, vague or fuzzy probability measure. In other words an event could have, for example, a probability of highly likely. 

Chance events such as random part failures and accidents usually occur as a result of actions from which more than one outcome is possible. Hence, it follows that the more complex the action, the greater the probabihty of chance or unplanned results. Chance events can also refer to those unplanned or unpredictable events that result from a combination of or an interaction between conditions and/or activities. Since accidents are basically unplanned or unpredictable events, they can generally be analyzed using probability theory and statistics. 

Starting from sixteen century onwards, the probability theory, calculus and mathematical formulations took over in the description of the natural real world system with uncertainty. It was assumed to follow the characteristics of random uncertainty, where the input and output variables of a system had numerical set of values with uncertain occurrences and magnitudes. This implied that the connection system of inputs to outputs was also random in behavior, i.e., the outcomes of such a system are strictly a matter of chance, and therefore, a sequence of event predictions is impossible. Not all uncertainty is random, and hence, cannot be modeled by the probability theory. At this junction, another uncertainty methodology, statistics comes into view, because a random process can be described precisely by the statistics of the long run averages, standard deviations, correlation coefficients, etc. Only numerical randomness can be described by the probability theory and statistics. 

The Reverend Thomas Bayes [1702-1761] was a British mathematician and Presbyterian minister. He is well known for his paper An essay towards solving a problem in the doctrine of chances [14], which was submitted by Richard Price two years after Bayes death. In this work, he interpreted probability of any event as the chance of the event expected upon its happening. There were ten propositions in his essay and Proposition 3,5 and 9 are particularly important. Proposition 3 stated that the probability of an event X conditional on another event Y is simply the ratio of the probability of both events to the probability of the event Y. This is the definition of conditional probability. In Proposition 5, he introduced the concept of conditional probability and showed that it can be expressed regardless of the order in which the events occur. Therefore, the concern in conditional probability and Bayes theorem is on correlation but not causality. The consequence of Proposition 3 and 5 is the Bayes theorem even though this was not what Bayes emphasized in his article. In Proposition 9, he used a billiard example to demonstrate his theory. The work was republished in modern notation by G. A. Barnard [13]. In 1774, Pierre-Simon Laplace extended the results by Bayes in his article Memoire sur la probabilite des causes par les evenements (in French). He treated probability as a tool for filling up the gap of knowledge. The Bayes theorem is one of the most frequently encountered eponyms in the literature of statistics. 

Stochastic chance-constrained programming is proposed by Charnes and Cooper [3] in 1959, which is an optimization theory in terms of probability. It is mainly for constraint conditions including random variables and the decisions must be made before random variables are observed. A principle is adopted with consideration that the decisions are made in the event of adverse situations which may not satisfy the constraints decisions are allowed not to meet the constraints in some degree, but the probability of constraints being satisfied should be kept not less than a confidence level a [1, 2]. 

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