Mathematics - iii (Differential Calculus) - Vol 3 | 3. Classical and Axiomatic Definitions of Probability by Abraham | Learn Smarter
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3. Classical and Axiomatic Definitions of Probability

Probability theory is essential in engineering, particularly in the context of Partial Differential Equations (PDEs). This unit delves into the Classical and Axiomatic definitions of probability, outlining their fundamental principles, applications, and limitations. Understanding these definitions enriches the study of stochastic PDEs and enhances modeling of real-world systems influenced by uncertainty.

Sections

  • 3

    Classical Definition Of Probability

    The Classical Definition of Probability assumes equal likelihood for all outcomes, providing a foundational understanding of probability theory.

    Learning Practice

  • 3.1.1

    Definition

    This section explores two foundational definitions of probability: the Classical Definition, which assumes equally likely outcomes, and the Axiomatic Definition, which provides a more rigorous mathematical framework.

    Learning Practice

  • 3.1.2

    Assumptions

    This section details the assumptions underpinning the Classical Definition of Probability, highlighting its foundational elements.

    Learning Practice

  • 3.1.3

    Examples

    This section provides illustrative examples of the Classical and Axiomatic definitions of probability with a focus on simple events.

    Learning Practice

  • 3.1.4

    Limitations

    This section discusses the limitations of the Classical Definition of Probability, highlighting scenarios where it fails.

    Learning Practice

  • 3.2

    Axiomatic Definition Of Probability

    The Axiomatic Definition of Probability provides a mathematical foundation for understanding probability, accommodating infinite sample spaces and non-uniform probabilities.

    Learning Practice

  • 3.2.1

    Overview

    This section discusses the Classical and Axiomatic definitions of probability, highlighting their principles, applications, and significance in engineering contexts.

    Learning Practice

  • 3.2.2

    Probability Space

    This section introduces the concept of probability space, defining its components and discussing the axiomatic approach to probability.

    Learning Practice

  • 3.2.3

    Kolmogorov’s Axioms

    Kolmogorov’s Axioms provide a rigorous mathematical framework for probability, enabling it to handle infinite sample spaces and non-uniform probabilities.

    Learning Practice

  • 3.2.5

    Advantages

    The Axiomatic Definition of Probability offers advantages over the Classical Definition by accommodating both finite and infinite sample spaces and handling diverse probabilities.

    Learning Practice

  • 3.2.6

    Relation To Classical Definition

    This section discusses how the Classical Definition of Probability is a special case of the Axiomatic Definition, specifically applicable when all outcomes are equally likely and the sample space is finite.

    Learning Practice

  • 3.3

    Applications In Engineering

    This section examines the roles of classical and axiomatic definitions of probability in engineering applications, particularly in relation to partial differential equations.

    Learning Practice

  • 3.4

    Summary

    This section discusses the Classical and Axiomatic definitions of probability, highlighting their principles, applications, and limitations.

    Learning Practice

References

unit 3 ch3.pdf

Class Notes

Memorization

What we have learnt

  • The Classical Definition of...
  • The Axiomatic Definition, i...
  • Both definitions play cruci...

Final Test

Revision Tests