Probability Theory

Abstract

Probability theory is a part of mathematics that aims to provide insights into phenomena that depend on chance or on uncertainty. A basic treatment of probability from a perspective of engineers who are going to use the probability theory as a support for the practical reliability analyses is presented. Probability can be defined in terms of frequency of occurrence, as a percentage of successes in a large number of similar situations. Or probability of the event may express the subjective belief about the event. The mathematical representation of probability theory starts with set theory and basic probability concepts. The definition of factorial and the Pascal triangle represents the background for the theory of combinations. The theory of combinations determines the possible grouping of objects. There are three processes of interest: (i) permutations, (ii) combination, and (iii) variations, which are actually a union of the other two. One could say a permutation is an ordered combination. When the objects of a group are arranged in a certain order, the arrangement is called a permutation. In a permutation, the order of the objects is very important. The conditional probability of an event is the probability given that another event has occurred. Bayes theorem can be seen as a way of understanding how the probability that a theory is true is affected by a new piece of evidence. A random variable is any variable determined by chance and with no predictable relationship to any other variable. The term random variable is presented in theory and examples. Distribution is a degree to which the outcomes of events are evenly spread over the possible values. Probability distribution function is a function that represents probabilities to which the outcomes of events are spread over the possible values. The probability distribution functions are presented. The bathtub failure rate concept is widely used to represent failure behavior of many engineering items. The term bathtub stems from the fact that the shape of the failure rate curve resembles a bathtub. The bathtub curve consists of three periods: (i) an infant mortality period with a decreasing failure rate, (ii) a normal life period or useful life period with a low and relatively constant failure rate, and (iii) a wear-out period that exhibits an increasing failure rate.

The probable is what usually happens

Aristotle