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Sets, Venn Diagrams, Probability, and Bayes’ Rule
Book chapter

Sets, Venn Diagrams, Probability, and Bayes’ Rule

P. Mohana Shankar
Probability, Random Variables, and Data Analytics with Engineering Applications, pp 5-76
09 Feb 2021
DOI: https://doi.org/10.1007/978-3-030-56259-5_2

Abstract

A priori and a posteriori probabilities Algebra of sets Bayes’ rule and total probability Bernoulli trials Binary transmission Conditional, marginal, joint and total probabilities Confusion matrix Continuous and discrete outcomes Counting methods Error rates Generalized Bernoulli trials Independent events Independent trials Permutations and combinations Positive predictive values Probabilities of miss and false alarm Probability Probability as a continuous function Probability as a geometric measure Probability of error Series and parallel connected systems Threshold values Transition matrix Tree charts Venn diagrams
This chapter begins with the elementary aspects of probability by starting with sets and Venn diagrams. The concepts of probability follow with appropriate descriptions of marginal, joint, conditional, and total probabilities, Bayes’ rule, Bernoulli trials, etc. Examples include those that examine the notion of continuous probability as a prelude to the presentation of random variables in the next chapter. Keeping with the theme of application-oriented content, the chapter contains topics in data analytics such as the estimation of a priori, conditional, and a posteriori probabilities associated with a given set of data collected from measurements. The presentation of the subject matter is organized to offer the reader the importance and relevance of the topics to present-day engineering problems. The association between transition matrix and confusion matrix is introduced to illustrate the connection of the probability concepts to data science. Examples and exercises include conceptual and data analytics-based ones.

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